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Main Page/PHYS 3220/new Excitation Potentials

2026-02-25T15:02:10Z

APouliot:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/neon/franck-hertz-supply-unit-5558801.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-anode voltage (U<sub>3</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>

<li>Hanne, G. F. “What Really Happens in the Franck–Hertz Experiment with Mercury?” American journal of physics 56.8 (1988): 696–700. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_scitation_primary_10_1119_1_15503 </li>
<li>Huebner, J. S. “Comment on the Franck–Hertz Experiment.” American journal of physics 44.3 (1976): 302–303. Web.
https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_10596</li>
<li>Liu, F. H. “Franck–Hertz Experiment with Higher Excitation Level Measurements.” American journal of physics 55.4 (1987): 366–369. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_15174</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 3220/Rutherford I

2026-02-25T14:55:29Z

APouliot:

<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 Rutherford by 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 <ref>H. Frauenfelder and E. Henley, ''Subatomic Physics'', Prentice Hall.</ref>
,<ref>A. Das and T. Ferbel, ''Introduction to Nuclear and Particle Physics'', J. Wiley.</ref>). 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 <ref name = "Melissinos">Melissinos, ''Experiments in Modern Physics'', Academic Press.</ref> 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 <ref>Leybold Rutherford Scattering Apparatus ''https://labdocs.leylab.de/doc/en/EXP/P/P6/P6521_e.pdf?hash=vn4AfD8G''</ref> 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:Ruth-vacuum.png]]
<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><b>When the foil is in the chamber, always set the scattering angle to zero when evacuating the chamber or returning the vacuum to the normal pressure.</b></p>

<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 box when not in use.</p>

<ol>
<li>Ensure foil scattering angle is set to zero degrees.</li>
<li>Ensure the lid is securely in place.</li>
<li>Open B (if it is not open).</li>
<li>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>'''Open A (NOTE:THIS STEP is very important. This is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<p><b>Note: DO NOT LEAVE THE PUMP RUNNING DURING EXPERIMENT - EVACUATE THE CHAMBER AND TURN OFF PUMP FOLLOWING STEPS 1 TO 8 DESCRIBED ABOVE.</b></p>

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

<p><b>When the foil is in the chamber, always set the scattering angle to zero when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Ensure the foil scattering angle is set to zero degrees.</li>
<li>Close A.</li>
<li>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 ref. <ref name = "Melissinos"/> but note our N0 is not the same as ref. <ref name = "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 data to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long (for the small angles you can probably take a few hundreds in a matter of seconds). However, as you reach the large angles, there will be no scattering into the detector given the chamber is under vacuum. Thus, you will have readings of zero counts for the larger angles. Hint: Think about the geometry of the source and detector at the large angles for the background measurements.</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>
<references />

<h1>See Also</h1>
<ol>
<li>Preston and Deitz, ''The Art of Experimental Physics'', Wiley and Sons.</li>
</ol>

Main Page/PHYS 3220/Rutherford I

2026-02-25T14:54:53Z

APouliot:

<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 Rutherford by 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 <ref>H. Frauenfelder and E. Henley, ''Subatomic Physics'', Prentice Hall.</ref>
,<ref>A. Das and T. Ferbel, ''Introduction to Nuclear and Particle Physics'', J. Wiley.</ref>). 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 <ref name = "Melissinos">Melissinos, ''Experiments in Modern Physics'', Academic Press.</ref> 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 <ref>Leybold Rutherford Scattering Apparatus ''https://labdocs.leylab.de/doc/en/EXP/P/P6/P6521_e.pdf?hash=vn4AfD8G''</ref> 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:Ruth-vacuum.png]
<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><b>When the foil is in the chamber, always set the scattering angle to zero when evacuating the chamber or returning the vacuum to the normal pressure.</b></p>

<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 box when not in use.</p>

<ol>
<li>Ensure foil scattering angle is set to zero degrees.</li>
<li>Ensure the lid is securely in place.</li>
<li>Open B (if it is not open).</li>
<li>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>'''Open A (NOTE:THIS STEP is very important. This is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<p><b>Note: DO NOT LEAVE THE PUMP RUNNING DURING EXPERIMENT - EVACUATE THE CHAMBER AND TURN OFF PUMP FOLLOWING STEPS 1 TO 8 DESCRIBED ABOVE.</b></p>

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

<p><b>When the foil is in the chamber, always set the scattering angle to zero when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Ensure the foil scattering angle is set to zero degrees.</li>
<li>Close A.</li>
<li>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 ref. <ref name = "Melissinos"/> but note our N0 is not the same as ref. <ref name = "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 data to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long (for the small angles you can probably take a few hundreds in a matter of seconds). However, as you reach the large angles, there will be no scattering into the detector given the chamber is under vacuum. Thus, you will have readings of zero counts for the larger angles. Hint: Think about the geometry of the source and detector at the large angles for the background measurements.</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>
<references />

<h1>See Also</h1>
<ol>
<li>Preston and Deitz, ''The Art of Experimental Physics'', Wiley and Sons.</li>
</ol>

