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Main Page/PHYS 4210/Fourier Optics

2013-03-19T18:05:22Z

Jlyons:

<h1>Fourier Optics</h1>
<p>In this experiment we investigate the diffractive (as opposed to refractive) properties of a lens, named Fraunhofer diffraction. A HeNe laser is used to illuminate an image or a grating whose Fourier image is generated in the focal plane of a lens. The Fourier image is modified by cutting away low or high components (orders of the diffraction pattern) and a second lens is used to view the altered image.</p>

<h2> Key Concepts</h2>
<table width=500>
<td width=250>
<ul>
<li>Fraunhoffer Diffraction</li>
<li>Focal Plane</li>
<li>Diffraction</li>
<li>Fourier Series</li>
<li>Fourier Transform</li>
<li>Ronchi Rulings</li>
</ul>
</td>
<td width=250>
<ul>
<li>Low Pass Filter</li>
<li>Condenser</li>
<li>High-Density Grating</li>
<li>Magnification</li>
<li>Resolution</li>
<li>Interference</li>
</ul>
</td>
</table>
<h1>Introduction</h1>
<p>Ray optics deals with the simple design of optical systems. While trying to take, e.g., a microscope to its limit, i.e., to improve the resolution, researchers in the middle of the 19th century (Abbé, in particular) found that the dark parts of the image inside an optical system contribute to the final image. Physical optics makes use of the wave nature of light to understand these phenomena. It turns out that a simple lens produces in its focal plane a diffraction pattern for the image. In ref. 1 detailed explanations of the mathematical description of imaging are provided. </p>
<p>The problem is summarized in Fig. 1, which shows how parallel light illuminates an object, and a symmetric set-up of two high-quality lenses. The first lens is used to construct a Fourier image in the focal plane, while the second lens forms back the image. The diffraction pattern in the focal plane (the drawing is rather artistic, i.e., inaccurate) is easily observed for monochromatic light (a mercury lamp with a colour filter, or a laser), and can be modified (filtered) to modify the image.</p>
<p>The description of the pattern in the focal plane becomes straightforward if we consider a simple image, such as, e.g., a grating. From previous optics demonstrations you may be familiar with two types of gratings that are characterized by their line density (in lines/inch): (i) simple rulings (e.g., Ronchi rulings for the measurement of the resolution of lenses and their spherical aberration) have an intensity profile that corresponds to a square-wave pattern; these rulings when illuminated by a monochromatic source generate a diffraction pattern of many equidistant dots along a line perpendicular to the rulings; (ii) gratings with a sinusoidal intensity patterns are often used to produce a single pair of diffraction maxima. This corresponds to the Fourier representations of a sinusoidal intensity pattern: a single cosine mode (central spot) and two sine-terms, while the Fourier series for case (i) has infinitely many contributions.</p>
<p>The location of the diffraction maxima depends on the spacing of the ruling ''d'' and the wavelength λ of the light. A useful parameter that appears in the intensity patterns of apertures is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Fou-eqn1.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<ol>
<li>The amplitude of the diffracted light is given by a Fourier transform of the aperture function ''F(y)'' which equals to one for perfect transmission and vanishes for a perfect blockage of light (ref. 4, p 131):
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Fou-eqn2.png|240px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>
</li>
<li>For some aperture functions the calculation of the diffracted light intensity is straightforward. A Maple worksheet is provided with examples of a diffraction grating (cosine-squared modulated transmission grating function ''F(y)'' and step-function pattern ''F(y)'' for a line ruling usually referred to as a Ronchi ruling).</li>
<li>The mathematical description of the transform of a sinusoidal or square-wave intensity pattern is straightforward and is done in standard courses on advanced calculus (see ref. 3). The pattern for a cosine-squared aperture function ''F(y)'' = cos<sup>2</sup>(π''y''/''d'') consists of a central peak and two side peaks located at
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Fou-eqn3.png|100px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>
</li>
</ol>
<p>The side peaks have an intensity of about a quarter of that for the central peak. For a square-wave aperture function (Ronchi ruling) peaks of nearly equal intensity occur at ''x'' = 0 and at ''x<sub>0</sub>'' = ±1/''d'' followed by peaks of decreasing intensity for ''x<sub>1</sub>'' = ±3/''d'' , ''x<sub>2</sub>'' = ±5/''d'' , etc. Weaker peaks arise at interspersed locations if the square-wave aperture is not symmetrical, i.e., if the rulings are of slightly different thickness than the gaps.</p>
<p>The amazing observation from figs. 1 a, & b is that a simple optical set-up provides instantaneous Fourier transforms. It is, therefore, referred to as a coherent optical computer.</p>

<table width=500 align=center><td>
<p align=justify>[[File:Fou-fig1a.png|500px|border|center]]
<b>Figure 1a </b>
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:Fou-fig1b.png|500px|border|center]]
<b>Figure 1b -</b> A lens provides a Fourier Transform of the image in its focal plane.
<br clear=right>
</p>
</td></table>
<p>One application of the Fourier image technique can be found in filtering of images for printing (e.g., the removal of graininess). The resolution of printers is typically in the range of 300-1200 dpi (dots per inch) for conventional laser printers and reaches 24000 dpi in professional typesetters. In the second part of the experiment we demonstrate the effect of removing high Fourier components of an image.</p>
<p>Some of the questions that should be understood are raised and answered in refs. 1 and 4: </p>
<ol style="list-style-type:lower-roman">
<li><h3>Comparison of strictly periodic vs. non-periodic images.</h3>
<p>The Fourier series representation that is valid for a grating (that is assumed to be composed of infinitely many lines, i.e., truly periodic) goes over into the Fourier transform, i.e., the spatial signal is not composed of discrete multiples of a basic frequency, but a continuous spectrum of spatial frequencies is present.</p></li>
<li><h3>Why do we observe a Fourier pattern in the focal plane? </h3>
<p>It is mathematically evident that a periodic square-wave signal (in our case in the spatial domain) can be decomposed into sine-contributions with multiples of a discrete frequency. Why does the image in the focal plane contain dots that correspond to the Fourier contributions (location and intensity are determined by the frequency and amplitude of the Fourier representation)? This is demonstrated in the chapter on diffraction theory: a Fraunhofer diffraction pattern of an aperture (or obstacle) is equivalent to the Fourier transform of the function describing the aperture. Ref. 4 uses a physically intuitive approach with phasor diagrams.</p></li>
<li><h3>What does one obtain for two-dimensional images?</h3>
<p>The previous discussion of rulings and gratings was simple in that they represent one-dimensional objects, i.e., there is no information content in the dimension along which the lines, in principle, are infinitely long. In general, however, an image does contain two-dimensional information. In the simplest case we cross two gratings and observe that the diffraction pattern becomes two-dimensional, namely a product of the two orthogonal patterns. Thus, for arbitrary images the lens produces a two-dimensional Fourier transform (Fourier series for periodic images).</p></li>
<li><h3>What is the effect of high-pass or low-pass filtering of images?</h3>
<p>One can consider for any image the question of information content at high and at low spatial frequencies. For half-tone printing (production of greyscales by printing black dots with varying density) one makes use of the limited resolution of our eyes: without magnification the eye perceives only the low-frequency components that carry the interesting information of, e.g., a photograph. Magnification of the image makes, however, the dots visible. It is possible to mask the image in the Fourier transform plane with a low-pass filter, i.e., to remove the high-frequency content to produce a soft image. Conversely, high-pass filtering and artificial modification (introduction of detail by blocking every second Fourier component, see ref. 2) is easily accomplished.</p></li>
</ol>

<h1>Experimental Procedure</h1>
<p><b>WARNING: Laser light can easily damage your eyes! Never look directly into the beam or at bright images of the spot that are produced on metallic or mirror surfaces! Wear protective glasses when operating the laser. While operating the laser a sign should be posted on the door to warn people entering the room about the presence of laser radiation.</b></p>

<p>First you should gain familiarity with gratings, rulings and the basic set-up. Two separate HeNe lasers are used in this experiment. The first one is used to direct its beam at gratings and Ronchi rulings with a screen for a manual recording of the pattern as well as detection of the intensity of dots using a photodetector/amplifier/voltmeter set-up. The second laser in our experiment is focussed onto a 10μm pinhole with adjustable micrometer screws. Three identical lenses are provided in order to realize the 4f-focussing set-up of Fig. 1. They are, in fact, good-quality two-element lenses with chromatic correction, a resolution of 32 lines/mm and have all a focal length of 350 mm (cf.. the Optikon specification sheet). The first lens acts as a condenser to provide a parallel uniformly illuminated beam. Adjustments of the three micrometer screws may be required to obtain a proper illumination of the image slide. Be careful, however, not to ‘lose’ the laser spot altogether. The second lens produces a Fourier image in the focal plane (where low-pass filtering can be applied using an aperture). The third lens is used to convert the (modified) Fourier image again into a normal (filtered) image. </p>
<ol>
<h3>Using laser set-up 1 (no pinhole):</h3>
<li>Observe the function of high-density gratings (square-wave and sinusoidal intensity patterns with thousands of lines per inch) as they are mounted in a laser beam (use the laser without a pinhole). Take measurements for slides A and B and determine their type and line density. Repeat the measurement using Ronchi rulings with hundreds of lines per inch. The wavelength of a HeNe laser equals 632.8 nm. A diode laser (e.g., in a laser pointer) has a wavelength of 675±10 nm. Record the spacings as produced on a screen at a fixed distance, calculate diffraction angles θn, and calculate d for the given grating. Comment on the observed pattern. A MATHEMATICA worksheet is provided in the Appendix for the calculation of diffraction patterns with a finite number of slits. </li>
<h3>Using laser set-up 2 (with pinhole):</h3>
<li> Observe the laser set-up with a pinhole. Place Ronchi rulings with tens of lines per inch or simple slides with rulings between the condenser lens and the first imaging lens. Place a screen into the focal plane and observe the diffraction patterns. Estimate the separation of the spots and compare your results with eqs. (1,3). Are there infinitely many Fourier components?</li>
<li> Place a grid-shaped ruling between the condenser and imaging lens. Observe the Fourier image as well as the regenerated image on the screen. Place slits of different widths in the focal plane to cut out high Fourier components first in the horizontal, then in the vertical planes. It is possible to completely remove one set of lines. Rotate the slit in the Fourier plane by 45 degrees. What do you observe? Explain this in your write-up.</li>
<li> Place a low resolution image slide into the holder and observe the results. Apply a low-pass filter to remove the graininess. Record images with and without filtering using a camera and provide explanations including some mathematical detail (Fourier-series analysis for periodic images in one dimension is sufficient but Fourier transform analysis can be done easily with Maple or Mathematica).</li>
</ol>

<h1>References</h1>
<ol>
<li>E. Hecht, Zajac, ''Optics'', Addison-Wesley. </li>
<li>Gillespie, ''Optical Information Processing'', [http://iopscience.iop.org/0031-9120/29/3/003 Phys. Educ. '''29''', 127-34 (1994)].</li>
<li>G.A. Arfken, ''Mathematical Methods for Physicists'', Academic Press.</li>
<li>F.G. Smith, J.H. Thomson, ''Optics'', 2nd ed.</li>
</ol>

<h1>Appendix</h1>
<ul>
<li><p>''MultipleSlitDiffractionPattern.nb'' (copy and paste into a Mathematica Notoebook)</p>
<p>Manipulate[</p>
<p> Plot[{((1/n) ChebyshevU[n - 1, Cos[b k/2]] Sinc[a k/2])^2, </p>
<p> Sinc[a k/2]^2}, {k, -kmax, kmax},</p>
<p> PlotRange -> {0, 1},</p>
<p> Axes -> False,</p>
<p> Frame -> True,</p>
<p> FrameTicks -> {True, False},</p>
<p> FrameLabel -> {"position", "intensity"},</p>
<p> PlotLabel -> ToString[n] <> " slit diffraction pattern",</p>
<p> Filling -> {1 -> Axis},</p>
<p> PlotStyle -> {Automatic, {Thick, Dashed}},</p>
<p> ImageSize -> {500, 350},</p>
<p> ImagePadding -> {{40, 20}, {40, 20}}],</p>
<p> {{a, 0.5, "slit width"}, 0, 1, Appearance -> "Labeled"},</p>
<p> {{b, 2.5, "slit spacing"}, 1, 5, Appearance -> "Labeled"},</p>
<p> {{n, 2, "number of slits"}, Range[10], SetterBar},</p>
<p>Delimiter,</p>
<p> {{kmax, 10, "horizontal range"}, 1, 100, Appearance -> "Labeled"}]</p>
</li>
</ul>

Main Page/PHYS 3220/Thermionic

2013-03-12T18:55:39Z

Jlyons:

<h1>Thermionic Emission</h1>

<p>It is well known that when a metal is heated to high temperatures it emits electrons, as in the case of the filament in an electron tube. This phenomenon of thermionic emission is called the Richardson Effect, and was discovered by Thomas Edison in 1883. It is explained by the electron theory of metals and is comparable to the evaporation of liquids. To obtain a thermionic current, both the emitter and the collector are placed in an evacuated vessel and an electric field is applied.</p>

<h1> Theory </h1>

<h2>Part A</h2>
<p>Consider a vacuum diode as shown in Fig. 1. The filament (usually a tungsten wire) is heated to a temperature in the vicinity of 2500<sup>o</sup> K. The electrons emitted by the filament (cathode) are drawn to the collector (anode) when the anode is raised to a high potential (0 to 400 Volts).</p>

<p>Typical current voltage characteristics I<sub>a</sub> vs. V<sub>a</sub> for various values of filament temperature are shown in Fig. 2.</p>

<table width=800 align=center><tr><td>
<p align=justify>[[File:Th-fig1v2.png|400px|border|center]]
<b>Figure 1 </b>
</p>
</td>
<td><p align=justify>[[File:Th-fig2.png|450px|border|center]]
<b>Figure 2</b>
<br clear=right>
</p>
</td></tr> </table>

<p>From Fig. 2, note two distinct regimes of operation: a region (known as the space-charge limited region) where I<sub>a</sub> increases rapidly with V<sub>a</sub>; and the saturation region (known as the temperature limited region). At low anode voltages the cathode is surrounded by a dense space-charge of electrons, which are being continually emitted and reabsorbed. Only those electrons which are well away from the cathode are drawn towards the anode, since the electrostatic potential of the electron cloud counteracts the accelerating potential. A separate discussion of the current-voltage characteristic for this region follows below (eq. 2). As the anode voltage is increased, more electrons are drawn out of the space charge region toward the anode. For sufficiently high voltages the space-charge disappears and the beam current is limited by the rate of emission from the cathode, which leads to a saturation of the current. This saturation current depends on the temperature of the emitter and its work function, and is given by the Richardson-Dushman equation (cf.. ref. 1), </p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Th-eqn1.png|200px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>where :
<ul>
<li>Jo = saturation current density</li>
<li>A = a proportionality constant given below</li>
<li>T = temperature of the emitter in K</li>
<li>W = work function of the metal of the emitter</li>
<li>k = Boltzmann constant</li>
</ul>
</p>

<table width=400 align=center>
<td>
<p align=justify>[[File:Th-eqn1b.png|300px|center]]
</p>
</td></table>

<p>where m and e are the electron mass and charge, and h is Planck's constant. This theoretical value of A should hold for a pure surface of a single crystal face. The Richardson-Dushman model as stated above oversimplifies the physical situation: e.g., one can introduce a quantum mechanical effect, i.e., the fact that electrons behave as waves and have a transmission and a reflection probability for leaving the surface (refs. 1-3). It is possible to use eq. (1) while treating ''A'' not as an absolute constant, but as a parameter that depends on the material and its crystal structure.</p>

<p>Within the space-charge-limited region one uses Poisson’s equation for the potential created by the electron cloud. The I-V characteristic is now given by the Child-Langmuir law. For a diode of cylindrical geometry this is, in MKSA units,</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Th-eqn2.png|240px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>where r<sub>a</sub> is the radius of the anode, ε0 is the permittivity constant, β is a correction factor to account for the non-zero radius of the cathode. The GRD7 tube in use has a value β<sup>2</sup> = 1.1.</p>

<h2>Part B </h2>

<p>If a magnetic field of strength B is applied parallel to the axis of the diode, i.e. parallel to the filament, the electrons do not move in straight line paths directly to the anode. Their motion becomes complicated as they are accelerated by both the electric and magnetic fields, but from the motion of a charged particle in a magnetic field alone it is clear that the electron trajectories become more curved when B increases. Fig. 3 shows schematically some trajectories of electrons in the tube. Fig. 4 displays the anode current as a function of B; the current goes to zero when B reaches B<sub>crit</sub> , corresponding to trajectory (d) in Fig. 3. To obtain a relation between the parameters and B<sub>crit</sub>, we note that when an electron is at the anode (or just inside it), the electron must have kinetic energy equal to eV, where V is the anode voltage.</p>

<p>This simplified analysis assumes that the electrons acquire a velocity towards the anode due to the electric field. Subsequently the effect of the magnetic field is estimated to compare the gyration radius for a given B with the size of the anode in our configuration.</p>

<table width=600 align=center><tr><td>
<p align=justify>[[File:Th-fig3new.png|350px|border|center]]
<b>Figure 3-</b> Trajectory of electrons in a magnetic field pointed out of the page.
</p>
</td>
<td><p align=justify>[[File:Th-fig4.png|290px|border|center]]
<b>Figure 4- </b>B<sub>crit</sub> corresponds to trajectory ‘d’
<br clear=right>
</p>
</td></tr> </table>

<p>If the electron follows a path that has a magnetic field controlled trajectory parallel to the anode, such as trajectory (d), it cannot be collected by the anode and no current can flow. Thus, we can relate the parameters as follows: when the electron's trajectories have a radius of curvature equal to half of the anode radius, the current stops, and the smallest B value at which this occurs is B<sub>crit</sub>. We can obtain the particle velocity v from its kinetic energy:</p>

<table width=400 align=center>
<td>
<p align=justify>[[File:Th-eqn2a.png|280px|center]]
</p>
</td></table>

<p>The magnetic field provides the centripetal force to keep it in orbit such that</p>

<table width=400 align=center>
<td>
<p align=justify>[[File:Th-eqn2bnew.png|360px|center]]
</p>
</td></table>

<table width=500 align=center>
<tr><td>
<p align=justify>[[File:Th-eqn3new.png|360px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<h1>Experiment </h1>

<p>The object of this experiment is to verify the three [(1), (2), (3)] expressions given above and thus understand the physical processes associated with the operation of the thermionic diode (Ferranti GRD7, shown in Fig. 5). The cathode is a tungsten filament in the form of a thin wire running along the axis of a surrounding cylindrical anode. The anode is divided into three sections. The two outer ones are called guard rings. They are maintained at the same potential as the central section and serve to reduce the fringing of the field within the central section. The necessary details of the diode are given below:</p>

<ul>
<li>Filament length, L = 1.45 cm</li>
<li>Filament radius, r<sub>c</sub> = 0.0065 cm</li>
<li>Anode radius, r<sub>a</sub> = 0.325 cm </li>
<li>anode length l<sub>a</sub> = 1.45 cm</li>
</ul>

<h2>Step 1</h2>

<p>Arrange the circuit as shown in the diagram Fig. 6. At fixed filament temperature, i.e. constant filament current (I<sub>f</sub>), the anode current (I<sub>a</sub>) is recorded for various accelerating potentials (V<sub>a</sub>) well into the saturation region. Repeat these observations for various filament temperatures. Check immediately whether you have enough data points in the space-charge limited region. The data should also span a wide range of accelerating voltages to cover the saturation regime!</p>

<p>
The filament temperatures are to be determined from the curve given in Fig. 7 that displays the temperature-filament current characteristic for the GRD7 diode. Use the optical pyrometer to confirm the temperature. Make sure to point the securely mounted pyrometer at the glowing filament, not it’s image that is visible on the anode. Understand how this instrument works and document this in your lab write-up.
</p>

<h2>Step 2</h2>
<p>Plot the anode current against the anode potential on a logarithmic scale: log Ia vs. log Va. Determine the slope for each curve (for each temperature) to check the power law. How consistent are these several slope determinations and how do they compare to the prediction given by equation (2)? Be sure to show on the graph where your slope is determined. Note that you should be in the space-charge-limited region.</p>

<p>Plot a graph of</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Th-eqn3b.png|100px|center]]
</p>
</td></table>

<p>and determine the slope (the saturation current I<sub>o</sub> is equal to the saturation current density J<sub>o</sub> x the surface area).</p>

<table width=600 align=center><tr><td>
<p align=justify>[[File:Th-fig5.png|300px|border|center]]
<b>Figure 5 </b>
</p>
</td>
<td><p align=justify>[[File:Th-fig6.png|380px|border|center]]
<b>Figure 6</b>
<br clear=right>
</p>
</td></tr> </table>