File:Ruth-vacuum.png

2026-02-25T14:23:11Z

APouliot: schematic of rutherford experiment vacuum system

== Summary ==
schematic of rutherford experiment vacuum system

Main Page/PHYS 3220/Rutherford I

2026-02-23T21:01:59Z

APouliot:

<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 Rutherford by 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 <ref>H. Frauenfelder and E. Henley, ''Subatomic Physics'', Prentice Hall.</ref>
,<ref>A. Das and T. Ferbel, ''Introduction to Nuclear and Particle Physics'', J. Wiley.</ref>). 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 <ref name = "Melissinos">Melissinos, ''Experiments in Modern Physics'', Academic Press.</ref> 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 <ref>Leybold Rutherford Scattering Apparatus ''https://labdocs.leylab.de/doc/en/EXP/P/P6/P6521_e.pdf?hash=vn4AfD8G''</ref> 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><b>When the foil is in the chamber, always set the scattering angle to zero when evacuating the chamber or returning the vacuum to the normal pressure.</b></p>

<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 box when not in use.</p>

<ol>
<li>Ensure foil scattering angle is set to zero degrees.</li>
<li>Ensure the lid is securely in place.</li>
<li>Open B (if it is not open).</li>
<li>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>'''Open A (NOTE:THIS STEP is very important. This is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<p><b>Note: DO NOT LEAVE THE PUMP RUNNING DURING EXPERIMENT - EVACUATE THE CHAMBER AND TURN OFF PUMP FOLLOWING STEPS 1 TO 8 DESCRIBED ABOVE.</b></p>

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

<p><b>When the foil is in the chamber, always set the scattering angle to zero when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Ensure the foil scattering angle is set to zero degrees.</li>
<li>Close A.</li>
<li>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 ref. <ref name = "Melissinos"/> but note our N0 is not the same as ref. <ref name = "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 data to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long (for the small angles you can probably take a few hundreds in a matter of seconds). However, as you reach the large angles, there will be no scattering into the detector given the chamber is under vacuum. Thus, you will have readings of zero counts for the larger angles. Hint: Think about the geometry of the source and detector at the large angles for the background measurements.</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>
<references />

<h1>See Also</h1>
<ol>
<li>Preston and Deitz, ''The Art of Experimental Physics'', Wiley and Sons.</li>
</ol>

Main Page/PHYS 3220/Rutherford I

2026-02-23T20:30:20Z

APouliot:

<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 Rutherford by 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 (ref. 5) 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><b>When the foil is in the chamber, always set the scattering angle to zero when evacuating the chamber or returning the vacuum to the normal pressure.</b></p>

<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 box when not in use.</p>

<ol>
<li>Ensure foil scattering angle is set to zero degrees.</li>
<li>Ensure the lid is securely in place.</li>
<li>Open B (if it is not open).</li>
<li>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>'''Open A (NOTE:THIS STEP is very important. This is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<p><b>Note: DO NOT LEAVE THE PUMP RUNNING DURING EXPERIMENT - EVACUATE THE CHAMBER AND TURN OFF PUMP FOLLOWING STEPS 1 TO 8 DESCRIBED ABOVE.</b></p>

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

<p><b>When the foil is in the chamber, always set the scattering angle to zero when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Ensure the foil scattering angle is set to zero degrees.</li>
<li>Close A.</li>
<li>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 data to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long (for the small angles you can probably take a few hundreds in a matter of seconds). However, as you reach the large angles, there will be no scattering into the detector given the chamber is under vacuum. Thus, you will have readings of zero counts for the larger angles. Hint: Think about the geometry of the source and detector at the large angles for the background measurements.</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>
<li>Leybold Rutherford Scattering Apparatus ''https://labdocs.leylab.de/doc/en/EXP/P/P6/P6521_e.pdf?hash=vn4AfD8G''</li>
</ol>

Main Page/PHYS 3220/Rutherford I

2026-02-23T20:23:06Z

APouliot:

<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 Rutherford by 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 (ref. 5) 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><b>When the foil is in the chamber, always set the scattering angle to zero when evacuating the chamber or returning the vacuum to the normal pressure.</b></p>

<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 box when not in use.</p>

<ol>
<li>Ensure foil scattering angle is set to zero degrees.</li>
<li>Ensure the lid is securely in place.</li>
<li>Open B (if it is not open).</li>
<li>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>'''Open A (NOTE:THIS STEP is very important. This is to avoid back pressure from the pump and oil from entering into the system).'''</li>
</ol>

<p><b>Note: DO NOT LEAVE THE PUMP RUNNING DURING EXPERIMENT - EVACUATE THE CHAMBER AND TURN OFF PUMP FOLLOWING STEPS 1 TO 8 DESCRIBED ABOVE.</b></p>

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

<p><b>When the foil is in the chamber, always set the scattering angle to zero when returning the vacuum to the normal pressure or evacuating the chamber.</b></p>

<ol>
<li>Ensure the foil scattering angle is set to zero degrees.</li>
<li>Close A.</li>
<li>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 data to correct for the above measurements. The small angles (<30<sup>o</sup>) should not take too long (for the small angles you can probably take a few hundreds in a matter of seconds). However, as you reach the large angles, there will be no scattering into the detector given the chamber is under vacuum. Thus, you will have readings of zero counts for the larger angles. Hint: Think about the geometry of the source and detector at the large angles for the background measurements.</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>
<li>Leybold Rutherford Scattering Apparatus ``https://labdocs.leylab.de/doc/en/EXP/P/P6/P6521_e.pdf?hash=vn4AfD8G'' <\li>
</ol>

Main Page/PHYS 3220/Digital Oscilloscope

2026-02-09T16:34:25Z

APouliot:

<h1>Digital Storage Oscilloscope</h1>

<p>In this experiment we use a function generator to produce rectangular pulses which are recorded and analyzed using a modern digital storage oscilloscope capable of performing a fast fourier transform (FFT) on a given signal. Then the behaviour of a simple RC (integrator) circuit fed by a square wave pulse is analyzed both in the steady-state and transient (turn-on) regimes. Finally the square-wave pulse is used to induce damped harmonic motion in an LC circuit.</p>

<h1>Introduction</h1>
<p>The behaviour of short time-varying signals can be investigated easily with a digital storage oscilloscope (DSO) that will allow you to trigger single events and to store them for any length of time. Periodic signals play an important role in many areas of physics. Periodic signals are conveniently analyzed in terms of harmonic (or frequency) content, either by means of a Fourier series or by a Fourier transform [1]. Typically, oscilloscopes display signals in the time domain. The DSO you are using here will allow you to process these signals so they can be displayed in the frequency domain by using an FFT signal processing module included in the DSO. For a finite wavetrain recorded at discrete time intervals two parameters impose practical limitations on acquiring knowledge about the frequency content of a pulse. The sampling rate ''Δt'' limits the maximum frequency that can be recorded (intuitively: a signal that changes sign at every ''t<sub>j</sub>'' = ''j Δt'' has the highest frequency that can be represented on the discrete time axis). The length of the recorded signal, ''T'', limits the frequency resolution, ''Δf'': the lowest frequency that can be recorded corresponds to a wave with a period that equals ''T''.</p>
<p>Thus, the Fourier transform that would be available in an ideal measurement (continuous sampling and infinite length of measurement)</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn1.png|220px|center]]
</p>
</td><td> <b>(1)</b></td></tr></table>
<p>this is replaced by a discrete Fourier transform: </p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn2.png|310px|center]]
</p>
</td><td> <b>(2)</b></td></tr></table>

<p>where the length of the signal in the time domain is given by ''T'' = ''N Δt'' , the maximum frequency (called the Nyquist frequency) is actually one-half of the inverse sampling rate ''f<sub>c</sub>'' = 1/(2 ''Δt'') as the range of frequencies ranges in principle from -''f<sub>c</sub>'' to ''f<sub>c</sub>'' (but the answer is symmetric in f so we are only interested in the range [0, ''f<sub>c</sub>'']). The frequency resolution equals ''Δf'' = 1/(''N Δt''). There are numerous problems associated with the replacement of eq (1) by eq (2), these problems are discussed in detail in [2] (and hinted at in the manual for the Tektronix TBS 1052B-EDU oscilloscope). To minimize some of the errors one usually multiplies the time signal with a window function whose main purpose is to provide a smooth switching of the pulse and thereby to eliminate artificial high-frequency components that would be associated with chopping a periodic signal at ''t'' = 0 and ''t'' = ''T'' and assuming periodicity (wrap-around).</p>
<p>In a digital storage oscilloscope (as in computer programs that analyze recorded data) an FT function is provided by an efficient algorithm (fast FT = FFT) [2]. In a computer interfaced experiment one usually chooses the sampling rate and temporal record length freely. However, with the DSO these parameters are not independent and are actually controlled simultaneously by the DSO time base control. Furthermore, the number of points used in memory (for display on the screen and transfer to a computer) for a single trace is fixed. The FFT algorithm requires this number to be a power of 2 (typical numbers are 1024, 2048, etc. ). </p>
<p>Consider a rectangular pulse, and a square-wave in particular. It can be thought of as a superposition of a sine wave with the same period τ, given by admixtures of sine waves with higher frequencies (or periods that are simple fractions of τ). This is known from Fourier series expansions of this periodic function. Only sine waves with odd multiples of the base frequency contribute (odd-order harmonics), and the coefficients in the linear combination can be calculated [1]. In part 1 of the experiment the Fourier spectrum (which could detect any frequency components, and not just multiples of the base frequency) is taken for rectangular pulses with different duty cycles. The objective is to show that for a pulse with duty cycle (τ/''n'')/τ = 1:''n'' (cf.. Fig. 1) the ''n''<sup>th</sup> harmonic is absent. For a square-wave pulse (1:2) only odd orders appear.</p>

<table width=500 align=center><td>
<p align=justify>[[File:Dso-fig1.png|500px|border|center]]
<b>Figure 1 -</b> A rectangular pulse with a duty cycle of 1:3.
<br clear=right>
</p>
</td></table>

<p>For a square wave pulse the harmonic coefficients can be calculated from the Fourier series expansion to be (we assume a signal that ranges between 0 and 1)</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn3.png|300px|center]]
</p>
</td><td> <b>(3)</b></td></tr></table>