<p>How can the work function W be determined from this plot? Determine W for the emitter and compare your value with the literature value for tungsten (ref. 2). Determine the constant ''A''.</p>

<h2>Step 3</h2>
<p> Place the given solenoid over the diode. Vary the current in the solenoid (this in turn varies the magnetic field B along the axis) to determine the cutoff field B<sub>crit</sub> for several values of the anode voltage V<sub>a</sub>. The magnetic field can be calculated from the solenoid current if one knows the number of windings per unit length (N) as B = μ I N, where μ is the permeability constant. Use the Gaussmeter to confirm your estimate of the magnetic field. Since the Hall probe of the magnetic flux meter cannot be placed at a proper angle inside the solenoid, make sure to understand how it measures a single component of the magnetic field, namely the one perpendicular to the probe. Use a cosine correction while measuring at some known angle with respect to the (hopefully) homogeneous field. Read up on how a Hall probe works and provide a brief explanation.</p>

<p>Plot B<sub>crit</sub><sup>2</sup> against V<sub>a</sub> to find the experimental relationship.</p>

<h1>References</h1>
<ol>
<li>S. Dushman, [http://rmp.aps.org/abstract/RMP/v2/i4/p381_1 Rev. Mod. Phys. '''2''', 381 (1930)].</li>
<li>Preston, Dietz, ''The Art of Experimental Physics'', 1st edition, pp. 143ff.</li>
<li>Harnwell-Livingwood, ''Experimental Atomic Physics'', pp. 189ff. </li>
</ol>

<h1>Appendix: Correction for the pyrometer reading. </h1>
<p>A pyrometer is a convenient means of measuring high temperatures, but it gives accurate reading <b>(only)</b> for a "black body"; our filament is not a black body, so a correction must be made. This is contained in the formula below relating the observed temperature to the true temperature, using the emissivity of the material.</p>
<table width=400 align=center>
<td>
<p align=justify>[[File:Th-eqn4.png|160px|center]]
</p>
</td></table>

<p>where</p>
<ul>
<li>T = true temperature in degree absolute</li>
<li>T<sub>obs</sub> = observed temperature in degree absolute</li>
<li>λ = 650 nm</li>
<li>E<sub>λ</sub> = 0.424, the spectral emissivity of Tungsten at λ = 650 nm</li>
<li>C<sub>2</sub> = 14.350 (second radiation constant) mm <sup>o</sup>K</li>
</ul>

<table width=400 align=left><td>
<p align=justify>[[File:Th-fig7.png|500px|border|center]]
<b>Figure 7</b>
<br clear=right>
</p>
</td></table>

File:Th-fig1v2.png

2013-03-12T18:54:59Z

Jlyons:

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:53:41Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 8- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 9- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>[[Media:MUONdetector.JPG|"''Muon Physics''" Scintillator]]</li>
<li>[[Media:MUONelec.JPG|"''Muon Physics''" Control Unit]]</li>
<li>[[Media:MUONfg.jpg|GW Function Generator (Model: GFG-8016G)]]</li>
<li>[[Media:MUONscope.jpg|Digital oscilloscope]]</li>
<li>[[Media:MUONterm.jpg|50-Ω terminators<ref>http://www.computercablestore.com/BNC_Terminator_Plug_50_Oh_PID939.aspx</ref> ]]</li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:51:32Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 8- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 9- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>[[Media:MUONdetector.JPG|"''Muon Physics''" Scintillator]]</li>
<li>[[Media:MUONelec.JPG|"''Muon Physics''" Control Unit]]</li>
<li>[[Media:MUONfg.jpg|GW Function Generator (Model: GFG-8016G)]]</li>
<li>[[Media:MUONscope.jpg|Digital oscilloscope]]</li>
<li>[[Media:MUONterm.jpg|50-Ω terminators ]]</li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

File:MUONterm.jpg

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Jlyons:

File:MUONscope.jpg

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Jlyons:

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:47:14Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 8- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 9- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
<li>[[Media:MUONdetector.JPG|"''Muon Physics''" Scintillator]]</li>
<li>[[Media:MUONelec.JPG|"''Muon Physics''" Control Unit]]</li>
<li>[[Media:MUONfg.jpg|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

File:MUONfg.jpg

2013-03-05T18:45:26Z

Jlyons:

File:MUONelec.JPG

2013-03-05T18:44:16Z

Jlyons:

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:43:49Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 8- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 9- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
<li>[[Media:MUONdetector.JPG|"''Muon Physics''" Scintillator]]</li>
<li>[[Media:ZEElectromagnet.JPG|Electromagnet]]</li>
<li>[[Media:ZEDischargePower.JPG|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

File:MUONdetector.JPG

2013-03-05T18:42:56Z

Jlyons:

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:41:37Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 8- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 9- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
<li>[[Media:ZEMagnetPower.JPG|Magnet Power Supply]]</li>
<li>[[Media:ZEElectromagnet.JPG|Electromagnet]]</li>
<li>[[Media:ZEDischargePower.JPG|Discharge Power Supply]]</li>
<li>[[Media:ZECCDCamera.JPG|CCD Camera]]</li>
<li>[[Media:ZELummer.JPG|Lummer-Gehrcke Plate]]</li>
<li>[[Media:ZEPolarizers.JPG|Polarizers and Waveplate]]</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:21:14Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 8- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 9- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-03-05T18:19:28Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 8 shows the output directly from the PMT into a 50Ω load. Figure
9 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 8- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 9- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 10- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 11- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine. Describe how you chose bin sizes for the time axis, and how signals due to background events were accounted for. </li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 3220/Rutherford I

2013-02-26T18:02:17Z

Jlyons:

<h1>Rutherford Scattering I</h1>

<h1>Introduction</h1>
<p>The Rutherford scattering experiment, where alpha particles (doubly charged Helium nuclei) were scattered off of a target (gold, aluminum, etc.) represents one of the most important experiments of this century. While the bulk of the alpha particles were scattered at small angles, indicating a soft collision process, a finite number of alpha particles however did scatter at very large angles. This could only have occurred though a collision with a massive object. From the distance of closest approach of the alpha with this object, and using information on the size of the whole atom, we came to know that the atom was mostly empty space. The results of this experiment formed the basis of subatomic structure, as we know it today – that the atom has a hard central core consisting of a tiny but massive core called the nucleus, surrounded by electrons, forming an electrically neutral system. </p>

<p>In this experiment we reproduce the results of Rutherfordby allowing alpha particles from a radioactive source (Am-241) to impinge on thin gold foil. We then compare the experimentally observed differential cross section (related to the number of detected alpha particles), as a function of the angle of scatter of the alpha particles off the target atoms. By comparing these particles to the theoretical expectations of elastic scattering of two particles, we can confirm that the alpha predominantly scatter off the nuclear core of the atom, and hence the structure of the atom is as Rutherford suggested.</p>

<h1>Theory</h1>

<table width=400 align=center><td>
<p align=justify>[[File:RuthI-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Particle scattering.
<br clear=right>
</p>
</td></table>

<p>The theoretical analysis of the scattering cross section can be based on classical or quantum mechanics. In classical mechanics the number of particles scattered at a certain angle (θ) is a unique function of the impact parameter (b). We assume that a pure Coulomb potential is valid, but the appropriateness of this assumption shall be discussed later. To obtain the scattering cross section classically, first one solves Newton's equation of motion to obtain the relationship between impact parameter and scattering angle, and the results is that <b>b α 1/θ</b>. Small impact parameters thus lead to close encounters of the two charged objects, and thus large scattering angles. Conversely, distant collisions lead to small scattering angles. In quantum mechanics this relationship is not unique, but interestingly, a probability distribution arises for the particles to reach deflection angles θ that are synonymous with the classical answer (this is a special feature of the Coulomb potential). Although this relation tells us we are on the right track intuitively, unfortunately it is not very useful since we cannot measure b in any given interaction. We have to relate the scattered angle to something we can measure: the number of alpha particles scattered into the solid angle of the detecting device. </p>

<p>The number of particles scattered at a given angle depends on the change of the impact parameter with the scattering angle, and this can be verified from any book on modern or subatomic physics (e.g. ref 3,4). If the incident charged particle has charge Z<sub>1</sub> and energy E, and hits a target of charge Z<sub>2</sub>, the Rutherford differential cross section can be written as</p>

<table width=500 align=center>
<tr><td>
<p align=justify>[[File:RuthI-eqn1.png|350px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p><b>Q:</b> Calculate the total cross section (hint: it diverges!!!). How is it that we continue to use this formalism when the prediction is infinity?</p>

<p>We have thus related the theoretical expectation (labelled Rutherford) to the measured variables. It is useful to review the article in Melissinos [1] as one can appreciate how careful experimental procedure is so crucial, and also the various factors that can affect the final result. There are several subtle (and some not so subtle) points one has to take into consideration for a meaningful comparison. </p>

<h1>Apparatus</h1>

<p><b>Read the Leybold manual first to make sure you understand the apparatus.</b></p>

<p>The apparatus consists of the '''scattering chamber''', the '''vacuum system''', the '''detector''', and the '''data acquisition system'''. The components inside the scattering chamber necessary to perform the experiment are '''the source''', two types of '''target foil''' (gold and aluminium) and '''two slits''' (1 mm and 5 mm wide). See the data sheets provided by Leybold for further details.</p>

<table width=400 align=center><td>
<p align=justify>[[File:RuthI-fig2.png|500px|border|center]]
<b>Figure 2 -</b> Schematic of experiment.
<br clear=right>
</p>
</td></table>

<h2>The source and slits</h2>

<p>To install a foil please contact the TA or Lab Technologist.</p>

<p>Treat this source as you would any '''radioactive source''': with care. Wear gloves when handling the source or when inserting the slits and target, and do not take it out of its protective container without supervision. The radiation from the source is emitted at all possible angles within the geometry of the container. To turn the radioactive source into a well-defined beam of particles one has to collimate the beam by using apertures. The source is placed in a metal holder so that the alpha particles emanate in a cone. A collimating slit is placed at a distance of about 2.8 cm from the front face of the metal holder, and a metal foil of a few microns thickness is placed in a holder in front of the slit, directly against it (the drawing is exaggerated), i.e. further away from the source than the slit. Here “front” is defined by the direction of the alpha particles if they were unimpeded in their direction of travel from the source through the collimator. The detector is 2.7 cm in front of the foil. The source-foil assembly can be rotated about an axis that passes through the foil, while the detector is fixed. This is equivalent to a fixed source-target assembly and rotating the detector. The central axis of the detector should point along this axis of the foil-source assembly at zero degrees. In practice, any errors of this may cause an overall shift in the result, as we shall discuss later.</p>

<h2>The detector</h2>

<p>The detector is a silicon (solid-state) device, and has a slit on its face of height ( height 7mm x width 1.5mm. The detector is connected to a pre-amplifier (to amplify the weak signal) and discriminator, and these are set to generate definite pulses when a pulse generated in the detector exceeds a certain threshold (as defined by the discriminator). A A digital counter records the shaped pulses. This way one can suppress unwanted background sources (although light hitting the detector could also trigger events). If one sets the threshold too high, one suppresses of course some events of interest, but this should at worst lead to a reduced count rate, but not necessarily affect the outcome of the experiment. Can the dis. threshold be adjusted?</p>

<h2>The vacuum system</h2>

<p>There are several sets of measurements using the chamber. In between each stage, the chamber must be returned to normal atmospheric pressure so that the lid may be removed. Hence the process of evacuating the chamber, and returning it to normal pressure, are described at first. Practice with no foil in the chamber and test that the lid of the chamber is secure when the chamber is evacuated.</p>

<table width=400 align=center><td>
<p align=justify>[[File:RuthI-fig3.png|400px|border|center]]
<b>Figure 3 -</b> Vacuum system Schematic.
<br clear=right>
</p>
</td></table>

<table width=400 align=center><td>
<p align=justify>[[File:Ruth1-fig4.jpg|400px|border|center]]
<b>Figure 4 -</b> Vacuum system.
<br clear=right>
</p>
</td></table>

<h2>Evacuating the scattering chamber</h2>

<p>Please be gentle with the valves at all times the '''foil is delicate and very expensive''' to replace. '''Never touch the foil directly as it is easily perforated''', and take it or out of the holder very gently. Carry out the vacuum operation slowly. Always replace the foil in its container bag in between steps.</p>

<ol>
<li>Open B check for secure lid slowly (if it is not open). Close A.</li>
<li>Turn on the roughing pump until the vacuum reaches 70 mm of Mercury.</li>
<li>Slowly close B. </li>
<li>Turn off the pump. </li>
<li>NOTE this is very important! '''Open A (this is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<h2>Releasing the chamber to atmospheric pressure</h2>

<p><b>If the foil is in the chamber, make sure that the foil is perpendicular to the chamber valve when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Close A. Open B slowly.</li>
<li>Open A slowly.</li>
</ol>

<h2>Data Acquisition System</h2>

<p>Note: There are two ways of recording the data: “rate” and “counts”. </p>

<p>A computer interfaced data acquisition system is provided, which can display on-line the Poisson statistical analysis of the recorded events. One might think that measuring the count rate 10 times (for a given time interval), and deducing the average and deviation would be sufficient. However, due to the statistical nature of the radioactive decay process we do not have a constant beam of particles; also the scattering itself is a probabilistic process. Such random events are obeying Poisson statistics. The computerized data acquisition system allows one to collect data in 10-second or 60-second intervals and to assemble a histogram (to be compared with a Poisson fit) and to display graphically the collected count rate. A function is provided to deduce the average count rate and deviation from the data. </p>

<h2>Required Components</h2>
<ol>
<li>[[Media:RSAngle.JPG|Protractor]]</li>
<li>[[Media:RSController.JPG|Control Box]]</li>
<li>[[Media:RSCounter.JPG|Counter]]</li>
<li>[[Media:RSDetector.JPG|Radioactive Detector]]</li>
<li>[[Media:RSGage.JPG|Vacuum Gauge]]</li>
<li>[[Media:RSSlits.JPG|Slits]]</li>
<li>[[Media:RSSource.JPG|Source]]</li>
<li>[[Media:RSVacuum.JPG|Vacuum Pump]]</li>
</ol>

<p>Here are some definitions that may prove to be useful:</p>

<p>The number of scattering centres in the target is related to the thickness of the foil and
the density of the material via ''' N0 = (a x d) x ρ x A<sub>0</sub>/ A''' , where
<ul>
<li>'''A<sub>0</sub>''' = Avogadro’s number = 6.0222 x 10<sup>23</sup> atoms /mole. </li>
<li>'''A''' = atomic weight of the target in g/mole. </li>
<li>'''ρ''' = density of the material (19.3 g/cm<sup>3</sup> for Au, 2.7 g/cm<sup>3</sup> for Al). </li>
<li>'''d''' = thickness of the target (2 micrometers = 2 microns for Au, 7 microns for Al).</li>
<li>'''a''' = area of the target intercepted by the beam.</li>
</ul>
</p>

<p>(N is the same as in Melissinos but note our N0 is not the same as Melissinos N0, which is the Avogadro’s number(A<sub>0</sub> here)).</p>

<p> The activity of the AM-241 alpha source is 330 kBq. </p>

<h1>Procedure</h1>

<p><b>We stress again that you must understand how the vacuum pump operates. Misuse can lead to destruction of the whole apparatus (oil flooding, tearing of the foils etc. ). </b></p>

<ol>
<li>Using a '''5mm slit''' and '''no foil''', find the “zero angle” (θ<sub>0</sub> ) of the apparatus by recording the counting rate of the alpha particles for several angles (-10<sup>o</sup> to +10<sup>o</sup>), starting from the zero on the dial angle indicator (chamber lid). Take measurements for negative and positive angles in increments of 2.5<sup>o</sup> (half of the smallest division on the dial).</li>

<li>Place the '''Gold foil''' together with the '''5mm slit''' on the mount and acquire measurements of counts for the following angles 10, 20, 30, 40, 50, 60 degrees with respect to θ<sub>0</sub>. Make sure you obtain at least 10 counts for each of these angles.</li>

<li>Repeat the above using the '''Aluminium foil''' with the '''5mm slit'''.</li>

<li>Use the '''5mm slit''' to take '''background''' measurements ('''do not place any of the foils''') for the same angles as in step 3 and use this date to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long. However, as you reach the large angles, the time necessary to obtain enough readings can be substantial. Make sure you take at least 1 count for each of the angles (for the small angles you can probably take a few hundreds in a matter of seconds).</li>
</ol>

<p>Plot the log(rate) vs. log(sin<sup>4</sup>θ/2) with appropriate conversion to radians. Derive the relationship between the observed rate and the cross section. What do you expect for the behaviour of this graph? Using the relationship between the rate and the Rutherford cross section formula (equation1), find the ratio of atomic numbers between Gold and Aluminium.</p>

<h1>References</h1>

<ol>
<li>Melissinos, ''Experiments in Modern Physics'', Academic Press.</li>
<li>Preston and Deitz, ''The Art of Experimental Physics'', Wiley and Sons.</li>
<li>H. Frauenfelder and E. Henley, ''Subatomic Physics'', Prentice Hall.</li>
<li>A. Das and T. Ferbel, ''Introduction to Nuclear and Particle Physics'', J. Wiley.</li>
</ol>

File:Ruth1-fig4.jpg

2013-02-26T18:01:06Z

Jlyons:

Main Page/PHYS 3220/Cavendish

2013-02-26T17:59:23Z

Jlyons:

<h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1>

<h1>Introduction</h1>
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p>
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm. bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p>
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Experiment setup (Top View).
<br clear=right>
</p>
</td></table>

<h1>Method</h1>
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p>
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p>
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p>
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p>
<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]
<b>Figure 2 -</b> Graph of small-mass oscillations.
<br clear=right>
</p>
</td></table>
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p>
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p>

<h1>Theory</h1>
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = Tz. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small mass’ rotating about their center of mass). The external torque in our case is provided by the angular form of Hooke's law as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which however is crucial to obtain a steady-state solution after long intervals).

If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed,
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn1.png|80px|center]]
</p>
</td></table>

i.e., Where C is the torque constant (cf.. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Cav-eqn2.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn3.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn4.png|80px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

where m = mass of each ball and 2d is the distance between them.

Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]
</p>
</td></table>

<p>where ''D'' is the distance between the mirror and the recording medium.</p>
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p>
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn5.png|130px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The torque (couple) exerted by the large masses is</p>
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn6.png|160px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p>
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn7.png|90px|center]]
</p>
</td></table>
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p>

<h1>Procedure</h1>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]
<b>Figure 3 -</b> Experimental Setup.
<br clear=right>
</p>
</td></table>

<h2>Data Collection</h2>
<ol>
<li>Run the program "Labview 8.2", and open the vi called "Cavendish.vi".</li>
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li>
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li>
<li>When the program runs, it leads you though the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li>
<table width=400 align=center><td>
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]
<b>Figure 4 -</b> The Labview control program.
<br clear=right>
</p>
</td></table>
<li>The data will be saved as two lists of numbers- one is the center pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li>
</ol>

<h2>Parameters of Apparatus</h2>
<table width=600>
<tr><td width=500> Diameter of lager spherical mass</td><td width=100> 6.386 cm</td></tr>
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr>
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr>
<tr><td> Mass of small spherical mass</td><td>20g</td></tr>
<tr><td> Thickness of Cavendish box enclouse</td><td>3.01cm</td></tr>
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr>
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr>
<tr><td> Number of pixels</td><td>128</td></tr>
<tr><td> Active area length</td><td> 10.2cm</td></tr>
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr>
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr>
</table>

<h2>Tasks</h2>
<ol>
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li>
<li>Discuss the effect of the attraction of the distant 1.5 kg. sphere for the small balls.</li>
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li>
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of the motion. From the data, find the damping constant. See textbook references on damped harmonic motion e.g. French's book.</li>
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li>
<li>Why do you need to measure the period of oscillation?</li>
</ol>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]
<b>Figure 5 -</b> Cavendish Beam Schematic.
<br clear=right>
</p>
</td></table>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]
<b>Figure 6 -</b> Cavendish Beam.
<br clear=right>
</p>
</td></table>

<h1>References</h1>
<ol>
<li> J.W. Beams, [http://physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 ''Physics Today'', May 1971, '''24''', pp. 32-40]. </li>
<li> For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70.</li>
</ol>

Main Page/PHYS 3220/Cavendish

2013-02-26T17:58:59Z

Jlyons:

<h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1>

<h1>Introduction</h1>
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p>
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm. bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p>
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p>
<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]
<b>Figure 1 -</b> Experiment setup (Top View).
<br clear=right>
</p>
</td></table>

<h1>Method</h1>
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p>
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p>
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p>
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p>
<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]
<b>Figure 2 -</b> Graph of small-mass oscillations.
<br clear=right>
</p>
</td></table>
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p>
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p>

<h1>Theory</h1>
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = Tz. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small mass’ rotating about their center of mass). The external torque in our case is provided by the angular form of Hooke's law as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which however is crucial to obtain a steady-state solution after long intervals).

If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed,
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn1.png|80px|center]]
</p>
</td></table>

i.e., Where C is the torque constant (cf.. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:Cav-eqn2.png|110px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn3.png|110px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn4.png|80px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

where m = mass of each ball and 2d is the distance between them.

Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]
</p>
</td></table>

<p>where ''D'' is the distance between the mirror and the recording medium.</p>
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p>
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn5.png|130px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The torque (couple) exerted by the large masses is</p>
<table width=200 align=center><tr><td>
<p align=justify>[[File:Cav-eqn6.png|160px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p>
<table width=200 align=center><td>
<p align=justify>[[File:Cav-eqn7.png|90px|center]]
</p>
</td></table>
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p>

<h1>Procedure</h1>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]
<b>Figure 3 -</b> Experimental Setup.
<br clear=right>
</p>
</td></table>

<h2>Data Collection</h2>
<ol>
<li>Run the program "Labview 8.2", and open the vi called "Cavendish.vi".</li>
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li>
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li>
<li>When the program runs, it leads you though the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li>
<table width=400 align=center><td>
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]
<b>Figure 4 -</b> The Labview control program.
<br clear=right>
</p>
</td></table>
<li>The data will be saved as two lists of numbers- one is the center pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li>
</ol>

<h2>Parameters of Apparatus</h2>
<table width=600>
<tr><td width=500> Diameter of lager spherical mass</td><td width=100> 6.386 cm</td></tr>
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr>
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr>
<tr><td> Mass of small spherical mass</td><td>20g</td></tr>
<tr><td> Thickness of Cavendish box enclouse</td><td>3.01cm</td></tr>
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr>
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr>
<tr><td> Number of pixels</td><td>128</td></tr>
<tr><td> Active area length</td><td> 10.2cm</td></tr>
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr>
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr>
</table>

<h2>Tasks</h2>
<ol>
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li>
<li>Discuss the effect of the attraction of the distant 1.5 kg. sphere for the small balls.</li>
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li>
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of the motion. From the data, find the damping constant. See textbook references on damped harmonic motion e.g. French's book.</li>
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li>
<li>Why do you need to measure the period of oscillation?</li>
</ol>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]
<b>Figure 5 -</b> Cavendish Beam Schematic.
<br clear=right>
</p>
</td></table>

<table width=400 align=center><td>
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]
<b>Figure 5 -</b> Cavendish Beam.
<br clear=right>
</p>
</td></table>

<h1>References</h1>
<ol>
<li> J.W. Beams, [http://physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 ''Physics Today'', May 1971, '''24''', pp. 32-40]. </li>
<li> For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70.</li>
</ol>

File:Cav-fig5.jpg

2013-02-26T17:57:55Z

Jlyons:

File:Vis-fig4v2.JPG

2013-02-26T15:27:09Z

Jlyons: uploaded a new version of "File:Vis-fig4v2.JPG"

Main Page/PHYS 3220/Viscosity

2013-02-26T15:24:46Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.JPG|300px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2v2.JPG|300px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3v2.JPG|300px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4v2.JPG|400px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

Main Page/PHYS 3220/Viscosity

2013-02-26T15:23:28Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.JPG|300px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2v2.JPG|300px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3v2.JPG|300px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4v2.JPG|300px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

Main Page/PHYS 3220/Viscosity

2013-02-26T15:22:48Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.JPG|300px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2v2.JPG|400px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3v2.JPG|400px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4v2.JPG|300px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

File:Vis-fig3v2.JPG

2013-02-26T15:21:34Z

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File:Vis-fig2v2.JPG

2013-02-26T15:20:55Z

Jlyons: uploaded a new version of "File:Vis-fig2v2.JPG"

File:Vis-fig1v2.JPG

2013-02-26T15:20:14Z

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Main Page/PHYS 3220/Viscosity

2013-02-19T18:40:26Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.JPG|400px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2v2.JPG|400px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3v2.JPG|400px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4v2.JPG|300px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

Main Page/PHYS 3220/Viscosity

2013-02-19T18:39:43Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.JPG|400px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2v2.JPG|400px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3v3.JPG|400px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4v2.JPG|300px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

File:Vis-fig4v2.JPG

2013-02-19T18:38:08Z

Jlyons:

File:Vis-fig3v2.JPG

2013-02-19T18:37:50Z

Jlyons:

File:Vis-fig2v2.JPG

2013-02-19T18:37:38Z

Jlyons:

Main Page/PHYS 3220/Viscosity

2013-02-19T18:37:12Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.JPG|400px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2.png|400px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3.png|400px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4.png|300px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

Main Page/PHYS 3220/Viscosity

2013-02-19T18:35:15Z

Jlyons:

<h1>Viscosity</h1>

<p>The purpose of this experiment is to determine the viscosity of a liquid and to find the variation of viscosity with temperature.</p>

<h1>Theory</h1>
<p>When a solid is subject to a shearing stress it deforms until the internal elastic forces of the solid exactly balance the external forces. Thus a finite force applied to a solid produces a finite deformation. If a similar force is applied to a liquid, however, the deformation increases indefinitely (the liquid flows). The flow can be imagined as the movement of adjacent layers over one another. The Newtonian friction caused by the relative motion between adjacent layers retards the flow and is called viscosity. This frictional force was assumed by Newton to be proportional to the velocity gradient perpendicular to the direction of the motion of the fluid, i.e., to dv/dr.</p>
<p>Consider an element of volume in a fluid as shown in the following diagram (ref. 1)</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig1v2.jpg|400px|border|center]]
<b>Figure 1 -</b> Element of volume of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>The shearing stress on this element of volume is F/A where F is the force on the upper surface and A is the cross section. <b>The shear is given by the ratio between the lateral displacement between the two surfaces to the separation between the surfaces.</b> Thus, if we assume that the upper surface is moving with a velocity, dv, greater than that of the lower surface, the amount of shear occurring in unit time is dv/dr. The coefficient of viscosity or simply <b>viscosity</b> is defined as follows (for streamline motion).</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>

<p>The c.g.s. unit of viscosity is called the <b>poise</b>; it represents the viscosity of a substance that acquires a unit velocity gradient under the influence of a shearing stress of 1 dyne/cm2.</p>
<p>One method of measuring viscosity is to determine the flow of a liquid through a capillary tube. Let us consider how to measure viscosity in this way.</p>

<p>When a liquid flows through a narrow tube so that each particle moves parallel to the axis of the tube with constant velocity, the motion is said to be regular or <b>streamlined</b>. In this case, liquid in contact with the walls is at rest while the velocity is a maximum at the centre of the tube.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig2.png|400px|border|center]]
<b>Figure 2 -</b> Streamlined flow of a liquid in a tube.
<br clear=right>
</p>
</td></table>

<p>We imagine the liquid as being divided up into a number of thin cylindrical shells, each shell sliding on the other. The viscous drag, f, per square cm. of one layer on the other is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn2.png|90px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>For a cylinder of radius, a, length l, through which a liquid is flowing under a pressure differential, p, the total viscous drag balances the force due to the pressure difference.</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn3.png|180px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<p>Assuming that there is no radial flow the pressure is constant over any given cross-section.
At r = a, v = 0, while at r = r, v = u, thus</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn4.png|120px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn5.png|100px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>

<p>Thus, by measuring the quantity of liquid transferred through a capillary tube per unit time subject to a specific pressure difference, the viscosity of the liquid may be determined.</p>
<p>The viscosity may also be determined using the concentric cylinder viscosimeter. Consider a liquid between two concentric cylinders as illustrated in Figure 3. </p>

<table width=400 align=center><td>
<p align=justify>[[File:Vis-fig3.png|400px|border|center]]
<b>Figure 3 -</b> Liquid between concentric cylinders.
<br clear=right>
</p>
</td></table>

<p>The outer cylinder is moving with angular velocity wB. If the liquid adheres to the walls of the cylinder, a shearing takes place in which concentric cylindrical layers of the liquid slip over each other with the angular velocity w increasing progressively from zero at the stationary cylinder to wB at the rotating one. Now, the rate of shear is given as</p>

<p>The quantity of liquid, Q, flowing through the tube per second is given by Poiseuille's formula,</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn6.png|250px|center]]
</p>
</td><td> <b>(6)</b></td></tr></table>

<p>Only the second term produces a viscosity effect, since a velocity gradient is needed to have relative slipping of layers and the associated friction. Thus, </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn7.png|130px|center]]
</p>
</td><td> <b>(7)</b></td></tr></table>

<p>If a torque L is applied to the rotating cylinder, (see Figure 3), the tangential force at the boundary SS' equals L/r and integration over r from A to B yields</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn8.png|140px|center]]
</p>
</td><td> <b>(8)</b></td></tr></table>

<p>Note that this formula is also valid if the inner cylinder is rotated with angular velocity w<sub>B</sub> while the outer cylinder is held stationary.</p>
<p>For cylinders closed at either end, equation (8) is modified to take into account the friction between the two ends of the cylinders according to</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn9.png|160px|center]]
</p>
</td><td> <b>(9)</b></td></tr></table>

<h1>Method</h1>
<ol>
<li> <h2>Capillary tube.</h2>
<p>For the determination of viscosity by use of a capillary tube we need to know the pressure p, this is given by </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn10.png|80px|center]]
</p>
</td><td> <b>(10)</b></td></tr></table>
<p>where
<ul><li>h = height of water column</li>
<li>d = density of water</li>
<li>g = acceleration due to gravity = 981 cm/sec<sup>2</sup></li>
</ul></p>
<p>The capillary tube is positioned at the bottom of a beaker, allowing for the monitoring of the water level. Using regular tap water, fill the beaker until the water starts going through the capillary tube. (Notice that the height of the water column is changing as water flows out, it is up to you to figure out a solution, consult your T.A. for ideas). </p>
<p>Measure the flow rate through the tube for various p values and calculate the viscosity. How does the result compare with the accepted viscosity value for water? Would you expect the same result if the capillary tube was slightly wider? Much wider? Comment on the reason for any potential discrepancies.</p>
<p>Record the water temperature. Repeat the measurements for at least two other temperatures. Assuming that the dependence resembles the “Arrhenius” equation </p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn11.png|110px|center]]
</p>
</td><td> <b>(11)</b></td></tr></table>
<p>where A can be viewed as a correction factor, E<sub>a</sub> plays a role similar to activation energy in chemical reactions and R is the gas constant. Plot a graph of the obtained values and fit to (11). Hint: it is possible to turn (11) into a linear graph. </p>
<p> Analyze the resulting fit with comments on the significance and the validity of the made assumption. </p>
</li>
<li><h2>Concentric cylinder viscosimeter</h2>
<p>This instrument is used to find the viscosity of vacuum pump oil and to determine its variation with temperature. The inner drum in the viscosimeter is mounted on two bearings and is free to rotate under a torque, L, supplied by a mass, m, falling through a given height. The torque is given by L = mgk where k is the radius of the pulley around which the string supporting the falling mass m, is wound. If the mass falls a distance s, in time t, the angular velocity w is given by</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn12.png|60px|center]]
</p>
</td><td> <b>(12)</b></td></tr></table>

<p>Therefore, Equation (9) becomes</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn13.png|160px|center]]
</p>
</td><td> <b>(13)</b></td></tr></table>
<p>or</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn14.png|70px|center]]
</p>
</td><td> <b>(14)</b></td></tr></table>
<p>where c is a constant</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Vis-eqn15.png|160px|center]]
</p>
</td><td> <b>(15)</b></td></tr></table>
<p>To find the absolute value of η, remove the centre cylinder (being careful not to damage the supporting bearings) and measure all pertinent dimensions. Re-assemble the apparatus and fill the container with oil until the level is about 1.5 cm above the lower surface of the inner cylinder. Measure the length l using a vernier calliper (notice that l is not simply the height of the oil level). Place the apparatus on the edge of the wall mount so that the suspended mass has an unrestricted fall of several feet.</p>
<table width=300 align=center><td>
<p align=justify>[[File:Vis-fig4.png|300px|border|center]]
<b>Figure 4 -</b> Viscosimeter.
<br clear=right>
</p>
</td></table>
<p>Record the time required for a mass m suspended from the string (wound around the pulley) to descend through a given distance. You will find that the downward velocity is not constant over the total drop but changes rapidly over the first few centimetres. It is therefore necessary to neglect the first few centimetres of drop, i.e., the stopwatch must be started once the descending mass is moving with constant velocity. Make several measurements of the time required to fall through a distance, s, and obtain an average time. Make a series of five determinations keeping the mass constant and increasing the effective length of the cylinder by adding liquid. The last observation should be made with the apparatus filled to the top of the inner cylinder.</p>

<p> To collect the data, we can observe the transit of the masses through the two [[media:Visc_sensor.jpg|sensor gates]] on an [[media:Visc_scope.jpg|oscilloscope]]. With the oscilloscope set for a long timebase, and the sensor box outputs attached to one channel, you will notice the abrupt signal change when anything passes through either sensor gate. You will need to use the "Run/Stop" button on the top right of oscilloscope to stop the trace from scrolling off the screen after the data is collected so you can measure or save the data. The scope can be used as a precise measuring tool for the time require for the mass to fall a particular distance.</p>

<p>With the level of the liquid at the top of the inner cylinder, take a series of observations of time of descent vs. mass.</p>
<p>To analyze your data, plot two graphs: one of the length '''l''' vs. time '''t''', and the other of the time, '''t''', vs. reciprocal mass'''1 /m'''. The first graph will give the value of the length correction '''e'''. The second graph should be a straight line of slope '''mt'''. '''η''' can then be obtained from Equation (12).</p>
<p>To determine the variation of viscosity with temperature, repeat the above fixed mass measurement at several temperatures. Be sure to allow for expansion of the liquid when the temperature is increased. Compare the obtained results to the previous section.</p>
</ol>
<h1>Units</h1>
<p>Give your results in both CGS and MKS units. <b>The values of the viscosities are available from handbooks</b>, and are usually quoted in centipoises (compare your results).</p>

<h1>Questions and Discussion</h1>
<p>Why does the moment of inertia of the rotating cylinder not enter the analysis of the problem? Discuss the sign of the length correction. Explain what the important sources of imprecision are, and elaborate on the importance of the constants e and c.</p>

<h1>References</h1>
<p>Any intermediate-level classical mechanics text, e.g.,</p>
<ol>
<li>K. Symon, ''Mechanics'', 3rd ed., pp. 345 ff.</li>
<li>R. Feynman, Leighton, Sands,''Lectures on Physics'', vol. II, chapter 41.</li>
</ol>

File:Vis-fig1v2.JPG

2013-02-19T18:34:41Z

Jlyons:

Main Page/PHYS 4210/He-Ne Lasers

2013-02-19T18:28:11Z

Jlyons:

<h1>He-Ne Lasers</h1>

<p>In this experiment we first align an open-ended laser. Then we set up some transverse mode patterns, and perform further exercises and experiments to understand how a laser works.</p>

<h2> Key Concepts</h2>
<table width=500>
<td width=250><ul>
<li>Stimulated Emission</li>
<li>Spontaneous Emission</li>
<li>Stimulated Emission</li>
<li>Incoherent/Coherent Radiation</li>
<li>Einstein Coefficients</li>
<li>Population Inversion</li>
<li>Forbidden Transitions</li>
<li>Metastable States</li>
<li>LS coupling</li>
<li>Electric dipole selection rules</li>
</ul>
</td>
<td width=250>
<ul>
<li>Axial Modes</li>
<li>TEM modes</li>
<li>Spectral Width</li>
<li>Atomic Lineshape</li>
<li>Loss/Gain Coefficient</li>
<li>Index of Refraction</li>
<li>Brewster’s Angle</li>
<li>Malus’s Law</li>
<li>Fresnel-Arago Law</li>
<li>Q-Switch</li>
</ul>
</td>
</table>

<h1>Reading and Exercises</h1>
<ul>
<li>Read pages 94 to 105 from Preston-Dietz. Carry out '''Exercise 1''' (pg. 100), '''Exercise 2''' (pg. 103), '''Exercise 3''' (pg. 104), and '''Exercise 4''' (pg. 104) and submit them as part of your report, either in the introduction or as an appendix as you deem appropriate.</li>
<li>Read pages 100 to 112, on laser cavity modes.</li>
<li>Do '''Exercise 1''' (pp 111-112). Do not forget to answer the last question of the exercise: Calculate the frequency difference between two adjacent axial modes TEM<sub>oom</sub> & TEM<sub>oo(m+1)</sub>.</li>
</ul>

<h1>Experiments</h1>
<h2>Aligning the laser</h2>
<p>Align the laser until it begins lasing. The TA will discuss techniques to accomplish this.</p>
<p>When you are successful with the aligning process, and lasing is achieved, try varying the distances between the mirrors (using the adjusting screws on the laser mount) that still supports lasing. Record, in your lab book, the aligning process used, maximum, minimum and 'best' distances between the mirrors.</p>
<p>What the radius of curvature of the mirrors which form the optical cavity? </p>
<p>Use the polarizers to determine the polarization of laser.</p>
<p>Assume that the He-Ne produces 3 mW of laser output power and that the electrical data given applies to your gas discharge tube. Compute the efficiency, in percent, for converting electrical energy to red laser light energy with this He-Ne laser. Discuss your results.</p>

<h2>Brewster's Angle</h2>
<p>Using the glass plate provided, find an approximate value for the Brewster's angle. You can determine this by rotating the glass plate until lasing stops. Only at the Brewster's angle does lasing resume.</p>
<p>Knowing that the tangent of the Brewster's angle is the ratio of the refractive indices of the lasing medium to air, determine the refractive index of the glass medium.</p>
<p>Every resonant laser cavity has a characteristic quality factor or Q that measures the internal losses. The higher the Q, the lower the losses.</p>
<p>A Q-switch pulse can be made by blocking one end of the mirror, then exciting the medium and then quickly unblocking the mirror. Stimulated emission will quickly drain the stored laser energy from the cavity in a short pulse with peak power much higher than the laser can produce. One can think of a Q-switch as a device that quickly switches from absorbing to transmitting, suddenly reducing cavity losses.</p>
<p>The Q-switch pulse length is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn1a.png|180px|center]]
</p>
</td></table>
<p>where t is the round trip time (back and forth in the cavity), and R is the output mirror reflectivity ( >98% ).</p>
<p>Therefore</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn2.png|210px|center]]
</p>
</td></table>
<p>where ''L'' is the distance between the mirrors, ''n<sub>1</sub>'' is the refractive index of the medium, and ''c'' is the speed of light. Pulse length can then be written as</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn3.png|180px|center]]
</p>
</td></table>
<p>Using the data from your laser, what is the theoretical value for the pulse length?</p>

<h2>TEM Modes</h2>
<p>Set up a camera with the screen at about 1 to 2 meters from the output mirror. Adjust the output coupler screws (or any other adjustments) to produce the TEM<sub>00m</sub>, TEM<sub>10m</sub>, TEM<sub>01m</sub>, .... modes. Photograph or sketch a few of them.</p>

<h2>Beam Profile of the TEM<sub>oom</sub> and the TEM<sub>10m</sub> modes</h2>
<p>Realign the beam to produce the TEM<sub>oom</sub> mode.</p>
<p>You will use a rotating mirror and a photodiode monitored on an oscilloscope to observe the profile of the laser beam.</p>
<p>Be sure to ensure the photodiode is not saturating when the laser is aligned onto it. If it is, switch the scale of the photodiode amplifier to a lower gain setting.</p>
<p>Repeat your observations for the TEM<sub>o1m</sub> mode. Remember that photodetectors are square-law detectors, i.e., the current density J is proportional to the square of the electric field. (See Preston for details). Sketch the beam profiles for both modes.</p>

<h2>Beam Profile or Shape</h2>
<p>A laser beam has a certain profile with most energy concentrated at the center. The beam has the following form</p>

<table width=400 align=center><td>
<p align=justify>[[File:HeNe-fig1v2.jpg|400px|border|center]]
<b>Figure 1 -</b> Amplitude distribution across laser beam oscillating in the TEM<sub>oo</sub> mode.
<br clear=right>
</p>
</td></table>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn4.png|120px|center]]
</p>
</td></table>