<h1>Procedure</h1>
<h2>Part 1: Harmonic Analysis of Rectangular Pulses</h2>


<h3>The Tektronix TBS 1052B-EDU Oscilloscope</h3>
<p>Connect the 50 Ω output of the function generator (WAVETEK, model 184) to Ch.1 of the DSO. Turn on the power switch of the function generator and familiarize yourself with the effect of varying the frequency settings of the function generator, and learn how to use the digital scope. Links to the user manuals: [[Media:DSOManual.pdf| DSO User Manual]] and [[Media:Wavetek184FGManual.pdf| Wavetek FG User Manual]]. Refer to section 3 of the WAVETEK instruction manual for a description of the function generator operation controls. There are several menus that control the DSO operation. For this part you need to use the menu that controls the first Y channel (CH 1), and possibly the TRIGGER menu. In the TRIGGER menu it is important that the <u>Source</u> of triggering be specified as CH 1 and the <u>Mode</u> of triggering be set on Auto. (Only for the measurement of the transient behaviour of an RC circuit will you need to switch to Single Mode). The 5 buttons to the right of the LCD display are used to toggle through the options indicated on the DSO screen.</p>
<p>The display of channel 1 or 2 can be turned on and off using the <u>CH1</u> and <u>CH2</u> menu buttons. Once the menu for a given channel is activated the 5 buttons to the right of the LCD display can be used to control:
<ol style="list-style-type:lower-latin">
<li> the ''Coupling'' (DC, AC, Ground).</li>
<li>''Bandwidth limit'' (suppress high frequencies if desired).</li>
<li>''Volts/Division'': coarse or fine (this controls whether the knobs carry out the usual large incremental voltage scale steps (1V, 2V, 5V, etc. ) or fine steps that actually simulate an infinitely variable scale control on an analogue scope).</li>
<li>''Probe'' (10X if the Tektronix probe is attached, 1X if a simple coaxial cable is used (this setting alters the display of scale for Volts/Division being measured - it is needed as the supplied probes step down the voltage by a 1:10 ratio, i.e., 1 V applied to the probe results in 0.1 V at the coaxial input to the scope).</li>
<li>''Invert (on/off)''. When “on” this will invert the displayed signal.</li>
</ol></p>

<p>The <u>MEASURE</u> menu can be used to find period and frequency of a periodic signal, in this case a pulse. a) The top button controls whether the lower 4 (of the 5 to the right of the LCD) provide control over the source or the type of measurement. ''Sources'' can be CH1 or CH2; ''Types'' of measurement are: Frequency, Period, Mean (average voltage), Peak-to-peak voltage, Cyc RMS(?), Rise and Fall times for a pulse, as well as positive and negative width (this is particularly useful for a rectangular wave form).</p>

<p>The <u>CURSOR</u> menu will give you control over the position of a set either of two vertical or two horizontal cursors that span the display area. These are controlled by the multipurpose knob and selected using the lower two buttons to the right of the LCD screen. A useful feature of the cursors is that one can read off the cursor positions indicated on the right of the screen as digital numbers, for the horizontal positions the difference indicates a time with likewise a vertical difference indicates signal amplititude its inverse (as frequency).</p>

<p>The <u>MATH</u> menu is invoked by pressing the red button labelled ''M'' to the left of CH 1. In math mode one can add, subtract or multiply CH 1 and CH 2 signals.</p>

<p>The <u>FFT</u> menu is invoked by pressing the yellow button labelled ''FFT'' located just below the math menu button. With it on can carry out a Fourier transform FFT of CH 1 or CH 2 separately; within the FFT menu one can choose a windowing function [<u>Rectangular</u> implies no windowing, <u>Hanning</u> [2] a standard function that compromises between accurate measurement of frequency amplitudes and the accuracy of frequency measurements (the windowing function introduces an artificial width: a single-frequency signal is broadened in frequency content), <u>Flattop</u> provides more accurate amplitudes in the Fourier spectrum. Finally one can zoom in on the FFT spectrum, since the display actually only uses a few hundred pixels in the horizontal (and vertical) directions, whereas an FFT is calculated with over 4000 points. Note that the FFT is calculated for incoming data, i.e., it is not possible on the TBS 1052B-EDU to record a pulse and then carry out the FFT for that pulse. </p>

<p>The vertical scale used for the FFT is calibrated in decibels. The logarithmic decibel scale is explained in 1<sup>st</sup> year physics texts in the context of sound waves. It is a relative scale, i.e., it depends on some reference strength (which in the case of the TBS 1052B-EDU is chosen to be: 0 dB = 1 V RMS amplitude). A drop by 3 dB corresponds approximately to a reduction by a factor of 2 in amplitude.</p>
<p>With the <u>RUN/STOP</u> button (top right on the scope) you can capture the display (helpful if the signal is shaky), the <u>HARDCOPY</u> button provides a screenshot output to a printer connected to the parallel port, while <u>AUTOSET</u> helps to find settings for some acceptable screen display if you have no idea how to set the vertical amplification and the timebase (horizontal) for your particular signal. It is customary to start by pressing autoset, observe the settings chosen by the scope, and subsequently fine tune the settings as appropriate.</p>