<p>where ''w'' is the radius of the beam. The Gaussian function, exp [- (''r''/''w'')<sup>2</sup> ] falls to 1/e, when ''r'' = ''w'', i.e.,</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn5.png|90px|center]]
</p>
</td></table>
<p>Since the energy is proportional to the square of the amplitude, the beam radius, or SPOT SIZE, ''w'', is defined as that distance from the axis where the power has dropped to 1/e<sup>2</sup> of its value at the center. Twice that distance, 2''w'', is the beam diameter.</p>
<p>The beam radius, ''w'', is the function of distance along the axis. If we call ''x'' the axial distance measured from the midpoint between the two (concave) mirrors, then the parameter ''w'' is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn6.png|160px|center]]
</p>
</td></table>
<p>where λ is the wavelength and ''w<sub>0</sub>'' is the minimum beam radius between mirrors.</p>
<table width=400 align=center><td>
<p align=justify>[[File:HeNe-fig2v2.jpg|400px|border|center]]
<br clear=right>
</p>
</td></table>

<p>Note that from ''w<sub>x</sub>'' above; at ''x'' = 0, ''w<sub>x</sub>'' = ''w<sub>o</sub>''.</p>
<p>From Preston (equation 21, p. 102),</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn7.png|160px|center]]
</p>
</td></table>
<p>Calculate ''w<sub>o</sub>'' and ''w<sub>x</sub>''.</p>
<p>From your observations of the beam profile for the TEM<sub>oom</sub>, determine ''w<sub>x</sub>'', the beam radius. How does your calculated value compare with the experimental value? Explain any differences.</p>

<h2>Malus's Law</h2>
<p>Malus’ law states that when a linearly polarized light beam of intensity ''I<sub>0</sub>'' passes through a linear polarizer with its axis rotated by angle ''A'' from the light beam polarization, the emergent intensity ''I'' is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn8.png|90px|center]]
</p>
</td></table>
<p>Use the rotatable polarizer and photodiode detector to verify this law quantitatively. Make detector readings at several values of angle ''A'' and record them in a neat table in your notebook. Graph your data to demonstrate the expected cos<sup>2</sup>''A''dependence.</p>

<h2>Verification of the Fresnel-Arago Law</h2>
<p>Fresnel-Arago law state that two coherent light rays which are polarized right angles to each other will not mutually interfere. </p>
<p>Use the laser to set up the Michelson interferometer as shown below to form an interference pattern. </p>
<p>Insert polarizers P<sub>1</sub> and P<sub>2</sub> such that their axes of polarization is in the same direction. You may have to make slight adjustments to retain the interference pattern. Now, rotate ONE of the polarizers through 90º. Verify that the law is true. Try to take pictures of the resulting effect and include them with your report.</p>

File:HeNe-fig2v2.jpg

2013-02-19T18:27:01Z

Jlyons:

Main Page/PHYS 4210/He-Ne Lasers

2013-02-19T18:26:24Z

Jlyons:

<h1>He-Ne Lasers</h1>

<p>In this experiment we first align an open-ended laser. Then we set up some transverse mode patterns, and perform further exercises and experiments to understand how a laser works.</p>

<h2> Key Concepts</h2>
<table width=500>
<td width=250><ul>
<li>Stimulated Emission</li>
<li>Spontaneous Emission</li>
<li>Stimulated Emission</li>
<li>Incoherent/Coherent Radiation</li>
<li>Einstein Coefficients</li>
<li>Population Inversion</li>
<li>Forbidden Transitions</li>
<li>Metastable States</li>
<li>LS coupling</li>
<li>Electric dipole selection rules</li>
</ul>
</td>
<td width=250>
<ul>
<li>Axial Modes</li>
<li>TEM modes</li>
<li>Spectral Width</li>
<li>Atomic Lineshape</li>
<li>Loss/Gain Coefficient</li>
<li>Index of Refraction</li>
<li>Brewster’s Angle</li>
<li>Malus’s Law</li>
<li>Fresnel-Arago Law</li>
<li>Q-Switch</li>
</ul>
</td>
</table>

<h1>Reading and Exercises</h1>
<ul>
<li>Read pages 94 to 105 from Preston-Dietz. Carry out '''Exercise 1''' (pg. 100), '''Exercise 2''' (pg. 103), '''Exercise 3''' (pg. 104), and '''Exercise 4''' (pg. 104) and submit them as part of your report, either in the introduction or as an appendix as you deem appropriate.</li>
<li>Read pages 100 to 112, on laser cavity modes.</li>
<li>Do '''Exercise 1''' (pp 111-112). Do not forget to answer the last question of the exercise: Calculate the frequency difference between two adjacent axial modes TEM<sub>oom</sub> & TEM<sub>oo(m+1)</sub>.</li>
</ul>

<h1>Experiments</h1>
<h2>Aligning the laser</h2>
<p>Align the laser until it begins lasing. The TA will discuss techniques to accomplish this.</p>
<p>When you are successful with the aligning process, and lasing is achieved, try varying the distances between the mirrors (using the adjusting screws on the laser mount) that still supports lasing. Record, in your lab book, the aligning process used, maximum, minimum and 'best' distances between the mirrors.</p>
<p>What the radius of curvature of the mirrors which form the optical cavity? </p>
<p>Use the polarizers to determine the polarization of laser.</p>
<p>Assume that the He-Ne produces 3 mW of laser output power and that the electrical data given applies to your gas discharge tube. Compute the efficiency, in percent, for converting electrical energy to red laser light energy with this He-Ne laser. Discuss your results.</p>

<h2>Brewster's Angle</h2>
<p>Using the glass plate provided, find an approximate value for the Brewster's angle. You can determine this by rotating the glass plate until lasing stops. Only at the Brewster's angle does lasing resume.</p>
<p>Knowing that the tangent of the Brewster's angle is the ratio of the refractive indices of the lasing medium to air, determine the refractive index of the glass medium.</p>
<p>Every resonant laser cavity has a characteristic quality factor or Q that measures the internal losses. The higher the Q, the lower the losses.</p>
<p>A Q-switch pulse can be made by blocking one end of the mirror, then exciting the medium and then quickly unblocking the mirror. Stimulated emission will quickly drain the stored laser energy from the cavity in a short pulse with peak power much higher than the laser can produce. One can think of a Q-switch as a device that quickly switches from absorbing to transmitting, suddenly reducing cavity losses.</p>
<p>The Q-switch pulse length is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn1a.png|180px|center]]
</p>
</td></table>
<p>where t is the round trip time (back and forth in the cavity), and R is the output mirror reflectivity ( >98% ).</p>
<p>Therefore</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn2.png|210px|center]]
</p>
</td></table>
<p>where ''L'' is the distance between the mirrors, ''n<sub>1</sub>'' is the refractive index of the medium, and ''c'' is the speed of light. Pulse length can then be written as</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn3.png|180px|center]]
</p>
</td></table>
<p>Using the data from your laser, what is the theoretical value for the pulse length?</p>

<h2>TEM Modes</h2>
<p>Set up a camera with the screen at about 1 to 2 meters from the output mirror. Adjust the output coupler screws (or any other adjustments) to produce the TEM<sub>00m</sub>, TEM<sub>10m</sub>, TEM<sub>01m</sub>, .... modes. Photograph or sketch a few of them.</p>

<h2>Beam Profile of the TEM<sub>oom</sub> and the TEM<sub>10m</sub> modes</h2>
<p>Realign the beam to produce the TEM<sub>oom</sub> mode.</p>
<p>You will use a rotating mirror and a photodiode monitored on an oscilloscope to observe the profile of the laser beam.</p>
<p>Be sure to ensure the photodiode is not saturating when the laser is aligned onto it. If it is, switch the scale of the photodiode amplifier to a lower gain setting.</p>
<p>Repeat your observations for the TEM<sub>o1m</sub> mode. Remember that photodetectors are square-law detectors, i.e., the current density J is proportional to the square of the electric field. (See Preston for details). Sketch the beam profiles for both modes.</p>

<h2>Beam Profile or Shape</h2>
<p>A laser beam has a certain profile with most energy concentrated at the center. The beam has the following form</p>

<table width=400 align=center><td>
<p align=justify>[[File:HeNe-fig1v2.jpg|400px|border|center]]
<b>Figure 1 -</b> Amplitude distribution across laser beam oscillating in the TEM<sub>oo</sub> mode.
<br clear=right>
</p>
</td></table>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn4.png|120px|center]]
</p>
</td></table>

<p>where ''w'' is the radius of the beam. The Gaussian function, exp [- (''r''/''w'')<sup>2</sup> ] falls to 1/e, when ''r'' = ''w'', i.e.,</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn5.png|90px|center]]
</p>
</td></table>
<p>Since the energy is proportional to the square of the amplitude, the beam radius, or SPOT SIZE, ''w'', is defined as that distance from the axis where the power has dropped to 1/e<sup>2</sup> of its value at the center. Twice that distance, 2''w'', is the beam diameter.</p>
<p>The beam radius, ''w'', is the function of distance along the axis. If we call ''x'' the axial distance measured from the midpoint between the two (concave) mirrors, then the parameter ''w'' is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn6.png|160px|center]]
</p>
</td></table>
<p>where λ is the wavelength and ''w<sub>0</sub>'' is the minimum beam radius between mirrors.</p>
<table width=400 align=center><td>
<p align=justify>[[File:HeNe-fig2.png|400px|border|center]]
<br clear=right>
</p>
</td></table>

<p>Note that from ''w<sub>x</sub>'' above; at ''x'' = 0, ''w<sub>x</sub>'' = ''w<sub>o</sub>''.</p>
<p>From Preston (equation 21, p. 102),</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn7.png|160px|center]]
</p>
</td></table>
<p>Calculate ''w<sub>o</sub>'' and ''w<sub>x</sub>''.</p>
<p>From your observations of the beam profile for the TEM<sub>oom</sub>, determine ''w<sub>x</sub>'', the beam radius. How does your calculated value compare with the experimental value? Explain any differences.</p>

<h2>Malus's Law</h2>
<p>Malus’ law states that when a linearly polarized light beam of intensity ''I<sub>0</sub>'' passes through a linear polarizer with its axis rotated by angle ''A'' from the light beam polarization, the emergent intensity ''I'' is given by</p>
<table width=200 align=center>
<td>
<p align=justify>[[File:HeNe-eqn8.png|90px|center]]
</p>
</td></table>
<p>Use the rotatable polarizer and photodiode detector to verify this law quantitatively. Make detector readings at several values of angle ''A'' and record them in a neat table in your notebook. Graph your data to demonstrate the expected cos<sup>2</sup>''A''dependence.</p>

<h2>Verification of the Fresnel-Arago Law</h2>
<p>Fresnel-Arago law state that two coherent light rays which are polarized right angles to each other will not mutually interfere. </p>
<p>Use the laser to set up the Michelson interferometer as shown below to form an interference pattern. </p>
<p>Insert polarizers P<sub>1</sub> and P<sub>2</sub> such that their axes of polarization is in the same direction. You may have to make slight adjustments to retain the interference pattern. Now, rotate ONE of the polarizers through 90º. Verify that the law is true. Try to take pictures of the resulting effect and include them with your report.</p>

File:HeNe-fig1v2.jpg

2013-02-19T18:25:39Z

Jlyons:

Main Page/PHYS 4210/Muon Lifetime

2013-01-30T20:28:44Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 8 shows the output directly from the PMT into a 50Ω load. Figure
9 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 8- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 9- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 10- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 11- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a new set of data (different from the one you used to determine the muon lifetime) of 2000 decay events. Group the data, chronologically, in sets of 50 points. (This leaves you with 40 sets of data containing fifty points.) Examine each data set and record how many events, or times, in each of the sets have a lifetime less than the lifetime you found out earlier. (On average this should be 31.5) Do this for all 40 of the data sets. Histogram the number of "successes." The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-29T21:21:12Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 8 shows the output directly from the PMT into a 50Ω load. Figure
9 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 8- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 9- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 10- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 11- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-29T21:18:56Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 4. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 5. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 6. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 7- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 8 shows the output directly from the PMT into a 50Ω load. Figure
9 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 8- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 9- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 10- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 11- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Disconnect the function generator and using the pulser on the detector, measure the time between successive rising edges
on the oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results vs. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-29T21:13:38Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 4- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 5- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 6- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 7 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Disconnect the function generator and using the pulser on the detector, measure the time between successive rising edges
on the oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results vs. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-29T21:11:51Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Disconnect the function generator and using the pulser on the detector, measure the time between successive rising edges
on the oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results vs. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-28T21:10:10Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h3>Time Dilation Effect</h3>
<p>A measurement of the muon stopping rate at two different altitudes can be used to
demonstrate the time dilation effect of special relativity. Although the detector
configuration is not optimal for demonstrating time dilation, a useful measurement can
still be preformed without additional scintillators or lead absorbers. Due to the finite size
of the detector, only muons with a typical total energy of about 160 MeV will stop inside
the plastic scintillator. The stopping rate is measured from the total number of observed
muon decays recorded by the instrument in some time interval. This rate in turn is
proportional to the flux of muons with total energy of about 160 MeV and this flux
decreases with diminishing altitude as the muons descend and decay in the atmosphere.
After measuring the muon stopping rate at one altitude, predictions for the stopping rate
at another altitude can be made with and without accounting for the time dilation effect of
special relativity. A second measurement at the new altitude distinguishes between
competing predictions.</p>
<p>A comparison of the muon stopping rate at two different altitudes should account for the
muon’s energy loss as it descends into the atmosphere, variations with energy in the
shape of the muon energy spectrum, and the varying zenith angles of the muons that stop
in the detector. Since the detector stops only low energy muons, the stopped muons
detected by the low altitude detector will, at the elevation of the higher altitude detector,
necessarily have greater energy. This energy difference ΔE(h) will clearly depend on the
pathlength between the two detector positions.</p>
<p>Vertically travelling muons at the position of the higher altitude detector that are
ultimately detected by the lower detector have an energy larger than those stopped and
detected by the upper detector by an amount equal to DE(h). If the shape of the muon
energy spectrum changes significantly with energy, then the relative muon stopping rates
at the two different altitudes will reflect this difference in spectrum shape at the two
different energies. (This is easy to see if you suppose muons do not decay at all.) This
variation in the spectrum shape can be corrected for by calibrating the detector in a
manner described below.</p>
<p>Like all charged particles, a muon loses energy through coulombic interactions with the
matter it traverses. The average energy loss rate in matter for singly charged particles
traveling close to the speed of light is approximately 2 MeV/g/cm<sup>2</sup>, where we measure
the thickness s of the matter in units of g/cm<sup>2</sup>. Here, ''s'' = ρx, where ρ is the mass density
of the material through which the particle is passing, measured in g/cm<sup>3</sup>, and the x is the
particle’s pathlength, measured in cm. (This way of measuring material thickness in
units of g/cm<sup>2</sup> allows us to compare effective thicknesses of two materials that might
have very different mass densities.) A more accurate value for energy loss can be
determined from the Bethe-Bloch equation.
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn7.png|300px|center]]
</p>
</td></tr></table>
Here N is the number of electrons in the stopping medium per cm<sup>3</sup>, ''e'' is the electronic
charge, ''z'' is the atomic number of the projectile, ''Z'' and ''A'' are the atomic number and
weight, respectively, of the stopping medium. The velocity of the projectile is ''β'' in units
of the speed ,c, of light and its corresponding Lorentz factor is ''γ''. The symbol ''I'' denotes the
mean excitation energy of the stopping medium atoms. Approximately, ''I''=''AZ'', where
''A''≈ 13 eV. More accurate values for ''I'', as well as corrections to the Bethe-Bloch equation,
can be found here<ref>Leo, W. R., "''Techniques for Nuclear and Particle Physics Experiments''", (1994,
Springer-Verlag, New York).</ref>.</p>

<p>A simple estimate of the energy lost ΔE by a muon as it travels a vertical distance H is
ΔE = 2 MeV/g/cm<sup>2</sup> * H * ρ<sub>air</sub>, where ρ<sub>air</sub> is the density of air, possibly averaged over
H using the density of air according to the “standard atmosphere.” Here the atmosphere
is assumed isothermal and the air pressure p at some height h above sea level is
parameterized by p = p<sub>0</sub> exp(-h/h<sub>0</sub>), where p<sub>0</sub> = 1030 g/cm<sup>2</sup> is the total thickness of the
atmosphere and h<sub>0</sub> = 8.4 km. The units of pressure may seem unusual to you but they are
completely acceptable. From hydrostatics, you will recall that the pressure P at the base
of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will
then recognize the units of the right hand side as g/cm<sup>2</sup>. The air density r, in familiar
units of g/cm<sup>3</sup>, is given by ρ = −dp/dh.</p>

<p>If the transit time for a particle to travel vertically from some height H down to sea level,
all measured in the lab frame, is denoted by t, then the corresponding time in the
particle’s rest frame is t’ and given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn8.png|200px|center]]
</p>
</td></tr></table>
Here β and γ have their usual relativistic meanings for the projectile and are measured in
the lab frame. Since relativistic muons lose energy at essentially a constant rate when
travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC<sub>0</sub> dh,
with C<sub>0</sub> = 2 MeV/(g/cm2). Also, from the Einstein relation, E = γmc<sup>2</sup>, dE = mc<sup>2</sup> dγ, so
dh = (mc<sup>2</sup>/ρC<sub>0</sub>) dγ. Hence,
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn9.png|220px|center]]
</p>
</td></tr></table>
Here γ<sub>1</sub> is the muon’s gamma factor at height H and γ<sub>2</sub> is its gamma factor just before it
enters the scintillator. We can take γ<sub>2</sub> = 1.5 since we want muons that stop in the scintillator and assume that on average stopped muons travel halfway into the scintillator,
corresponding to a distance s = 10 g/cm<sup>2</sup>. The entrance muon momentum is then taken
from range-momentum graphs at the Particle Data Group WWW site and the
corresponding γ<sub>2</sub> computed. The lower limit of integration is given by γ<sub>1</sub> = E1/mc<sup>2</sup>, where
E<sub>1</sub> = E<sub>2</sub> + ΔE, with E<sub>2</sub> =160 MeV. The integral can be evaluated numerically. (See, for example, <ref>[http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html]</ref>)</p>
<p>Hence, the ratio R of muon stopping rates for the same detector at two different positions
separated by a vertical distance H, and ignoring for the moment any variations in the
shape of the energy spectrum of muons, is just R = exp(− t’/τ ), where τ is the muon
proper lifetime.</p>
<p>When comparing the muon stopping rates for the detector at two different elevations, we
must remember that muons that stop in the lower detector have, at the position of the
upper detector, a larger energy. If, say, the relative muon abundance grows dramatically
with energy, then we would expect a relatively large stopping rate at the lower detector
simply because the starting flux at the position of the upper detector was so large, and not
because of any relativistic effects. Indeed, the muon momentum spectrum does peak, at
around p = 500 MeV/c or so, although the precise shape is not known with high accuracy.
See figure 4.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig4.png|400px|border|center]]
<b>Figure 4 <ref>Greider, P.K.F., "''Cosmic Rays at Earth''", (2001, Elsevier, Amsterdam).</ref>- </b>Muon momentum spectrum at sea level. The curves are fits to various data sets
(shown as geometric shapes).
<br clear=right>
</p>
</td></table>
<p>We therefore need a way to correct for variations in the shape of the muon energy
spectrum in the region from about 160 MeV – 800 MeV. (Corresponding to
momentums’s p = 120 MeV/c – 790 MeV/c.) We do this by first measuring the muon
stopping rate at two different elevations (Δh = 3008 meters between Taos, NM and
Dallas, TX) and then computing the ratio R<sub>raw</sub> of raw stopping rates. (R<sub>raw</sub> = Dallas/Taos
= 0.41 ± 0.05) Next, using the above expression for the transit time between the two
elevations, we compute the transit time in the muon’s rest frame (t’ = 1.32τ) for vertically
travelling muons and calculate the corresponding theoretical stopping rate ratio
R = exp(− t’/τ ) = 0.267. We then compute the double ratio R<sub>0</sub> = R<sub>raw</sub> /R = 1.5 ± 0.2 of the
measured stopping rate ratio to this theoretical rate ratio and interpret this as a correction
factor to account for the increase in muon flux between about E =160 MeV and
E = 600 MeV. This correction is to be used in all subsequent measurements for any pair
of elevations.</p>
<p>To verify that the correction scheme works, we take a new stopping rate measurement at
a different elevation (h = 2133 meters a.s.l. at Los Alamos, NM), and compare a new
stopping rate ratio measurement with our new, corrected theoretical prediction for the
stopping rate ratio R<sub>μ</sub> = R<sub>0</sub> R = 1.6exp(− t’/τ). We find t’ = 1.06τ and R<sub>μ</sub> = 0.52 ± 0.06.
The raw measurements yield R<sub>raw</sub> = 0.56 ± 0.01, showing good agreement.</p>

<p>For your own time dilation experiment, you could first measure the raw muon stopping
rate at an upper and lower elevation. Accounting for energy loss between the two
elevations, you first calculate the transit time t’ in the muon’s rest frame and then a naïve
theoretical lower elevation stopping rate. This naïve rate should then be multiplied by the
muon spectrum correction factor 1.5 ± 0.2 before comparing it to the measured rate at the
lower elevation. Alternatively, you could measure the lower elevation stopping rate,
divide by the correction factor, and then account for energy loss before predicting what
the upper elevation stopping rate should be. You would then compare your prediction
against a measurement.</p>

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an integrated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Disconnect the function generator and using the pulser on the detector, measure the time between successive rising edges
on the oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results vs. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-22T15:17:21Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = λexp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h3>Time Dilation Effect</h3>
<p>A measurement of the muon stopping rate at two different altitudes can be used to
demonstrate the time dilation effect of special relativity. Although the detector
configuration is not optimal for demonstrating time dilation, a useful measurement can
still be preformed without additional scintillators or lead absorbers. Due to the finite size
of the detector, only muons with a typical total energy of about 160 MeV will stop inside
the plastic scintillator. The stopping rate is measured from the total number of observed
muon decays recorded by the instrument in some time interval. This rate in turn is
proportional to the flux of muons with total energy of about 160 MeV and this flux
decreases with diminishing altitude as the muons descend and decay in the atmosphere.
After measuring the muon stopping rate at one altitude, predictions for the stopping rate
at another altitude can be made with and without accounting for the time dilation effect of
special relativity. A second measurement at the new altitude distinguishes between
competing predictions.</p>
<p>A comparison of the muon stopping rate at two different altitudes should account for the
muon’s energy loss as it descends into the atmosphere, variations with energy in the
shape of the muon energy spectrum, and the varying zenith angles of the muons that stop
in the detector. Since the detector stops only low energy muons, the stopped muons
detected by the low altitude detector will, at the elevation of the higher altitude detector,
necessarily have greater energy. This energy difference ΔE(h) will clearly depend on the
pathlength between the two detector positions.</p>
<p>Vertically travelling muons at the position of the higher altitude detector that are
ultimately detected by the lower detector have an energy larger than those stopped and
detected by the upper detector by an amount equal to DE(h). If the shape of the muon
energy spectrum changes significantly with energy, then the relative muon stopping rates
at the two different altitudes will reflect this difference in spectrum shape at the two
different energies. (This is easy to see if you suppose muons do not decay at all.) This
variation in the spectrum shape can be corrected for by calibrating the detector in a
manner described below.</p>
<p>Like all charged particles, a muon loses energy through coulombic interactions with the
matter it traverses. The average energy loss rate in matter for singly charged particles
traveling close to the speed of light is approximately 2 MeV/g/cm<sup>2</sup>, where we measure
the thickness s of the matter in units of g/cm<sup>2</sup>. Here, ''s'' = ρx, where ρ is the mass density
of the material through which the particle is passing, measured in g/cm<sup>3</sup>, and the x is the
particle’s pathlength, measured in cm. (This way of measuring material thickness in
units of g/cm<sup>2</sup> allows us to compare effective thicknesses of two materials that might
have very different mass densities.) A more accurate value for energy loss can be
determined from the Bethe-Bloch equation.
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn7.png|300px|center]]
</p>
</td></tr></table>
Here N is the number of electrons in the stopping medium per cm<sup>3</sup>, ''e'' is the electronic
charge, ''z'' is the atomic number of the projectile, ''Z'' and ''A'' are the atomic number and
weight, respectively, of the stopping medium. The velocity of the projectile is ''β'' in units
of the speed ,c, of light and its corresponding Lorentz factor is ''γ''. The symbol ''I'' denotes the
mean excitation energy of the stopping medium atoms. Approximately, ''I''=''AZ'', where
''A''≈ 13 eV. More accurate values for ''I'', as well as corrections to the Bethe-Bloch equation,
can be found here<ref>Leo, W. R., "''Techniques for Nuclear and Particle Physics Experiments''", (1994,
Springer-Verlag, New York).</ref>.</p>

<p>A simple estimate of the energy lost ΔE by a muon as it travels a vertical distance H is
ΔE = 2 MeV/g/cm<sup>2</sup> * H * ρ<sub>air</sub>, where ρ<sub>air</sub> is the density of air, possibly averaged over
H using the density of air according to the “standard atmosphere.” Here the atmosphere
is assumed isothermal and the air pressure p at some height h above sea level is
parameterized by p = p<sub>0</sub> exp(-h/h<sub>0</sub>), where p<sub>0</sub> = 1030 g/cm<sup>2</sup> is the total thickness of the
atmosphere and h<sub>0</sub> = 8.4 km. The units of pressure may seem unusual to you but they are
completely acceptable. From hydrostatics, you will recall that the pressure P at the base
of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will
then recognize the units of the right hand side as g/cm<sup>2</sup>. The air density r, in familiar
units of g/cm<sup>3</sup>, is given by ρ = −dp/dh.</p>

<p>If the transit time for a particle to travel vertically from some height H down to sea level,
all measured in the lab frame, is denoted by t, then the corresponding time in the
particle’s rest frame is t’ and given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn8.png|200px|center]]
</p>
</td></tr></table>
Here β and γ have their usual relativistic meanings for the projectile and are measured in
the lab frame. Since relativistic muons lose energy at essentially a constant rate when
travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC<sub>0</sub> dh,
with C<sub>0</sub> = 2 MeV/(g/cm2). Also, from the Einstein relation, E = γmc<sup>2</sup>, dE = mc<sup>2</sup> dγ, so
dh = (mc<sup>2</sup>/ρC<sub>0</sub>) dγ. Hence,
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn9.png|220px|center]]
</p>
</td></tr></table>
Here γ<sub>1</sub> is the muon’s gamma factor at height H and γ<sub>2</sub> is its gamma factor just before it
enters the scintillator. We can take γ<sub>2</sub> = 1.5 since we want muons that stop in the scintillator and assume that on average stopped muons travel halfway into the scintillator,
corresponding to a distance s = 10 g/cm<sup>2</sup>. The entrance muon momentum is then taken
from range-momentum graphs at the Particle Data Group WWW site and the
corresponding γ<sub>2</sub> computed. The lower limit of integration is given by γ<sub>1</sub> = E1/mc<sup>2</sup>, where
E<sub>1</sub> = E<sub>2</sub> + ΔE, with E<sub>2</sub> =160 MeV. The integral can be evaluated numerically. (See, for example, <ref>[http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html]</ref>)</p>
<p>Hence, the ratio R of muon stopping rates for the same detector at two different positions
separated by a vertical distance H, and ignoring for the moment any variations in the
shape of the energy spectrum of muons, is just R = exp(− t’/τ ), where τ is the muon
proper lifetime.</p>
<p>When comparing the muon stopping rates for the detector at two different elevations, we
must remember that muons that stop in the lower detector have, at the position of the
upper detector, a larger energy. If, say, the relative muon abundance grows dramatically
with energy, then we would expect a relatively large stopping rate at the lower detector
simply because the starting flux at the position of the upper detector was so large, and not
because of any relativistic effects. Indeed, the muon momentum spectrum does peak, at
around p = 500 MeV/c or so, although the precise shape is not known with high accuracy.
See figure 4.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig4.png|400px|border|center]]
<b>Figure 4 <ref>Greider, P.K.F., "''Cosmic Rays at Earth''", (2001, Elsevier, Amsterdam).</ref>- </b>Muon momentum spectrum at sea level. The curves are fits to various data sets
(shown as geometric shapes).
<br clear=right>
</p>
</td></table>
<p>We therefore need a way to correct for variations in the shape of the muon energy
spectrum in the region from about 160 MeV – 800 MeV. (Corresponding to
momentums’s p = 120 MeV/c – 790 MeV/c.) We do this by first measuring the muon
stopping rate at two different elevations (Δh = 3008 meters between Taos, NM and
Dallas, TX) and then computing the ratio R<sub>raw</sub> of raw stopping rates. (R<sub>raw</sub> = Dallas/Taos
= 0.41 ± 0.05) Next, using the above expression for the transit time between the two
elevations, we compute the transit time in the muon’s rest frame (t’ = 1.32τ) for vertically
travelling muons and calculate the corresponding theoretical stopping rate ratio
R = exp(− t’/τ ) = 0.267. We then compute the double ratio R<sub>0</sub> = R<sub>raw</sub> /R = 1.5 ± 0.2 of the
measured stopping rate ratio to this theoretical rate ratio and interpret this as a correction
factor to account for the increase in muon flux between about E =160 MeV and
E = 600 MeV. This correction is to be used in all subsequent measurements for any pair
of elevations.</p>
<p>To verify that the correction scheme works, we take a new stopping rate measurement at
a different elevation (h = 2133 meters a.s.l. at Los Alamos, NM), and compare a new
stopping rate ratio measurement with our new, corrected theoretical prediction for the
stopping rate ratio R<sub>μ</sub> = R<sub>0</sub> R = 1.6exp(− t’/τ). We find t’ = 1.06τ and R<sub>μ</sub> = 0.52 ± 0.06.
The raw measurements yield R<sub>raw</sub> = 0.56 ± 0.01, showing good agreement.</p>

<p>For your own time dilation experiment, you could first measure the raw muon stopping
rate at an upper and lower elevation. Accounting for energy loss between the two
elevations, you first calculate the transit time t’ in the muon’s rest frame and then a naïve
theoretical lower elevation stopping rate. This naïve rate should then be multiplied by the
muon spectrum correction factor 1.5 ± 0.2 before comparing it to the measured rate at the
lower elevation. Alternatively, you could measure the lower elevation stopping rate,
divide by the correction factor, and then account for energy loss before predicting what
the upper elevation stopping rate should be. You would then compare your prediction
against a measurement.</p>

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an interrogated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box
input. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Disconnect the function generator and using the pulser on the detector, measure the time between successive rising edges
on the oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results vs. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-02T16:16:53Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = l exp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h3>Time Dilation Effect</h3>
<p>A measurement of the muon stopping rate at two different altitudes can be used to
demonstrate the time dilation effect of special relativity. Although the detector
configuration is not optimal for demonstrating time dilation, a useful measurement can
still be preformed without additional scintillators or lead absorbers. Due to the finite size
of the detector, only muons with a typical total energy of about 160 MeV will stop inside
the plastic scintillator. The stopping rate is measured from the total number of observed
muon decays recorded by the instrument in some time interval. This rate in turn is
proportional to the flux of muons with total energy of about 160 MeV and this flux
decreases with diminishing altitude as the muons descend and decay in the atmosphere.
After measuring the muon stopping rate at one altitude, predictions for the stopping rate
at another altitude can be made with and without accounting for the time dilation effect of
special relativity. A second measurement at the new altitude distinguishes between
competing predictions.</p>
<p>A comparison of the muon stopping rate at two different altitudes should account for the
muon’s energy loss as it descends into the atmosphere, variations with energy in the
shape of the muon energy spectrum, and the varying zenith angles of the muons that stop
in the detector. Since the detector stops only low energy muons, the stopped muons
detected by the low altitude detector will, at the elevation of the higher altitude detector,
necessarily have greater energy. This energy difference ΔE(h) will clearly depend on the
pathlength between the two detector positions.</p>
<p>Vertically travelling muons at the position of the higher altitude detector that are
ultimately detected by the lower detector have an energy larger than those stopped and
detected by the upper detector by an amount equal to DE(h). If the shape of the muon
energy spectrum changes significantly with energy, then the relative muon stopping rates
at the two different altitudes will reflect this difference in spectrum shape at the two
different energies. (This is easy to see if you suppose muons do not decay at all.) This
variation in the spectrum shape can be corrected for by calibrating the detector in a
manner described below.</p>
<p>Like all charged particles, a muon loses energy through coulombic interactions with the
matter it traverses. The average energy loss rate in matter for singly charged particles
traveling close to the speed of light is approximately 2 MeV/g/cm<sup>2</sup>, where we measure
the thickness s of the matter in units of g/cm<sup>2</sup>. Here, ''s'' = ρx, where ρ is the mass density
of the material through which the particle is passing, measured in g/cm<sup>3</sup>, and the x is the
particle’s pathlength, measured in cm. (This way of measuring material thickness in
units of g/cm<sup>2</sup> allows us to compare effective thicknesses of two materials that might
have very different mass densities.) A more accurate value for energy loss can be
determined from the Bethe-Bloch equation.
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn7.png|300px|center]]
</p>
</td></tr></table>
Here N is the number of electrons in the stopping medium per cm<sup>3</sup>, ''e'' is the electronic
charge, ''z'' is the atomic number of the projectile, ''Z'' and ''A'' are the atomic number and
weight, respectively, of the stopping medium. The velocity of the projectile is ''β'' in units
of the speed ,c, of light and its corresponding Lorentz factor is ''γ''. The symbol ''I'' denotes the
mean excitation energy of the stopping medium atoms. Approximately, ''I''=''AZ'', where
''A''≈ 13 eV. More accurate values for ''I'', as well as corrections to the Bethe-Bloch equation,
can be found here<ref>Leo, W. R., "''Techniques for Nuclear and Particle Physics Experiments''", (1994,
Springer-Verlag, New York).</ref>.</p>

<p>A simple estimate of the energy lost ΔE by a muon as it travels a vertical distance H is
ΔE = 2 MeV/g/cm<sup>2</sup> * H * ρ<sub>air</sub>, where ρ<sub>air</sub> is the density of air, possibly averaged over
H using the density of air according to the “standard atmosphere.” Here the atmosphere
is assumed isothermal and the air pressure p at some height h above sea level is
parameterized by p = p<sub>0</sub> exp(-h/h<sub>0</sub>), where p<sub>0</sub> = 1030 g/cm<sup>2</sup> is the total thickness of the
atmosphere and h<sub>0</sub> = 8.4 km. The units of pressure may seem unusual to you but they are
completely acceptable. From hydrostatics, you will recall that the pressure P at the base
of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will
then recognize the units of the right hand side as g/cm<sup>2</sup>. The air density r, in familiar
units of g/cm<sup>3</sup>, is given by ρ = −dp/dh.</p>

<p>If the transit time for a particle to travel vertically from some height H down to sea level,
all measured in the lab frame, is denoted by t, then the corresponding time in the
particle’s rest frame is t’ and given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn8.png|200px|center]]
</p>
</td></tr></table>
Here β and γ have their usual relativistic meanings for the projectile and are measured in
the lab frame. Since relativistic muons lose energy at essentially a constant rate when
travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC<sub>0</sub> dh,
with C<sub>0</sub> = 2 MeV/(g/cm2). Also, from the Einstein relation, E = γmc<sup>2</sup>, dE = mc<sup>2</sup> dγ, so
dh = (mc<sup>2</sup>/ρC<sub>0</sub>) dγ. Hence,
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn9.png|220px|center]]
</p>
</td></tr></table>
Here γ<sub>1</sub> is the muon’s gamma factor at height H and γ<sub>2</sub> is its gamma factor just before it
enters the scintillator. We can take γ<sub>2</sub> = 1.5 since we want muons that stop in the scintillator and assume that on average stopped muons travel halfway into the scintillator,
corresponding to a distance s = 10 g/cm<sup>2</sup>. The entrance muon momentum is then taken
from range-momentum graphs at the Particle Data Group WWW site and the
corresponding γ<sub>2</sub> computed. The lower limit of integration is given by γ<sub>1</sub> = E1/mc<sup>2</sup>, where
E<sub>1</sub> = E<sub>2</sub> + ΔE, with E<sub>2</sub> =160 MeV. The integral can be evaluated numerically. (See, for example, <ref>[http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html]</ref>)</p>
<p>Hence, the ratio R of muon stopping rates for the same detector at two different positions
separated by a vertical distance H, and ignoring for the moment any variations in the
shape of the energy spectrum of muons, is just R = exp(− t’/τ ), where τ is the muon
proper lifetime.</p>
<p>When comparing the muon stopping rates for the detector at two different elevations, we
must remember that muons that stop in the lower detector have, at the position of the
upper detector, a larger energy. If, say, the relative muon abundance grows dramatically
with energy, then we would expect a relatively large stopping rate at the lower detector
simply because the starting flux at the position of the upper detector was so large, and not
because of any relativistic effects. Indeed, the muon momentum spectrum does peak, at
around p = 500 MeV/c or so, although the precise shape is not known with high accuracy.
See figure 4.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig4.png|400px|border|center]]
<b>Figure 4 <ref>Greider, P.K.F., "''Cosmic Rays at Earth''", (2001, Elsevier, Amsterdam).</ref>- </b>Muon momentum spectrum at sea level. The curves are fits to various data sets
(shown as geometric shapes).
<br clear=right>
</p>
</td></table>
<p>We therefore need a way to correct for variations in the shape of the muon energy
spectrum in the region from about 160 MeV – 800 MeV. (Corresponding to
momentums’s p = 120 MeV/c – 790 MeV/c.) We do this by first measuring the muon
stopping rate at two different elevations (Δh = 3008 meters between Taos, NM and
Dallas, TX) and then computing the ratio R<sub>raw</sub> of raw stopping rates. (R<sub>raw</sub> = Dallas/Taos
= 0.41 ± 0.05) Next, using the above expression for the transit time between the two
elevations, we compute the transit time in the muon’s rest frame (t’ = 1.32τ) for vertically
travelling muons and calculate the corresponding theoretical stopping rate ratio
R = exp(− t’/τ ) = 0.267. We then compute the double ratio R<sub>0</sub> = R<sub>raw</sub> /R = 1.5 ± 0.2 of the
measured stopping rate ratio to this theoretical rate ratio and interpret this as a correction
factor to account for the increase in muon flux between about E =160 MeV and
E = 600 MeV. This correction is to be used in all subsequent measurements for any pair
of elevations.</p>
<p>To verify that the correction scheme works, we take a new stopping rate measurement at
a different elevation (h = 2133 meters a.s.l. at Los Alamos, NM), and compare a new
stopping rate ratio measurement with our new, corrected theoretical prediction for the
stopping rate ratio R<sub>μ</sub> = R<sub>0</sub> R = 1.6exp(− t’/τ). We find t’ = 1.06τ and R<sub>μ</sub> = 0.52 ± 0.06.
The raw measurements yield R<sub>raw</sub> = 0.56 ± 0.01, showing good agreement.</p>

<p>For your own time dilation experiment, you could first measure the raw muon stopping
rate at an upper and lower elevation. Accounting for energy loss between the two
elevations, you first calculate the transit time t’ in the muon’s rest frame and then a naïve
theoretical lower elevation stopping rate. This naïve rate should then be multiplied by the
muon spectrum correction factor 1.5 ± 0.2 before comparing it to the measured rate at the
lower elevation. Alternatively, you could measure the lower elevation stopping rate,
divide by the correction factor, and then account for energy loss before predicting what
the upper elevation stopping rate should be. You would then compare your prediction
against a measurement.</p>

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an interrogated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>

<p>You will be using an oscilloscope for the following exercises. Note that every connection into the oscilloscope should be terminated using the provided 50Ω terminator.</p>