<h3>Measurement of the rectangular pulses produced by the function generator outputs</h3>

<p>Begin detailed measurements by setting a (1:''n'') duty cycle at some frequency with the function generator. Carry out the harmonic analysis using the FFT menu functions. Save the data of the time signal and the FFT using a USB stick to include with your report. Provide your observations about the harmonic spectrum.</p>
<p>Measure rise and fall times for the square wave pulse. For this purpose start with a display of a few cycles on the screen. Now move one of the edges (first a rising, later a falling edge) to the center of the display. Turn the (horizontal) timebase switch to display shorter segments of the pulse until the initially vertical line acquires the characteristic shape for a charging capacitor (discharging in the case of a falling edge), i.e., it becomes an exponential function. Note the time scale at which this happens. Use the Measure menu to obtain a measurement of the rise and fall times (it will depend somewhat on the segment displayed on the screen, since it measures between 90 and 10 % of the signal displayed; cf. the manual for the TBS 1052B-EDU). You should spend some time thinking about clocks used in computers, how fast they have to be (PC chips run internally at speeds in the low GHz range these days, while the entire computer (the bus) can be clocked at up to 66 MHz), and how the rise and fall times are important since during those times the logical state (0 or 1) is really undetermined. There can be no ‘perfect’ square-wave or rectangular pulse.</p>

<p>In the following exercise observe how a simple logic gate can be used to shape the pulse. Use the TTL output of the function generator on Ch. 2 and look at the same characteristics as with the 50 Ohm output. Compare the measured rise and fall times of the two outputs. You can overlap the two signals and zoom in to observe the two signals in detail. </p>


<p>To deepen your understanding of the idea of triggering on a signal it is interesting to display the signals from two uncorrelated generators: use the signal from the one of the function generator output (50 Ohm output or TTL output) on one channel and connect the internal 1 kHz generator to the other. (See TBS 1052B-EDU scope diagram – probe compensation terminal lugs located in the DSO manual.) From the TRIGGER menu choose to trigger on either CH1 or CH2. What do you observe? Use the RUN/STOP button repeatedly to capture still images and explain the behaviour.</p>

<h2>Part 2: Charging and Discharging a Capacitor</h2>

<p>Set the TTL output of the function generator to produce a square-wave signal of a given frequency (rectangular pulse with duty cycle 1:2). Connect an RC circuit such that the capacitor C is charged via the variable resistor box R. Measure the voltage as produced by the function generator (use a BNC Tee to split the TTL output) on CH1 and the voltage across C on CH2. Note that CH1 and CH2 have a common ground: think before making the connections. The physics of charging and discharging a capacitor is explained in first-year physics texts. Familiarize yourself with the material, we provide no equations here. You need to realize that during the ‘low’ output the square-wave generator acts as a short, i.e., it discharges the capacitor through the resistor R (the internal resistance of the function generator is often in the 50 Ohms range which is negligible if R is in the kΩ range).</p>
<p>For a fixed choice of R and C (calculate the time constant T) make measurements for three different settings of the square-wave frequency. Adjust the period τ  (to have a 1:2 duty cycle) such that the three cases cover a period τ less than ''T'', comparable to ''T'', and bigger than ''T''. Use DC coupling on both channels, and observe the steady-state behaviour of the sequence of periodically charging and discharging. Provide explanations to compare the three cases. Why is the voltage across the capacitor periodic? In the next section you will measure the transient or turn-on behaviour. Fig. 3 is provided as an illustration of the case where ''τ'' < ''T'' such that the capacitor does not fully charge or discharge during one of the half-periods τ/2. Each segment of the curve is obtained from the solution to the differential equation describing the charge or discharge regimes, and the correct initial condition is being applied (we assume no charges on the capacitor plates at ''t''=0).</p>
<table width=500 align=center><td>
<p align=justify>[[File:Dso-fig3.png|500px|border|center]]
<b>Figure 3 -</b> RC circuit response to a periodic signal.
<br clear=right>
</p>
</td></table>
<p>An interesting quantity to measure in the charging/discharging RC circuit is the voltage across the resistor, which according to Ohm’s law provides a measure of the current in the circuit. Thus, it is possible to observe how the current changes sign and jumps from some low (possibly near-zero) value at the end of a cycle to ±U/R. Mathematically the answer is discontinuous, and this is interesting to investigate in real life. From a measurement point of view the matter can be straightforward: if one uses an external function generator, one can simply connect a probe across R. Note, however, that one cannot also connect the other probe across the capacitor (or to measure the signal coming from the function generator), since CH1 and CH2 have a common ground (and the internal 1 kHz generator uses this ground as well). One trick, however, is to perform the same measurement as before on CH1 and CH2, and to display the difference between the two channels. This is done by using the CH1 + CH2 option on the MATH menu while inverting one of the channels. Perform such a measurement with several cycles shown on the display. There is an apparent discontinuity in the voltage across R (obtained as the difference between the voltage produced by the function generator and the voltage across C). While a discontinuity is located at the center of the screen zoom in using the TIMEBASE adjustment. At what times scale do you resolve the discontinuity, and why? </p>