<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave (using the function generator) to the input of the electronics box
input. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Make sure to terminate this connection with a 50Ω terminator as well. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Disconnect the function generator and using the pulser on the detector, measure the time between successive rising edges
on the oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results vs. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise some patience since the counting rate is “slowish.”) </p>
</li>
<li>What high voltage (HV) should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using the scope.<b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement. This is easily done by connecting the PMT output on the detector to the PMT input on the electronics box. You may disconnect the oscilloscope as it is not needed for this part of the experiment.</p>
<p>Start and observe the decay time spectrum. The longer this experiment runs for, the more accurate your data will be. We suggest that you collect data over night (or over a weekend) for the best results.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-02T15:55:23Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = l exp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h3>Time Dilation Effect</h3>
<p>A measurement of the muon stopping rate at two different altitudes can be used to
demonstrate the time dilation effect of special relativity. Although the detector
configuration is not optimal for demonstrating time dilation, a useful measurement can
still be preformed without additional scintillators or lead absorbers. Due to the finite size
of the detector, only muons with a typical total energy of about 160 MeV will stop inside
the plastic scintillator. The stopping rate is measured from the total number of observed
muon decays recorded by the instrument in some time interval. This rate in turn is
proportional to the flux of muons with total energy of about 160 MeV and this flux
decreases with diminishing altitude as the muons descend and decay in the atmosphere.
After measuring the muon stopping rate at one altitude, predictions for the stopping rate
at another altitude can be made with and without accounting for the time dilation effect of
special relativity. A second measurement at the new altitude distinguishes between
competing predictions.</p>
<p>A comparison of the muon stopping rate at two different altitudes should account for the
muon’s energy loss as it descends into the atmosphere, variations with energy in the
shape of the muon energy spectrum, and the varying zenith angles of the muons that stop
in the detector. Since the detector stops only low energy muons, the stopped muons
detected by the low altitude detector will, at the elevation of the higher altitude detector,
necessarily have greater energy. This energy difference ΔE(h) will clearly depend on the
pathlength between the two detector positions.</p>
<p>Vertically travelling muons at the position of the higher altitude detector that are
ultimately detected by the lower detector have an energy larger than those stopped and
detected by the upper detector by an amount equal to DE(h). If the shape of the muon
energy spectrum changes significantly with energy, then the relative muon stopping rates
at the two different altitudes will reflect this difference in spectrum shape at the two
different energies. (This is easy to see if you suppose muons do not decay at all.) This
variation in the spectrum shape can be corrected for by calibrating the detector in a
manner described below.</p>
<p>Like all charged particles, a muon loses energy through coulombic interactions with the
matter it traverses. The average energy loss rate in matter for singly charged particles
traveling close to the speed of light is approximately 2 MeV/g/cm<sup>2</sup>, where we measure
the thickness s of the matter in units of g/cm<sup>2</sup>. Here, ''s'' = ρx, where ρ is the mass density
of the material through which the particle is passing, measured in g/cm<sup>3</sup>, and the x is the
particle’s pathlength, measured in cm. (This way of measuring material thickness in
units of g/cm<sup>2</sup> allows us to compare effective thicknesses of two materials that might
have very different mass densities.) A more accurate value for energy loss can be
determined from the Bethe-Bloch equation.
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn7.png|300px|center]]
</p>
</td></tr></table>
Here N is the number of electrons in the stopping medium per cm<sup>3</sup>, ''e'' is the electronic
charge, ''z'' is the atomic number of the projectile, ''Z'' and ''A'' are the atomic number and
weight, respectively, of the stopping medium. The velocity of the projectile is ''β'' in units
of the speed ,c, of light and its corresponding Lorentz factor is ''γ''. The symbol ''I'' denotes the
mean excitation energy of the stopping medium atoms. Approximately, ''I''=''AZ'', where
''A''≈ 13 eV. More accurate values for ''I'', as well as corrections to the Bethe-Bloch equation,
can be found here<ref>Leo, W. R., "''Techniques for Nuclear and Particle Physics Experiments''", (1994,
Springer-Verlag, New York).</ref>.</p>

<p>A simple estimate of the energy lost ΔE by a muon as it travels a vertical distance H is
ΔE = 2 MeV/g/cm<sup>2</sup> * H * ρ<sub>air</sub>, where ρ<sub>air</sub> is the density of air, possibly averaged over
H using the density of air according to the “standard atmosphere.” Here the atmosphere
is assumed isothermal and the air pressure p at some height h above sea level is
parameterized by p = p<sub>0</sub> exp(-h/h<sub>0</sub>), where p<sub>0</sub> = 1030 g/cm<sup>2</sup> is the total thickness of the
atmosphere and h<sub>0</sub> = 8.4 km. The units of pressure may seem unusual to you but they are
completely acceptable. From hydrostatics, you will recall that the pressure P at the base
of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will
then recognize the units of the right hand side as g/cm<sup>2</sup>. The air density r, in familiar
units of g/cm<sup>3</sup>, is given by ρ = −dp/dh.</p>

<p>If the transit time for a particle to travel vertically from some height H down to sea level,
all measured in the lab frame, is denoted by t, then the corresponding time in the
particle’s rest frame is t’ and given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn8.png|200px|center]]
</p>
</td></tr></table>
Here β and γ have their usual relativistic meanings for the projectile and are measured in
the lab frame. Since relativistic muons lose energy at essentially a constant rate when
travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC<sub>0</sub> dh,
with C<sub>0</sub> = 2 MeV/(g/cm2). Also, from the Einstein relation, E = γmc<sup>2</sup>, dE = mc<sup>2</sup> dγ, so
dh = (mc<sup>2</sup>/ρC<sub>0</sub>) dγ. Hence,
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn9.png|220px|center]]
</p>
</td></tr></table>
Here γ<sub>1</sub> is the muon’s gamma factor at height H and γ<sub>2</sub> is its gamma factor just before it
enters the scintillator. We can take γ<sub>2</sub> = 1.5 since we want muons that stop in the scintillator and assume that on average stopped muons travel halfway into the scintillator,
corresponding to a distance s = 10 g/cm<sup>2</sup>. The entrance muon momentum is then taken
from range-momentum graphs at the Particle Data Group WWW site and the
corresponding γ<sub>2</sub> computed. The lower limit of integration is given by γ<sub>1</sub> = E1/mc<sup>2</sup>, where
E<sub>1</sub> = E<sub>2</sub> + ΔE, with E<sub>2</sub> =160 MeV. The integral can be evaluated numerically. (See, for example, <ref>[http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html]</ref>)</p>
<p>Hence, the ratio R of muon stopping rates for the same detector at two different positions
separated by a vertical distance H, and ignoring for the moment any variations in the
shape of the energy spectrum of muons, is just R = exp(− t’/τ ), where τ is the muon
proper lifetime.</p>
<p>When comparing the muon stopping rates for the detector at two different elevations, we
must remember that muons that stop in the lower detector have, at the position of the
upper detector, a larger energy. If, say, the relative muon abundance grows dramatically
with energy, then we would expect a relatively large stopping rate at the lower detector
simply because the starting flux at the position of the upper detector was so large, and not
because of any relativistic effects. Indeed, the muon momentum spectrum does peak, at
around p = 500 MeV/c or so, although the precise shape is not known with high accuracy.
See figure 4.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig4.png|400px|border|center]]
<b>Figure 4 <ref>Greider, P.K.F., "''Cosmic Rays at Earth''", (2001, Elsevier, Amsterdam).</ref>- </b>Muon momentum spectrum at sea level. The curves are fits to various data sets
(shown as geometric shapes).
<br clear=right>
</p>
</td></table>
<p>We therefore need a way to correct for variations in the shape of the muon energy
spectrum in the region from about 160 MeV – 800 MeV. (Corresponding to
momentums’s p = 120 MeV/c – 790 MeV/c.) We do this by first measuring the muon
stopping rate at two different elevations (Δh = 3008 meters between Taos, NM and
Dallas, TX) and then computing the ratio R<sub>raw</sub> of raw stopping rates. (R<sub>raw</sub> = Dallas/Taos
= 0.41 ± 0.05) Next, using the above expression for the transit time between the two
elevations, we compute the transit time in the muon’s rest frame (t’ = 1.32τ) for vertically
travelling muons and calculate the corresponding theoretical stopping rate ratio
R = exp(− t’/τ ) = 0.267. We then compute the double ratio R<sub>0</sub> = R<sub>raw</sub> /R = 1.5 ± 0.2 of the
measured stopping rate ratio to this theoretical rate ratio and interpret this as a correction
factor to account for the increase in muon flux between about E =160 MeV and
E = 600 MeV. This correction is to be used in all subsequent measurements for any pair
of elevations.</p>
<p>To verify that the correction scheme works, we take a new stopping rate measurement at
a different elevation (h = 2133 meters a.s.l. at Los Alamos, NM), and compare a new
stopping rate ratio measurement with our new, corrected theoretical prediction for the
stopping rate ratio R<sub>μ</sub> = R<sub>0</sub> R = 1.6exp(− t’/τ). We find t’ = 1.06τ and R<sub>μ</sub> = 0.52 ± 0.06.
The raw measurements yield R<sub>raw</sub> = 0.56 ± 0.01, showing good agreement.</p>

<p>For your own time dilation experiment, you could first measure the raw muon stopping
rate at an upper and lower elevation. Accounting for energy loss between the two
elevations, you first calculate the transit time t’ in the muon’s rest frame and then a naïve
theoretical lower elevation stopping rate. This naïve rate should then be multiplied by the
muon spectrum correction factor 1.5 ± 0.2 before comparing it to the measured rate at the
lower elevation. Alternatively, you could measure the lower elevation stopping rate,
divide by the correction factor, and then account for energy loss before predicting what
the upper elevation stopping rate should be. You would then compare your prediction
against a measurement.</p>

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. (A FPGA ,or a field programmable gate array, is an interrogated circuit chip that can be programmed by the experiment designer for any specific use. In this experiment the FPGA is used as the microprocessor for the muon lifetime experiment.) A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>
<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave to the input of the electronics box
input. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Using the pulser on the detector, measure the time between successive rising edges
on an oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results v. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise needs
a digital scope and some patience since the counting rate is “slowish.”) </p>
</li>
<li>What HV should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using a scope. (A digital scope works best.) <b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement.</p>
<p>Start and observe the decay time spectrum.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2013-01-02T15:48:03Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''", MATPHYS LLC., [http://www.matphys.com/ www.matphys.com] </ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p> Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm<sup>2</sup> (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = l exp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h3>Time Dilation Effect</h3>
<p>A measurement of the muon stopping rate at two different altitudes can be used to
demonstrate the time dilation effect of special relativity. Although the detector
configuration is not optimal for demonstrating time dilation, a useful measurement can
still be preformed without additional scintillators or lead absorbers. Due to the finite size
of the detector, only muons with a typical total energy of about 160 MeV will stop inside
the plastic scintillator. The stopping rate is measured from the total number of observed
muon decays recorded by the instrument in some time interval. This rate in turn is
proportional to the flux of muons with total energy of about 160 MeV and this flux
decreases with diminishing altitude as the muons descend and decay in the atmosphere.
After measuring the muon stopping rate at one altitude, predictions for the stopping rate
at another altitude can be made with and without accounting for the time dilation effect of
special relativity. A second measurement at the new altitude distinguishes between
competing predictions.</p>
<p>A comparison of the muon stopping rate at two different altitudes should account for the
muon’s energy loss as it descends into the atmosphere, variations with energy in the
shape of the muon energy spectrum, and the varying zenith angles of the muons that stop
in the detector. Since the detector stops only low energy muons, the stopped muons
detected by the low altitude detector will, at the elevation of the higher altitude detector,
necessarily have greater energy. This energy difference ΔE(h) will clearly depend on the
pathlength between the two detector positions.</p>
<p>Vertically travelling muons at the position of the higher altitude detector that are
ultimately detected by the lower detector have an energy larger than those stopped and
detected by the upper detector by an amount equal to DE(h). If the shape of the muon
energy spectrum changes significantly with energy, then the relative muon stopping rates
at the two different altitudes will reflect this difference in spectrum shape at the two
different energies. (This is easy to see if you suppose muons do not decay at all.) This
variation in the spectrum shape can be corrected for by calibrating the detector in a
manner described below.</p>
<p>Like all charged particles, a muon loses energy through coulombic interactions with the
matter it traverses. The average energy loss rate in matter for singly charged particles
traveling close to the speed of light is approximately 2 MeV/g/cm<sup>2</sup>, where we measure
the thickness s of the matter in units of g/cm<sup>2</sup>. Here, ''s'' = ρx, where ρ is the mass density
of the material through which the particle is passing, measured in g/cm<sup>3</sup>, and the x is the
particle’s pathlength, measured in cm. (This way of measuring material thickness in
units of g/cm<sup>2</sup> allows us to compare effective thicknesses of two materials that might
have very different mass densities.) A more accurate value for energy loss can be
determined from the Bethe-Bloch equation.
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn7.png|300px|center]]
</p>
</td></tr></table>
Here N is the number of electrons in the stopping medium per cm<sup>3</sup>, ''e'' is the electronic
charge, ''z'' is the atomic number of the projectile, ''Z'' and ''A'' are the atomic number and
weight, respectively, of the stopping medium. The velocity of the projectile is ''β'' in units
of the speed ,c, of light and its corresponding Lorentz factor is ''γ''. The symbol ''I'' denotes the
mean excitation energy of the stopping medium atoms. Approximately, ''I''=''AZ'', where
''A''≈ 13 eV. More accurate values for ''I'', as well as corrections to the Bethe-Bloch equation,
can be found here<ref>Leo, W. R., "''Techniques for Nuclear and Particle Physics Experiments''", (1994,
Springer-Verlag, New York).</ref>.</p>

<p>A simple estimate of the energy lost ΔE by a muon as it travels a vertical distance H is
ΔE = 2 MeV/g/cm<sup>2</sup> * H * ρ<sub>air</sub>, where ρ<sub>air</sub> is the density of air, possibly averaged over
H using the density of air according to the “standard atmosphere.” Here the atmosphere
is assumed isothermal and the air pressure p at some height h above sea level is
parameterized by p = p<sub>0</sub> exp(-h/h<sub>0</sub>), where p<sub>0</sub> = 1030 g/cm<sup>2</sup> is the total thickness of the
atmosphere and h<sub>0</sub> = 8.4 km. The units of pressure may seem unusual to you but they are
completely acceptable. From hydrostatics, you will recall that the pressure P at the base
of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will
then recognize the units of the right hand side as g/cm<sup>2</sup>. The air density r, in familiar
units of g/cm<sup>3</sup>, is given by ρ = −dp/dh.</p>

<p>If the transit time for a particle to travel vertically from some height H down to sea level,
all measured in the lab frame, is denoted by t, then the corresponding time in the
particle’s rest frame is t’ and given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn8.png|200px|center]]
</p>
</td></tr></table>
Here β and γ have their usual relativistic meanings for the projectile and are measured in
the lab frame. Since relativistic muons lose energy at essentially a constant rate when
travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC<sub>0</sub> dh,
with C<sub>0</sub> = 2 MeV/(g/cm2). Also, from the Einstein relation, E = γmc<sup>2</sup>, dE = mc<sup>2</sup> dγ, so
dh = (mc<sup>2</sup>/ρC<sub>0</sub>) dγ. Hence,
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn9.png|220px|center]]
</p>
</td></tr></table>
Here γ<sub>1</sub> is the muon’s gamma factor at height H and γ<sub>2</sub> is its gamma factor just before it
enters the scintillator. We can take γ<sub>2</sub> = 1.5 since we want muons that stop in the scintillator and assume that on average stopped muons travel halfway into the scintillator,
corresponding to a distance s = 10 g/cm<sup>2</sup>. The entrance muon momentum is then taken
from range-momentum graphs at the Particle Data Group WWW site and the
corresponding γ<sub>2</sub> computed. The lower limit of integration is given by γ<sub>1</sub> = E1/mc<sup>2</sup>, where
E<sub>1</sub> = E<sub>2</sub> + ΔE, with E<sub>2</sub> =160 MeV. The integral can be evaluated numerically. (See, for example, <ref>[http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html]</ref>)</p>
<p>Hence, the ratio R of muon stopping rates for the same detector at two different positions
separated by a vertical distance H, and ignoring for the moment any variations in the
shape of the energy spectrum of muons, is just R = exp(− t’/τ ), where τ is the muon
proper lifetime.</p>
<p>When comparing the muon stopping rates for the detector at two different elevations, we
must remember that muons that stop in the lower detector have, at the position of the
upper detector, a larger energy. If, say, the relative muon abundance grows dramatically
with energy, then we would expect a relatively large stopping rate at the lower detector
simply because the starting flux at the position of the upper detector was so large, and not
because of any relativistic effects. Indeed, the muon momentum spectrum does peak, at
around p = 500 MeV/c or so, although the precise shape is not known with high accuracy.
See figure 4.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig4.png|400px|border|center]]
<b>Figure 4 <ref>Greider, P.K.F., "''Cosmic Rays at Earth''", (2001, Elsevier, Amsterdam).</ref>- </b>Muon momentum spectrum at sea level. The curves are fits to various data sets
(shown as geometric shapes).
<br clear=right>
</p>
</td></table>
<p>We therefore need a way to correct for variations in the shape of the muon energy
spectrum in the region from about 160 MeV – 800 MeV. (Corresponding to
momentums’s p = 120 MeV/c – 790 MeV/c.) We do this by first measuring the muon
stopping rate at two different elevations (Δh = 3008 meters between Taos, NM and
Dallas, TX) and then computing the ratio R<sub>raw</sub> of raw stopping rates. (R<sub>raw</sub> = Dallas/Taos
= 0.41 ± 0.05) Next, using the above expression for the transit time between the two
elevations, we compute the transit time in the muon’s rest frame (t’ = 1.32τ) for vertically
travelling muons and calculate the corresponding theoretical stopping rate ratio
R = exp(− t’/τ ) = 0.267. We then compute the double ratio R<sub>0</sub> = R<sub>raw</sub> /R = 1.5 ± 0.2 of the
measured stopping rate ratio to this theoretical rate ratio and interpret this as a correction
factor to account for the increase in muon flux between about E =160 MeV and
E = 600 MeV. This correction is to be used in all subsequent measurements for any pair
of elevations.</p>
<p>To verify that the correction scheme works, we take a new stopping rate measurement at
a different elevation (h = 2133 meters a.s.l. at Los Alamos, NM), and compare a new
stopping rate ratio measurement with our new, corrected theoretical prediction for the
stopping rate ratio R<sub>μ</sub> = R<sub>0</sub> R = 1.6exp(− t’/τ). We find t’ = 1.06τ and R<sub>μ</sub> = 0.52 ± 0.06.
The raw measurements yield R<sub>raw</sub> = 0.56 ± 0.01, showing good agreement.</p>

<p>For your own time dilation experiment, you could first measure the raw muon stopping
rate at an upper and lower elevation. Accounting for energy loss between the two
elevations, you first calculate the transit time t’ in the muon’s rest frame and then a naïve
theoretical lower elevation stopping rate. This naïve rate should then be multiplied by the
muon spectrum correction factor 1.5 ± 0.2 before comparing it to the measured rate at the
lower elevation. Alternatively, you could measure the lower elevation stopping rate,
divide by the correction factor, and then account for energy loss before predicting what
the upper elevation stopping rate should be. You would then compare your prediction
against a measurement.</p>

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, "''Data Reduction and Error Analysis for the
Physical Sciences, 2ed.''", (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>
<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave to the input of the electronics box
input. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. Over what frequency range does the amplifier operate?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Using the pulser on the detector, measure the time between successive rising edges
on an oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results v. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise needs
a digital scope and some patience since the counting rate is “slowish.”) </p>
</li>
<li>What HV should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using a scope. (A digital scope works best.) <b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement.</p>
<p>Start and observe the decay time spectrum.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>

Main Page/PHYS 4210/Muon Lifetime

2012-11-16T15:27:21Z

Jlyons:

<h1>Muon Lifetime</h1>

<h1>Introduction <ref> Most of the information contained herein was taken directly from the manual supplied with the apparatus, Coan, T.E. and Ye, J. "''Muon Physics''"</ref></h1>
<p>The muon is one of nature’s fundamental “building blocks of matter” and acts in many
ways as if it were an unstable heavy electron, for reasons no one fully understands.
Discovered in 1937 by C.W. Anderson and S.H. Neddermeyer when they exposed a
cloud chamber to cosmic rays, its finite lifetime was first demonstrated in 1941 by F.
Rasetti. The instrument described in this manual permits you to measure the charge
averaged mean muon lifetime in plastic scintillator, to measure the relative flux of muons
as a function of height above sea-level and to demonstrate the time dilation effect of
special relativity. The instrument also provides a source of genuinely random numbers
that can be used for experimental tests of standard probability distributions.</p>

<h3>Our Muon Source</h3>
<p>The top of earth's atmosphere is bombarded by a flux of high energy charged particles
produced in other parts of the universe by mechanisms that are not yet fully understood.
The composition of these "primary cosmic rays" is somewhat energy dependent but a
useful approximation is that 98% of these particles are protons or heavier nuclei and 2%
are electrons. Of the protons and nuclei, about 87% are protons, 12% helium nuclei and
the balance are still heavier nuclei that are the end products of stellar nucleosynthesis.
<ref>Simpson, J.A., "<i>Elemental and Isotopic Composition of the Galactic Cosmic Rays</i>",
in [http://www.annualreviews.org/doi/abs/10.1146/annurev.ns.33.120183.001543 Rev. Nucl. Part. Sci., <b>33</b>, pp. 323.]</ref></p>

<p>The primary cosmic rays collide with the nuclei of air molecules and produce a shower of
particles that include protons, neutrons, pions (both charged and neutral), kaons, photons,
electrons and positrons. These secondary particles then undergo electromagnetic and
nuclear interactions to produce yet additional particles in a cascade process. Figure 1
indicates the general idea. Of particular interest is the fate of the charged pions produced
in the cascade. Some of these will interact via the strong force with air molecule nuclei
but others will spontaneously decay (indicated by the arrow) via the weak force into a
muon plus a neutrino or antineutrino:

<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn1.png|100px|center]]
</p>
</td></tr></table>

The muon does not interact with matter via the strong force but only through the weak
and electromagnetic forces. It travels a relatively long instance while losing its kinetic
energy and decays by the weak force into an electron plus a neutrino and antineutrino.
We will detect the decays of some of the muons produced in the cascade. (Our detection
efficiency for the neutrinos and antineutrinos is utterly negligible.)</p>

<table width=500 align=center><td>
<p align=justify>[[File:MD_fig1.png|300px|border|center]]
<b>Figure 1- </b>Cosmic ray cascade induced by a cosmic ray proton striking an air molecule
nucleus.
<br clear=right>
</p>
</td></table>

<p>Not all of the particles produced in the cascade in the upper atmosphere survive down to
sea-level due to their interaction with atmospheric nuclei and their own spontaneous
decay. The flux of sea-level muons is approximately 1 per minute per cm2 (see
<ref name="PDG">[http://pdg.lbl.gov Particle Data Group]</ref> for more precise numbers) with a mean kinetic energy of about
4 GeV.</p>

<p>Careful study <ref name="PDG"/> shows that the mean production height in the atmosphere of
the muons detected at sea-level is approximately 15 km. Travelling at the speed of light,
the transit time from production point to sea-level is then 50 μsec. Since the lifetime of
at-rest muons is more than a factor of 20 smaller, the appearance of an appreciable sealevel
muon flux is qualitative evidence for the time dilation effect of special relativity.</p>

<h3>Muon Decay Time Distribution</h3>
<p>The decay times for muons are easily described mathematically. Suppose at some time t
we have N(t) muons. If the probability that a muon decays in some small time interval dt
is λdt, where λ is a constant “decay rate” that characterizes how rapidly a muon decays,
then the change dN in our population of muons is just dN = −N(t)λ dt, or dN/N(t) = −λdt.
Integrating, we have N(t) = N<sub>0</sub>exp(−λ t), where N(t) is the number of surviving muons at
some time t and N<sub>0</sub> is the number of muons at t = 0. The "lifetime" τ of a muon is the
reciprocal of λ, τ = 1/λ. This simple exponential relation is typical of radioactive decay.</p>

<p>Now, we do not have a single clump of muons whose surviving number we can easily
measure. Instead, we detect muon decays from muons that enter our detector at
essentially random times, typically one at a time. It is still the case that their decay time
distribution has a simple exponential form of the type described above. By decay time
distribution D(t), we mean that the time-dependent probability that a muon decays in the
time interval between t and t + dt is given by D(t)dt. If we had started with N<sub>0</sub> muons,
then the fraction −dN/N<sub>0</sub> that would on average decay in the time interval between t and
t + dt is just given by differentiating the above relation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn2.png|200px|center]]
</p>
</td></tr></table>
The left-hand side of the last equation is nothing more than the decay probability we
seek, so D(t) = l exp(−λ t). This is true regardless of the starting value of N<sub>0</sub>. That is, the
distribution of decay times, for new muons entering our detector, is also exponential with
the very same exponent used to describe the surviving population of muons. Again, what
we call the muon lifetime is τ = 1/λ.</p>
<p>Because the muon decay time is exponentially distributed, it does not matter that the
muons whose decays we detect are not born in the detector but somewhere above us in
the atmosphere. An exponential function always “looks the same” in the sense that
whether you examine it at early times or late times, its e-folding time is the same.</p>

<h3>Detector Physics</h3>
<p>The active volume of the detector is a plastic scintillator in the shape of a right circular
cylinder of 15 cm diameter and 12.5 cm height placed at the bottom of the black anodized
aluminum alloy tube. Plastic scintillator is transparent organic material made by mixing
together one or more fluors with a solid plastic solvent that has an aromatic ring structure.
A charged particle passing through the scintillator will lose some of its kinetic energy by
ionization and atomic excitation of the solvent molecules. Some of this deposited energy
is then transferred to the fluor molecules whose electrons are then promoted to excited
states. Upon radiative de-excitation, light in the blue and near-UV portion of the
electromagnetic spectrum is emitted with a typical decay time of a few nanoseconds. A
typical photon yield for a plastic scintillator is 1 optical photon emitted per 100 eV of
deposited energy. The properties of the polyvinyltoluene-based scintillator used in the
muon lifetime instrument are summarized in table 1.</p>
<p>To measure the muon's lifetime, we are interested in only those muons that enter, slow,
<i>stop</i> and then <i>decay</i> inside the plastic scintillator. Figure 2 summarizes this process. Such
muons have a total energy of only about 160 MeV as they enter the tube. As a muon
slows to a stop, the excited scintillator emits light that is detected by a photomultiplier
tube (PMT), eventually producing a logic signal that triggers a timing clock. (See the
electronics section below for more detail.) A stopped muon, after a bit, decays into an
electron, a neutrino and an anti-neutrino. (See the next section for an important
qualification of this statement.) Since the electron mass is so much smaller that the muon
mass, m<sub>μ</sub>/m<sub>e</sub> ~ 210, the electron tends to be very energetic and to produce scintillator
light essentially all along its pathlength. The neutrino and anti-neutrino also share some
of the muon's total energy but they entirely escape detection. This second burst of
scintillator light is also seen by the PMT and used to trigger the timing clock. The
distribution of time intervals between successive clock triggers for a set of muon decays
is the physically interesting quantity used to measure the muon lifetime.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig2.png|400px|border|center]]
<b>Figure 2- </b>Schematic showing the generation of the two light pulses (short arrows) used in
determining the muon lifetime. One light pulse is from the slowing muon (dotted line)
and the other is from its decay into an electron or positron (wavey line).
<br clear=right>
</p>
</td></table>
<table width=200 align=center><td>
<p align=justify>[[File:MD_tab1.png|400px|border|center]]
<b>Table 1- </b>General Scintillator Properties.
<br clear=right>
</p>
</td></table>

<h3>Interaction of μ<sup>−</sup>’s with matter</h3>
<p>The muons whose lifetime we measure necessarily interact with matter. Negative muons
that stop in the scintillator can bind to the scintillator's carbon and hydrogen nuclei in
much the same way as electrons do. Since the muon is not an electron, the Pauli
exclusion principle does not prevent it from occupying an atomic orbital already filled
with electrons. Such bound negative muons can then interact with protons
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn3.png|120px|center]]
</p>
</td></tr></table>
before they spontaneously decay. Since there are now two ways for a negative muon to
disappear, the effective lifetime of negative muons in matter is somewhat less than the
lifetime of positively charged muons, which do not have this second interaction
mechanism. Experimental evidence for this effect is shown in figure 3 where
“disintegration” curves for positive and negative muons in aluminum are shown <ref name="Rossi">Rossi, B.,<i>High-Energy Particles</i>, (1952, Prentice-Hall, Inc., New York).</ref>. The abscissa is the time interval t between the arrival of a muon in the
aluminum target and its decay. The ordinate, plotted logarithmically, is the number of
muons greater than the corresponding abscissa. These curves have the same meaning as
curves representing the survival population of radioactive substances. The slope of the
curve is a measure of the effective lifetime of the decaying substance. The muon lifetime
we measure with this instrument is an average over both charge species so the mean
lifetime of the detected muons will be somewhat less than the free space value
τ<sub>μ</sub> = 2.19703 ± 0.00004 μsec.</p>

<p>The probability for nuclear absorption of a stopped negative muon by one of the
scintillator nuclei is proportional to Z<sup>4</sup>, where Z is the atomic number of the nucleus
<ref name="Rossi"/>. A stopped muon captured in an atomic orbital will make transitions down
to the K-shell on a time scale short compared to its time for spontaneous decay
<ref>Wheeler, J.A.,"<i>Some Consequences of the Electromagnetic Interaction between μ<sup>-</sup>-Mesons and Nuclei</i> [http://rmp.aps.org/abstract/RMP/v21/i1/p133_1 Rev. Mod. Phys. <b>21</b>, 133 (1949)] </ref> . Its Bohr radius is roughly 200 times smaller than that for an electron due to its
much larger mass, increasing its probability for being found in the nucleus. From our
knowledge of hydrogenic wavefunctions, the probability density for the bound muon to
be found inside the nucleus is proportional to Z<sup>3</sup>. Once inside the nucleus, a muon’s
probability for encountering a proton is proportional to the number of protons there and
so scales like Z. The net effect is for the overall absorption probability to scale like Z<sup>4</sup>.
Again, this effect is relevant only for negatively charged muons.</p>

<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 3 <ref name="Rossi"/>- </b>Disintegration curves for positive and negative muons in aluminum. The
ordinates at t = 0 can be used to determine the relative numbers of negative and positive
muons that have undergone spontaneous decay. The slopes can be used to determine the
decay time of each charge species.
<br clear=right>
</p>
</td></table>

<h3>μ<sup>+</sup>/μ<sup>−</sup> Charge Ratio at Ground Level</h3>

<p>Our measurement of the muon lifetime in plastic scintillator is an average over both
negatively and positively charged muons. We have already seen that μ−’s have a lifetime
somewhat smaller than positively charged muons because of weak interactions between
negative muons and protons in the scintillator nuclei. This interaction probability is
proportional to Z<sup>4</sup>, where Z is the atomic number of the nuclei, so the lifetime of negative
muons in scintillator and carbon should be very nearly equal. This latter lifetime τ<sub>c</sub> is
measured to be τ<sub>c</sub> = 2.043 ± 0.003 μsec.<ref>Reiter, R.A. et al.,"<i>Precise Measurements of the Mean Lives of μ<sup>+</sup> and μ<sup>-</sup> Mesons in Carbon</i>" [http://prl.aps.org/abstract/PRL/v5/i1/p22_1 Phys. Rev. Lett. <b>5</b>, 22 (1960)]</ref>. </p>

<p>It is easy to determine the expected average lifetime τ<sub>obs</sub> of positive and negative
muons in plastic scintillator. Let λ<sup>−</sup> be the decay rate per negative muon in plastic
scintillator and let λ<sup>+</sup> be the corresponding quantity for positively charged muons. If we
then let N<sup>-</sup> and N<sup>+</sup> represent the number of negative and positive muons incident on the
scintillator per unit time, respectively, the average observed decay rate <λ> and its
corresponding lifetime τ<sub>obs</sub> are given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn4.png|300px|center]]
</p>
</td></tr></table>
where ρ ≡ N<sup>+</sup>/N<sup>−</sup>, τ<sup>−</sup>≡(λ<sup>−</sup>)<sup>−1</sup> is the lifetime of negative muons in scintillator and τ<sup>+</sup>≡(λ<sup>+</sup>)<sup>−1</sup> is the corresponding quantity for positive muons.</p>
<p>Due to the Z<sup>4</sup> effect, τ<sup>−</sup>= τ<sub>c</sub> for plastic scintillator, and we can set τ<sup>+</sup> equal to the free
space lifetime value τ<sub>μ</sub> since positive muons are not captured by the scintillator nuclei.
Setting ρ=1 allows us to estimate the average muon lifetime we expect to observe in the
scintillator.</p>
<p>We can measure ρ for the momentum range of muons that stop in the scintillator by
rearranging the above equation:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn5.png|200px|center]]
</p>
</td></tr></table>
</p>

<h3>Backgrounds</h3>
<p>The detector responds to any particle that produces enough scintillation light to trigger its
readout electronics. These particles can be either charged, like electrons or muons, or
neutral, like photons, that produce charged particles when they interact inside the
scintillator. Now, the detector has no knowledge of whether a penetrating particle stops
or not inside the scintillator and so has no way of distinguishing between light produced
by muons that stop and decay inside the detector, from light produced by a pair of
through-going muons that occur one right after the other. This important source of
background events can be dealt with in two ways. First, we can restrict the time interval
during which we look for the two successive flashes of scintillator light characteristic of
muon decay events. Secondly, we can estimate the background level by looking at large
times in the decay time histogram where we expect few events from genuine muon
decay.</p>

<h3>Fermi Coupling Constant G<sub>F</sub></h3>
<p>Muons decay via the weak force and the Fermi coupling constant G<sub>F</sub> is a measure of the
strength of the weak force. To a good approximation, the relationship between the muon
lifetime τ and G<sub>F</sub> is particularly simple:
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn6.png|100px|center]]
</p>
</td></tr></table>
</p>
where m is the mass of the muon and the other symbols have their standard meanings.
Measuring t with this instrument and then taking m from, say, the Particle Data Group<ref>[http://pdg.lbl.gov Particle Data Group]</ref> produces a value for G<sub>F</sub>.

<h3>Time Dilation Effect</h3>
<p>A measurement of the muon stopping rate at two different altitudes can be used to
demonstrate the time dilation effect of special relativity. Although the detector
configuration is not optimal for demonstrating time dilation, a useful measurement can
still be preformed without additional scintillators or lead absorbers. Due to the finite size
of the detector, only muons with a typical total energy of about 160 MeV will stop inside
the plastic scintillator. The stopping rate is measured from the total number of observed
muon decays recorded by the instrument in some time interval. This rate in turn is
proportional to the flux of muons with total energy of about 160 MeV and this flux
decreases with diminishing altitude as the muons descend and decay in the atmosphere.
After measuring the muon stopping rate at one altitude, predictions for the stopping rate
at another altitude can be made with and without accounting for the time dilation effect of
special relativity. A second measurement at the new altitude distinguishes between
competing predictions.</p>
<p>A comparison of the muon stopping rate at two different altitudes should account for the
muon’s energy loss as it descends into the atmosphere, variations with energy in the
shape of the muon energy spectrum, and the varying zenith angles of the muons that stop
in the detector. Since the detector stops only low energy muons, the stopped muons
detected by the low altitude detector will, at the elevation of the higher altitude detector,
necessarily have greater energy. This energy difference ΔE(h) will clearly depend on the
pathlength between the two detector positions.</p>
<p>Vertically travelling muons at the position of the higher altitude detector that are
ultimately detected by the lower detector have an energy larger than those stopped and
detected by the upper detector by an amount equal to DE(h). If the shape of the muon
energy spectrum changes significantly with energy, then the relative muon stopping rates
at the two different altitudes will reflect this difference in spectrum shape at the two
different energies. (This is easy to see if you suppose muons do not decay at all.) This
variation in the spectrum shape can be corrected for by calibrating the detector in a
manner described below.</p>
<p>Like all charged particles, a muon loses energy through coulombic interactions with the
matter it traverses. The average energy loss rate in matter for singly charged particles
traveling close to the speed of light is approximately 2 MeV/g/cm<sup>2</sup>, where we measure
the thickness s of the matter in units of g/cm<sup>2</sup>. Here, ''s'' = ρx, where ρ is the mass density
of the material through which the particle is passing, measured in g/cm<sup>3</sup>, and the x is the
particle’s pathlength, measured in cm. (This way of measuring material thickness in
units of g/cm<sup>2</sup> allows us to compare effective thicknesses of two materials that might
have very different mass densities.) A more accurate value for energy loss can be
determined from the Bethe-Bloch equation.
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn7.png|300px|center]]
</p>
</td></tr></table>
Here N is the number of electrons in the stopping medium per cm<sup>3</sup>, ''e'' is the electronic
charge, ''z'' is the atomic number of the projectile, ''Z'' and ''A'' are the atomic number and
weight, respectively, of the stopping medium. The velocity of the projectile is ''β'' in units
of the speed ,c, of light and its corresponding Lorentz factor is ''γ''. The symbol ''I'' denotes the
mean excitation energy of the stopping medium atoms. Approximately, ''I''=''AZ'', where
''A''≈ 13 eV. More accurate values for ''I'', as well as corrections to the Bethe-Bloch equation,
can be found here<ref>Leo, W. R., "''Techniques for Nuclear and Particle Physics Experiments''", (1994,
Springer-Verlag, New York).</ref>.</p>

<p>A simple estimate of the energy lost ΔE by a muon as it travels a vertical distance H is
ΔE = 2 MeV/g/cm<sup>2</sup> * H * ρ<sub>air</sub>, where ρ<sub>air</sub> is the density of air, possibly averaged over
H using the density of air according to the “standard atmosphere.” Here the atmosphere
is assumed isothermal and the air pressure p at some height h above sea level is
parameterized by p = p<sub>0</sub> exp(-h/h<sub>0</sub>), where p<sub>0</sub> = 1030 g/cm<sup>2</sup> is the total thickness of the
atmosphere and h<sub>0</sub> = 8.4 km. The units of pressure may seem unusual to you but they are
completely acceptable. From hydrostatics, you will recall that the pressure P at the base
of a stationary fluid is P = ρgh. Dividing both sides by g yields P/g = ρh, and you will
then recognize the units of the right hand side as g/cm<sup>2</sup>. The air density r, in familiar
units of g/cm<sup>3</sup>, is given by ρ = −dp/dh.</p>

<p>If the transit time for a particle to travel vertically from some height H down to sea level,
all measured in the lab frame, is denoted by t, then the corresponding time in the
particle’s rest frame is t’ and given by
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn8.png|200px|center]]
</p>
</td></tr></table>
Here β and γ have their usual relativistic meanings for the projectile and are measured in
the lab frame. Since relativistic muons lose energy at essentially a constant rate when
travelling through a medium of mass density ρ, dE/ds = C0, so we have dE = ρC<sub>0</sub> dh,
with C<sub>0</sub> = 2 MeV/(g/cm2). Also, from the Einstein relation, E = γmc<sup>2</sup>, dE = mc<sup>2</sup> dγ, so
dh = (mc<sup>2</sup>/ρC<sub>0</sub>) dγ. Hence,
<table width=200 align=center>
<tr><td>
<p align=justify>[[File:MD_eqn9.png|220px|center]]
</p>
</td></tr></table>
Here γ<sub>1</sub> is the muon’s gamma factor at height H and γ<sub>2</sub> is its gamma factor just before it
enters the scintillator. We can take γ<sub>2</sub> = 1.5 since we want muons that stop in the scintillator and assume that on average stopped muons travel halfway into the scintillator,
corresponding to a distance s = 10 g/cm<sup>2</sup>. The entrance muon momentum is then taken
from range-momentum graphs at the Particle Data Group WWW site and the
corresponding γ<sub>2</sub> computed. The lower limit of integration is given by γ<sub>1</sub> = E1/mc<sup>2</sup>, where
E<sub>1</sub> = E<sub>2</sub> + ΔE, with E<sub>2</sub> =160 MeV. The integral can be evaluated numerically. (See, for example, <ref>[http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/integral/integral.html]</ref>)</p>
<p>Hence, the ratio R of muon stopping rates for the same detector at two different positions
separated by a vertical distance H, and ignoring for the moment any variations in the
shape of the energy spectrum of muons, is just R = exp(− t’/τ ), where τ is the muon
proper lifetime.</p>
<p>When comparing the muon stopping rates for the detector at two different elevations, we
must remember that muons that stop in the lower detector have, at the position of the
upper detector, a larger energy. If, say, the relative muon abundance grows dramatically
with energy, then we would expect a relatively large stopping rate at the lower detector
simply because the starting flux at the position of the upper detector was so large, and not
because of any relativistic effects. Indeed, the muon momentum spectrum does peak, at
around p = 500 MeV/c or so, although the precise shape is not known with high accuracy.
See figure 4.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig3.png|300px|border|center]]
<b>Figure 4 <ref>Greider, P.K.F., Cosmic Rays at Earth, (2001, Elsevier, Amsterdam).</ref>- </b>Muon momentum spectrum at sea level. The curves are fits to various data sets
(shown as geometric shapes).
<br clear=right>
</p>
</td></table>
<p>We therefore need a way to correct for variations in the shape of the muon energy
spectrum in the region from about 160 MeV – 800 MeV. (Corresponding to
momentums’s p = 120 MeV/c – 790 MeV/c.) We do this by first measuring the muon
stopping rate at two different elevations (Δh = 3008 meters between Taos, NM and
Dallas, TX) and then computing the ratio R<sub>raw</sub> of raw stopping rates. (R<sub>raw</sub> = Dallas/Taos
= 0.41 ± 0.05) Next, using the above expression for the transit time between the two
elevations, we compute the transit time in the muon’s rest frame (t’ = 1.32τ) for vertically
travelling muons and calculate the corresponding theoretical stopping rate ratio
R = exp(− t’/τ ) = 0.267. We then compute the double ratio R0 = R<sub>raw</sub> /R = 1.5 ± 0.2 of the
measured stopping rate ratio to this theoretical rate ratio and interpret this as a correction
factor to account for the increase in muon flux between about E =160 MeV and
E = 600 MeV. This correction is to be used in all subsequent measurements for any pair
of elevations.</p>
<p>To verify that the correction scheme works, we take a new stopping rate measurement at
a different elevation (h = 2133 meters a.s.l. at Los Alamos, NM), and compare a new
stopping rate ratio measurement with our new, corrected theoretical prediction for the
stopping rate ratio R<sub>μ</sub> = R<sub>0</sub> R = 1.6exp(− t’/τ). We find t’ = 1.06τ and R<sub>μ</sub> = 0.52 ± 0.06.
The raw measurements yield R<sub>raw</sub> = 0.56 ± 0.01, showing good agreement.</p>