<h2>Part 3: Transient Behaviour (cf.. Fig. 3)</h2>
<p>Use the same connections as in Part 2, i.e., consider the charging and discharging of a capacitor C through a resistor R. In this part we are interested in the first 10-20 cycles of a square-wave pulse to see how the RC circuit approaches its nearly-periodic behaviour observed in the previous part. For this purpose one has to set the triggering onto the mode single pulses and then adjust the trigger level for a small positive voltage. The RUN/STOP button is used to acquire the signal. The trigger level is set to such a value that when the square-wave generator is disconnected, the scope indicates readiness to record but is not triggered. As the square wave is applied by flipping a switch a single trace is recorded. It is possible to adjust the timing of the single trace with the horizontal adjustment, and by repeating the measurement.</p>
<p>Make observations for settings of the square-wave time constants τ < ''T'', and comment on your results. Why is the other limit τ >'' T'' uninteresting?</p>

<h2>Part 4: Damped Oscillations in an LC circuit</h2>
<p>Here we use square-wave pulses to induce damped oscillations in a circuit that consists of an inductance L and a capacitance C connected in series. Consult a first-year physics text (e.g., [4]) on the physics of storing electrical energy in the form of an electric field in C and in the form of a magnetic field in L, and how they are exchanged in an RLC circuit. Note that a resistance is present even if no resistor is put into the circuit, as there is no ideal coil, i.e., it always has an Ohmic resistance. The differential equation that can be derived from Kirchoff’s law together with the properties of a capacitor, an inductance and a resistor is comparable to that of a damped harmonic oscillator. For the capacitor charge ''q(t)'' we obtain:</p>

<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn4.png|210px|center]]
</p>
</td><td> <b>(4)</b></td></tr></table>

<p>Here L plays the role of inertia (mass), R appears as a friction constant, and the capacitance C plays the role of the spring constant. Understand why the application of a square-wave pulse corresponds to kicking a damped harmonic oscillator: ''q''(0) = 0, ''q’''(0) ≠ 0.</p>
<p>In the case of weak damping (undercritical damping) the solution to eq. (3) has the form:</p>
<table width=400 align=center>
<tr><td>
<p align=justify>[[File:Dso-eqn5.png|280px|center]]
</p>
</td><td> <b>(5)</b></td></tr></table>
<p>Record time signals for an LC circuit with known capacitance and inductance, and compare with the solution to the differential equation. Is the measured frequency consistent with the solution? Estimate the resistance from the decrease in the envelope. You can measure the resistance R with a digital voltmeter.


Note, the commercial function generator has a typical internal resistance of 50 Ω. Perform measurements with a variable resistor box R included in the circuit and adjust R to find critical damping (in addition to the undercritically damped situation described above). Critical damping is obtained when the time constant of the damping ''T'' = 2L/R equals the inverse of the natural (circular) frequency ω.</p>

<h1>References</h1>
<ol>
<li>G.A. Arfken, ''Mathematical Methods for Physicists'', Academic Press.</li>
<li>W.P. Press, S. Teukolsky, Vetterling, Flannery, ''Numerical Recipes'', Chapter 12, Cambridge University Press.</li>
<li>Horowitz Hill, ''The Art of Electronics'', chapter 5.14, Cambridge University Press.</li>
<li>R. Wolfson, J.M. Pasachoff, ''Physics'', 2nd ed., chapter 33-3, Harper Collins, New York 1995.</li>
</ol>

Main Page/PHYS 3220/new Excitation Potentials

2026-01-27T17:48:02Z

APouliot:

<h1>The Franck-Hertz Experiment: Excitation Potentials of Mercury and Neon</h1>

<h1>Introduction</h1>

<p>One of the most direct proofs of the existence of discrete energy states within the atom was first demonstrated in experiments on critical potentials, performed initially by Franck and Hertz in the early 1900's. Studying the way electrons lose energy in collisions with mercury vapour, they laid the basis for the quantum theory of atoms by observing that the electrons give energy to internal motion of mercury atoms in discrete units only.</p>
<p>The collision of a neutral atom with a fast particle (e.g., an electron) may result in the excitation or ionization of the atom. A slow electron in an elastic collision can give very little of its kinetic energy to the translational motion of a mercury atom (without changing the energy state of the atom) - just as a ping-pong ball cannot effectively move a billiard ball. If a moderately slow electron has enough kinetic energy to overcome an atomic excitation threshold (several eV) the collision may be inelastic and much of the energy of the electron can go into exciting a higher state of the atom. The energy in electron volts (eV) necessary to raise an atom from its normal ("ground") state to a given excited state is called the excitation potential for that state. For sufficiently high scattering energy of the impinging electron even ionization may occur.</p>
<p>The energy levels of mercury (Hg) are shown in Fig. 1; it is easy to see that the internal structure is complicated - a consequence of the many electrons in the atom. The diagram gives considerable information you need to know for this experiment. The numbers associated with the lines drawn between the energy levels are wavelengths (in Angstroms Å). In the present experiment we explore only the energy levels 6<sup>3</sup>P on the diagram, the first group of excited states. The electrons do not acquire enough energy to excite many of the other levels.</p>
<p>The Franck-Hertz apparatus consists of an evacuated glass envelope containing a cathode, screen, plate and a small drop of mercury, which can be vaporized by heating. The plate is always kept slightly negative with respect to the grid (that acts as an anode, i.e. accelerates the electrons) and both are set at various positive voltages with respect to the cathode. As the grid potential is raised, the plate current increases accordingly. For accelerating voltages below 5V all collisions with mercury atoms will be elastic (kinetic energy below about 5 eV). Hence, these electrons are energetic enough to overcome the negative plate-grid potential and are collected by the plate. The current flowing in the tube depends upon both the number of charged carriers (electrons) and their velocities (j = nev). Thus a significant change in the particle velocity can affect the size of the current. Once electrons with more than about 5eV energy excite a mercury atom, they slow down and the current flowing in the tube drops. If there is a larger voltage across the tube so that the electron can be re-accelerated to ~ 5 eV after giving it up once in the first collision, then we can see decreases in the current at higher voltages corresponding to a repeated inelastic collision. This process can yield a cyclic rise and fall of the current with the voltage.</p>