<p>For your own time dilation experiment, you could first measure the raw muon stopping
rate at an upper and lower elevation. Accounting for energy loss between the two
elevations, you first calculate the transit time t’ in the muon’s rest frame and then a naïve
theoretical lower elevation stopping rate. This naïve rate should then be multiplied by the
muon spectrum correction factor 1.5 ± 0.2 before comparing it to the measured rate at the
lower elevation. Alternatively, you could measure the lower elevation stopping rate,
divide by the correction factor, and then account for energy loss before predicting what
the upper elevation stopping rate should be. You would then compare your prediction
against a measurement.</p>

<h1>Electronics</h1>
<p>A block diagram of the readout electronics is shown in figure 5. The logic of the signal
processing is simple. Scintillation light is detected by a photomultiplier tube (PMT)
whose output signal feeds a two-stage amplifier. The amplifier output then feeds a
voltage comparator (“discriminator”) with adjustable threshold. This discriminator
produces a TTL output pulse for input signals above threshold and this TTL output pulse
triggers the timing circuit of the FPGA. A second TTL output pulse arriving at the FPGA
input within a fixed time interval will then stop and reset the timing circuit. (The reset
takes about 1 msec during which the detector is disabled.) The time interval between the
start and stop timing pulses is the data sent to the PC via the communications module that
is used to determine the muon lifetime. If a second TTL pulse does not arrive within the
fixed time interval, the timing circuit is reset automatically for the next measurement.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig5.png|500px|border|center]]
<b>Figure 5- </b>Block diagram of the readout electronics. The amplifier and discriminator
outputs are available on the front panel of the electronics box. The HV supply is inside
the detector tube.
<br clear=right>
</p>
</td></table>
<p>The front panel of the electronics box is shown in figure 6. The amplifier output is
accessible via the BNC connector labeled Amplifier output. Similarly, the comparator
output is accessible via the connector labeled Discriminator output. The voltage level
against which the amplifier output is compared to determine whether the comparator triggers can be adjusted using the “Threshold control” knob. The threshold voltage is monitored by using the red and black connectors that accept standard multimeter probe
leads. The toggle switch controls a beeper that sounds when an amplifier signal is above
the discriminator threshold. The beeper can be turned off.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig6.png|500px|border|center]]
<b>Figure 6- </b>Front of the electronics box.
<br clear=right>
</p>
</td></table>
<p>The back panel of the electronics box is shown is figure 7. An extra fuse is stored inside
the power switch.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig7.png|500px|border|center]]
<b>Figure 7- </b>Rear of electronics box. The communications ports are on the left. Use only
one.
<br clear=right>
</p>
</td></table>
<p>Figure 8 shows the top of the detector cylinder. DC power to the electronics inside the
detector tube is supplied from the electronics box through the connector ''DC Power''. The
high voltage (HV) to the PMT can be adjusted by turning the potentiometer located at the
top of the detector tube. The HV level can be measured by using the pair of red and black
connectors that accept standard multimeter probes. The HV monitor output is 1/100 times
the HV applied to the PMT.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig8.png|500px|border|center]]
<b>Figure 8- </b>Top view of the detector lid. The HV adjustment potentiometer and monitoring
ports for the PMT are located here.
<br clear=right>
</p>
</td></table>
<p>A pulser inside the detector tube can drive a light emitting diode (LED) imbedded in the
scintillator. It is turned on by the toggle switch at the tube top. The pulser produces pulse
pairs at a fixed repetition rate of 100 Hz while the time between the two pulses
comprising a pair is adjusted by the knob labeled ''Time Adj''. The pulser output voltage is
accessible at the connector labeled ''Pulse Output''.</p>
<p>For reference, Figure 9 shows the output directly from the PMT into a 50Ω load. Figure
10 shows the corresponding amplifier and discriminator output pulses.</p>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig9.png|500px|border|center]]
<b>Figure 9- </b>Output pulse directly from PMT into a 50Ω load. Horizontal scale is 20 ns/div
and vertical scale is 100 mV/div.
<br clear=right>
</p>
</td></table>
<table width=500 align=center><td>
<p align=justify>[[File:MD_fig10.png|500px|border|center]]
<b>Figure 10- </b>Amplifier output pulse from the input signal from figure 9 and the resulting
discriminator output pulse. Horizontal scale is 20 ns/div and the vertical scale is 100
mV/div (amplifier output) and 200 mV/div (discriminator output).
<br clear=right>
</p>
</td></table>

<h1>Software and User Interface</h1>
<p>Software is used to both help control the instrument and to record and process the raw
data. There is also software to simulate muon decay data. All software is contained on the
CD that accompanies the instrument and can also be freely downloaded from
www.muon.edu. (Both Microsoft and Linux operating systems are supported.) Source
code for the user interface and the data fitting software is written in the Tcl/Tk scripting
language and is provided.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig11.png|400px|border|center]]
<b>Figure 11- </b>User Interface.
<br clear=right>
</p>
</td></table>
<p>There are 5 sections to the main display panel:
<ul>
<li>Control</li>
<li>Muon Decay Time Histogram</li>
<li>Monitor</li>
<li>Rate Meter</li>
<li>Muons through detector</li>
</ul>
</p>

<h3>Control</h3>
<p>The ''Configure'' sub-menu is shown in Figure 12. This menu allows you to specify which
communications port (''com1'', ''com2'', ''com3'' or ''com4'') that you will connect to the
electronics box. Select either ''com1'' or ''com2'' if you will use a serial port for
communication. Typically, you will have only a single serial port on your PC so in this
case you would select ''com1''. (The serial port on your PC is the D-shaped connector with
9 pins.) If you select the wrong port, an error message will eventually appear after you try
to start the data acquisition (see below), telling you that the port you selected cannot be
opened.</p>
<p>If you wish to use the USB port, then connect to the USB port on your computer, select
''com2'' and follow the instructions below for starting the program. If your PC cannot find
the USB port, then ''com2'' is not the correct port selection or you lack the USB driver in
the first place. To correct the first situation, examine the folder “/system/hardware
devices/communications” and find out what port other than com1 exists. Choose this port.</p>
<p>If you need to install the USB driver, then the Windows operating system will inform you
of such and ask you where it can find it. In this case just enter data into the pop-up
window pointing to the location of the driver, contained in the USB driver folder on the
included CD. The Windows operating system will then automatically assign a port name
that you can determine by examining the folder
“/system/hardware devices/communications”.</p>
<p>The maximum x-axis value for the histogram of the muon decay times and the number of
data bins is also set here. There are also controls for reading back all ready collected data.</p>
<p>The blue colored ''Save/Exit'' switch is used to finalize all your communication and
histogramming selections.</p>
<table width=400 align=center><td>
<p align=justify>[[File:MD_fig12.png|400px|border|center]]
<b>Figure 12- </b>Configure Sub-Menu.
<br clear=right>
</p>
</td></table>
<p>The ''Start'' button in the user interface initiates a measurement using the settings selected
from the configure menu. After selecting it, you will see the “Rate Meter” and the
“Muons through detector” graphs show activity.</p>
<p>The ''Pause'' button temporarily suspends data acquisition so that the three graphs stop
being updated. Upon selection, the button changes its name to ''Resume''. Data taking
resumes when the button is selected a second time.</p>
<p>The ''Fit'' button when selected will prompt the user for a password. (The instructor can
change the password.) If the correct password is entered, the data displayed in the decay
time histogram is fit and the results displayed in the upper right hand corner of the graph.
Data continues to be collected and displayed. The fit curve drawn through the data points
disappears once a new data point is collected but results of the fit remain.</p>
<p>The ''View Raw Data'' button opens a window that allows you to display the timing data for
a user selected number of events, with the most recent events read in first. Here an event
is any signal above the discriminator threshold so it includes data from both through
going muons as well as signals from muons that stop and decay inside the detector. Each
raw data record contains two fields of information. The first is a time, indicating the year,
month, day, hour, minute and second, reading left to right, in which the data was
recorded. The second field is an integer that encodes two kinds of information. If the
integer is less than 40000, it is the time between two successive flashes, in units of
nanoseconds. If the integer is greater than or equal to than 40000, then the units position
indicates the number of “time outs,” (instances where a second scintillator flash did not
occur within the preset timing window opened by the first flash). See the data file format
below for more information. Typically, viewing raw data is a diagnostic operation and is
not needed for normal data taking.</p>
<p>The ''Quit'' button stops the measurement and asks you whether you want to save the data.
Answering No writes the data to a file that is named after the date and time the
measurement was originally started, i.e., 03-07-13-17-26.data. Answering ''Yes'' appends
the data to the file muon.data. The file muon.data is intended as the main data file.</p>
<h3>Data file format</h3>
<p>Timing information about each signal above threshold is written to disk and is contained
either in the file muon.data or a file named with the date of the measurement session.
Which file depends on how the data is saved at the end of a measurement session.</p>
<p>The first field is an encoded positive integer that is either the number of nanoseconds
between successive signals that triggered the readout electronics, or the number of
“timeouts” in the one-second interval identified by the corresponding data in the second
column. An integer '''less''' than 40000 is the time, measured in nanoseconds, between
successive signals and, background aside, identifies a muon decay. Only data of this type
is entered automatically into the decay time histogram.</p>
<p>An integer '''greater''' than or equal to 40000 corresponds to the situation where the time
between successive signals exceeded the timing circuit’s maximum number of 40000
clock cycles. A non-zero number in the units place indicates the number of times this
‘timeout” situation occurred in the particular second identified by the data in the first
field. For example, the integer 40005 in the first field indicates that the readout circuit
was triggered 5 times in a particular second but that each time the timing circuit reached
its maximum number of clock cycles before the next signal arrived.</p>
<p>The second field is the number of seconds, as measured by the PC, from the beginning of
1 January 1970 (i.e., 00:00:00 1970-01-01 UTC), a date conventional in computer
programming.</p>
<h3>Monitor</h3>
<p>This panel shows rate-related information for the current measurement. The elapsed time
of the current measurement is shown along with the accumulated number of times from
the start of the measurement that the readout electronics was triggered (''Number of
Muons''). The ''Muon Rate'' is the number of times the readout electronics was triggered in
the previous second. The number of pairs of successive signals, where the time interval
between successive signals is less than the maximum number of clock cycles of the
timing circuit, is labeled ''Muon Decays'', even though some of these events may be
background events and not real muon decays. Finally, the number of muon decays per
minute is displayed as ''Decay Rate''.</p>
<h3>Rate Meter</h3>
<p>This continuously updated graph plots the number of signals above discriminator
threshold versus time. It is useful for monitoring the overall trigger rate.</p>
<h3>Muons through Detector</h3>
<p>This graph shows the time history of the number of signals above threshold. Its time scale
is automatically adjusted and is intended to show time scales much longer than the rate
meter. This graph is useful for long term monitoring of the trigger rate. Strictly speaking,
it includes signals from not only through going muons but any source that might produce
a trigger. The horizontal axis is time, indicated down to the second. The scale is sliding
so that the far left-hand side always corresponds to the start of the measurement session.
The bin width is indicated in the upper left-hand portion of the plot.</p>
<h3>Muon Decay Time Histogram</h3>
<p>This plot is probably the most interesting one to look at. It is a histogram of the time
difference between successive triggers and is the plot used to measure the muon lifetime.
The horizontal scale is the time difference between successive triggers in units of
microseconds. Its maximum displayed value is set by the ''Configure'' menu. (All time
differences less than 20 μsec are entered into the histogram but may not actually be
displayed due to menu choices.) You can also set the number of horizontal bins using the
same menu. The vertical scale is the number of times this time difference occurred and is
adjusted automatically as data is accumulated. A button (''Change y scale Linear/Log'')
allows you to plot the data in either a linear-linear or log-linear fashion. The horizontal
error bars for the data points span the width of each timing bin and the vertical error bars
are the square root of the number of entries for each bin.</p>
<p>The upper right hand portion of the plot shows the number of data points in the
histogram. Again, due to menu selections not all points may be displayed. If you have
selected the ''Fit'' button then information about the fit to the data is displayed. The muon
lifetime is returned, assuming muon decay times are exponentially distributed, along with
the chi-squared per degree of freedom ratio, a standard measure of the quality of the fit.
(For more details<ref>Bevington, P.R. and D.K. Robinson, Data Reduction and Error Analysis for the
Physical Sciences, 2ed., (1992, McGraw-Hill, New York).</ref>.)</p>
<p>A ''Screen capture'' button allows you to produce a plot of the display. Select the button
and then open the ''Paint'' utility (in Windows) and execute the ''Paste'' command under the
''Edit'' pull-down menu.</p>

<h3>The Lifetime Fitter</h3>
<p>The included muon lifetime fitter for the decay time histogram assumes that the
distribution of times is the sum of an exponential distribution and a flat distribution. The
exponential distribution is attributed to real muon decays while the flat distribution is
attributed to background events. The philosophy of the fitter is to first estimate the flat
background from the data at large nominal decays times and to then subtract this
estimated background from the original distribution to produce a new distribution that
can then be fit to a pure exponential.</p>
<p>The background estimation is a multi-step process. Starting with the raw distribution of
decay times, we fit the distribution with an exponential to produce a tentative lifetime τ’.
We then fit that part of the raw distribution that have times greater than 5t’ with a
straight line of slope zero. The resulting number is our first estimate of the background.
We next subtract this constant number from all bins of the original histogram to produce
a new distribution of decay times. Again, we fit to produce a tentative lifetime τ’’ and fit
again that part of this new distribution that have times greater than 5τ’’. The tentative
background level is subtracted from the previous distribution to produce a new
distribution and the whole process is repeated again for a total of 3 background
subtraction steps.</p>

<h3>Muon Decay Simulation</h3>
<p>Simulated muon decay data can be generated using the program ''muonsimu'' found in the
''muon_simu'' folder. Its interface and its general functionality are very similar to the
program ''muon'' in the ''muon_data'' folder. The simulation program ''muonsimu'' lets you
select the decay time of the muon and the number of decays to simulate. Simulated data
is stored in exactly the same format as real data.</p>

<h3>Utility Software</h3>
<p>The folder ''muon_util'' contains several useful programs that ease the analysis of decay
data. The executable file sift sifts through a raw decay data file and writes to a file of your choosing only those records that describe possible muon decays. It ignores records that
describe timing data inconsistent with actual muon decay.</p>
<p>The executable file ''merge'' merges two data files of your choosing into a single file of
your choosing. The data records are time ordered according to the date of original
recording so that the older the record the earlier it occurs in the merged file.</p>
<p>The executable file ''ratecalc'' calculates the average trigger rate (per second) and the muon
decay rate (per minute) from a data file of your choosing. The returned errors are
statistical.</p>
<p>The executable ''freewrap'' is the compiler for any Tcl/Tk code that your write or modify. If
you modify a Tcl/Tk script, you need to compile it before running it. On a Windows
machine you do this by opening a DOS window, and going to the ''muon_util'' directory.
You then execute the command freewrap ''your_script.tcl'', where ''your_script.tcl'' is the
name of your Tcl/Tk script. Do not forget the tcl extension!</p>

<h1>Exercises</h1>
<h2>Apparatus</h2>
<ol>
<li>"''Muon Physics''" Scintillator and Control Unit <ref>For a detailed analysis of the performance specifications see: Coan, T.E., Liu, T. and Ye, J. "<i>A compact apparatus for muon lifetime measurement and time dilation
demonstration in the undergraduate laboratory</i>", [http://ajp.aapt.org/resource/1/ajpias/v74/i2 Am. J. Phys. <b>74</b>, 161 (2006)].</ref></li>
<li> Digital oscilloscope </li>
<li> Function generator </li>
<li> 50-Ω terminator </li>
<li> Control computer and software </li>
<li> assorted cables</li>
</ol>

<h2>Testing the Electronics</h2>
<ol>
<li><p>Measure the gain of the 2-stage amplifier using a sine wave.</p>
<p>Apply a 100kHz 100mV peak-to-peak sine wave to the input of the electronics box
input. Measure the amplifier output and take the ratio V<sub>out</sub>/V<sub>in</sub>. Due to attenuation
resistors inside the electronics box inserted between the amplifier output and the front
panel connector, you will need to multiply this ratio by the factor 1050/50 = 21 to
determine the real amplifier gain.</p>
<p>Q: Increase the frequency. How good is the frequency response of the amp?</p>
<p>Q: Estimate the maximum decay rate you could observe with the instrument.</p>
</li>
<li><p>Measure the saturation output voltage of the amp.</p>
<p>Increase the magnitude of the input sine wave and monitor the amplifier output.</p>
<p>Q: Does a saturated amp output change the timing of the FPGA? What are the
implications for the size of the light signals from the scintillator?</p>
</li>
<li><p>Examine the behavior of the discriminator by feeding a sine wave to the box input and
adjusting the discriminator threshold. Monitor the discriminator output and describe its
shape.</p></li>
<li><p>Measure the timing properties of the FPGA:</p>
<ol style="list-style-type:lower-latin">
<li>Using the pulser on the detector, measure the time between successive rising edges
on an oscilloscope. Compare this number with the number from software display.</li>
<li>Measure the linearity of the FPGA:
Alter the time between rising edges and plot scope results v. FPGA results;
Can use time between 1 μs and 20 μs in steps of 2 μs.</li>
<li><p>Determine the timeout interval of the FPGA by gradually increasing the time between
successive rising edges of a double-pulse and determine when the FPGA no longer
records results;</p>
<p>Q: What does this imply about the maximum time between signal pulses?</p>
</li>
<li><p>Decrease the time interval between successive pulses and try to determine/bound the
FPGA internal timing bin width.</p>
<p>Q: What does this imply about the binning of the data?</p>
<p>Q: What does this imply about the minimum decay time you can observe?</p>
</li>
</ol>
</li>
<li><p>Adjust (or misadjust) discriminator threshold.</p>
<p>Increase the discriminator output rate as measured by the scope or some other means.
Observe the raw muon count rate and the spectrum of "decay" times. (This exercise needs
a digital scope and some patience since the counting rate is “slowish.”) </p>
</li>
<li>What HV should you run at? Adjust/misadjust HV and observe amp output. (We know
that good signals need to be at about 200 mV or so before discriminator, so set
discriminator before hand.) With fixed threshold, alter the HV and watch raw muon count
rate and decay spectrum.</li>
<li>Connect the output of the detector can to the input of the electronics box. Look at the
amplifier output using a scope. (A digital scope works best.) <b>Be sure that the scope
input is terminated at 50Ω.</b> What do you see? Now examine the discriminator
output simultaneously. Again, be certain to terminate the scope input at 50Ω. What do
you see?</li>
</ol>

<h2>Muon Lifetime Measurement</h2>
<ol>
<li><p>Set up the instrument for a muon lifetime measurement.</p>
<p>Start and observe the decay time spectrum.</p>
<p>Q: The muons whose decays we observe are born outside the detector and therefore
spend some (unknown) portion of their lifetime outside the detector. So, we never
measure the actual lifetime of any muon. Yet, we claim we are measuring the lifetime of
muons. How can this be?</p>
</li>
<li>Fit the decay time histogram with with your own fitting routine.</li>
<li>From your measurement of the muon lifetime and a value of the muon mass from
some trusted source, calculate the value of Fermi coupling constant G<sub>F</sub>. Compare your
value with that from a trusted source.</li>
<li>Using the approach outlined above, measure the charge ratio ρ of positive to
negative muons at ground level.</li>
<li>Once the muon lifetime is determined, compare the theoretical binomial distribution
with an experimental distribution derived from the random lifetime data of individual
muon decays. For example, let p be the (success) probability of decay within 1 lifetime,
p = 0.63. The probability of failure q = 1 − p. Take a fresh data sample of 2000 good
decay events. For each successive group of 50 events, count how many have a decay time
less than 1 lifetime. (On average this is 31.5.) Histogram the number of "successes." This
gives you 40 experiments to do. The plot of 40 data points should have a mean at 50*0.63
with a variance σ<sup>2</sup> = Npq = 50*0.63*0.37 = 11.6. Are the experimental results consistent
with theory?</li>
</ol>

<h1>References</h1>
<references/>