<table width=400 align=center><td>
<p align=justify>[[File:Fh-fig1.png|800px|border|center]]
<b>Figure 1 -</b> Energy Levels of Mercury.
<br clear=right>
</p>
</td></table>

<h1>Procedure</h1>

<h2>Required Components</h2>

<p> For this experiment you will be using equipment provided by Leybold®. Go to https://www.leybold-shop.com/physics/physics-equipment/atomic-and-nuclear-physics/franck-hertz-experiments/neon/franck-hertz-supply-unit-5558801.html and click on <b>Related Documents</b>. Read through the instruction sheet for the Franck-Hertz Supply Unit (pay attention to sections 1-4, 5.1, 5.2 and 5.6) and the Experiment Descriptions for Hg (P6.2.4.1) and Ne (P6.2.4.3). These leaflets provide useful information on how to use the equipment and optimize the Franck-Hertz signal. THE EXPERIMENTAL SETUP OF THE TEMPERATURE PROBE IS CRITICAL - THE PROBE MUST BE INSERTED INTO THE BLIND HOLE OF THE COPPER TUBE of the oven. Ensure the temperature sensor is properly connected and IS NOT TOUCHING THE Hg Franck-Hertz tube. </p>

<h3>Observing the Signal</h3>

<p>Electrons liberated from the filament and accelerated to the detector plate which do not collide with an Hg atom will register as a current. This current is amplified by the supply unit and be viewed on the oscilloscope. Note that evidence of collisions with Hg atoms will result in a deficit of current at specific accelerating voltages. This will be observed as dips on the oscilloscope trace. It is the origin and properties of these dips which is the focus of this experiment.</p>

<p>When the temperature is stable you can record the current-voltage characteristic of the Franck-Hertz tube. The current-voltage trace can be observed using the oscilloscope in XY mode. Make sure the signal you observe does not have horizontal clipping (the peaks cut off); see the leaflets for guidance on how to optimize your signal. (What is the meaning of the vertical cut-off?)</p>

<p>Record an I-V curve for an initial temperature of about 180ºC. Set the oscilloscope display mode to XY and the persist to 2 seconds to best visually observe the oscilloscope signal on the screen. </p>

<h3>Saving the Scope Traces</h3>
<p>Use a USB to save your optimized Franck-Hertz signals using the following procedure. (For more information on saving in XY mode refer to the user manual for the oscilloscope.) </p>
<ol>
<li>Connect your USB device to the oscilloscope. </li>
<li>Switch the oscilloscope display mode from XY to YT.</li>
<li>Push the Save/Recall button on the oscilloscope to activate the save menu. </li>
<li>Push the "Print" button to save all files to your USB drive. (The "Print" button is set to "Save All Files". This will save waveforms on Ch.1 and Ch.2 and a picture of the waveforms. </li>
</ol>

<h2>Measurements</h2>

<p>Find as many values as you can of the excitation energy ("excitation potential") for Hg from your record. Repeat these measurements for 5 different temperature values ranging from 140ºC to 195ºC. <b>DO NOT EXCEED a setpoint temperature of 195ºC on the supply unit.</b> Comment on the effect of the Hg pressure in the tube. Perform a full error analysis and compare your results with the expected values.</p>
<p> The neon tube is operated at room temperature. Optimize and record the Ne Franck-Hertz curve and find the excitation energy values for Ne. Compare with the expected values. Can you see the luminous layers in the neon tube? (Hint: Use the MAN operating mode to manually adjust the accelerating voltage and turn off the lights in the room.)

<h1>Questions</h1>
<ol>
<li>Explain the effect of changing the grid-to-plate voltage (V<sub>1</sub>)?</li>
<li>Find out what is meant by "contact potential" in the Franck-Hertz tube and explain how it could be determined. Can you estimate it from your record?</li>
<li>Determine from simple classical mechanics (using a head-on collision with recoil at 180 degrees) what fraction of an electron's kinetic energy can be transferred to a mercury atom in an '''elastic''' collision. Derive an approximate value of the fraction. Repeat for a neon atom. </li>
<li>Why are the other levels not observed? (e.g. 6<sup>3</sup>P<sub>2</sub>, 6<sup>3</sup>P<sub>o</sub>, 6<sup>1</sup>P<sub>1</sub>.)</li>
</ol>

<h1>References</h1>
<ol>
<li>Brehm, J., Mullin W, ''Modern Physics'', p. 168</li>
<li>Halliday, D., Resnick, R., ''Physics I'', pp. 522-24.</li>

<li>Hanne, G. F. “What Really Happens in the Franck–Hertz Experiment with Mercury?” American journal of physics 56.8 (1988): 696–700. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_scitation_primary_10_1119_1_15503 </li>
<li>Huebner, J. S. “Comment on the Franck–Hertz Experiment.” American journal of physics 44.3 (1976): 302–303. Web.
https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_10596</li>
<li>Liu, F. H. “Franck–Hertz Experiment with Higher Excitation Level Measurements.” American journal of physics 55.4 (1987): 366–369. Web. https://ocul-yor.primo.exlibrisgroup.com/permalink/01OCUL_YOR/sqt9v/cdi_crossref_primary_10_1119_1_15174</li>
<li>Preston, D., Dietz, E.,'' The Art of Experimental Physics'', pp. 197ff.</li>
</ol>

Main Page/PHYS 4210 & 4211

2025-12-18T18:20:24Z

APouliot: Update available experiments for 2026

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://eclass.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2026 PHYS 42110/4211 Lab Scheduling procedures will be announced via the eClass course website http://eclass.yorku.ca/ prior to the beginning of term.


<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>












<h2> List of Available Experiments </h2>

<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Bell's Inequalities|Bell's Inequalities*]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Sonoluminescence|Sonoluminescence]]</td><td> 111 PSE</td><td></td></tr>




















<p> </p>

Main Page/PHYS 4210 & 4211

2025-12-18T17:39:14Z

APouliot:

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://eclass.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2026 PHYS 42110/4211 Lab Scheduling procedures will be announced via the eClass course website http://eclass.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>



<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>


<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>


<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Bell's Inequalities|Bell's Inequalities*]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Sonoluminescence|Sonoluminescence]]</td><td> 111 PSE</td><td></td></tr>
























<p> </p>

MediaWiki:Sidebar

2025-12-15T20:19:43Z

APouliot:

Main Page/PHYS 4211

2025-12-15T19:59:18Z

APouliot: Completing merge of this page with 4210 page

#REDIRECT [[Main Page/PHYS 4210 & 4211]]

Talk:Main Page/PHYS 3220

2025-12-15T19:46:56Z

APouliot: Blanked the page

Main Page/PHYS 3220

2025-12-15T19:40:50Z

APouliot: Updated room numbers and Moodle ->eClass

<h1>PHYS 3220 3.0 Experiments in Modern Physics </h1>
<p>A selection of experiments in fluid mechanics,
electromagnetism, optics, and atomic, nuclear, and
particle physics. Analysis of the data and detailed
write-ups are required. One lecture hour which is
devoted to techniques of data analysis and three
laboratory hours per week.</p>

<h1>Laboratory Manual</h1>
<p>'''IMPORTANT NOTE: Winter 2026 PHYS 3220 Lab Scheduling procedures will be announced via the eClass course website http://eclass.yorku.ca/ prior to the beginning of term.'''</p>

<ul>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>
<tr><td>[[Main Page/PHYS 3220/Cavendish|Measurement of the Gravitation Constant G: The Cavendish Experiment]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Speed of Light|A Measurement of the Velocity of Light: The Foucault-Michelson Experiment]] </td><td> 209 PSE </td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Millikan|Determination of the Electric Charge Unit ''e'' : The Millikan Oil Drop Experiment]] </td><td> 111 PSE </td><td></td></tr>

<tr><td> [[Main Page/PHYS 3220/Thermionic|Thermionic Emission]]</td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Semiconductors I|Semiconductors I]] </td><td> 150 PSE </td><td></td></tr>

</table>
</ul>

Main Page/PHYS 4210 & 4211

2025-12-15T19:38:24Z

APouliot: Updated references/ links to moodle -> eClass

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://eclass.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2026 PHYS 42110/4211 Lab Scheduling procedures will be announced via the eClass course website http://eclass.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 150 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>

Main Page/PHYS 4210 & 4211

2025-12-15T19:35:41Z

APouliot: Corrected room numbers and updated functionality of Mass Spec and Zeeman effect labs

<h1>PHYS 4210 & 4211 3.0 </h1>

<p>Selected advanced experiments in physics related to
topics in solid state physics, atomic spectroscopy, microwaves, low-noise measurements, superconductivity, and nuclear and particle physics.

Prerequisites: SC/PHYS 3220 3.00; registration in a Bachelor or Honours Program in physics and astronomy or in biophysics. Co-requisite: SC/PHYS 3040 6.00.</p>

<p> See the course [http://moodle.yorku.ca eClass] for all administrative details. </p>

<h1>Laboratory Manual</h1>

IMPORTANT NOTE: Winter 2021 PHYS 4211 Lab Scheduling procedures will be announced via the Moodle eClass course website http://moodle.yorku.ca/ prior to the beginning of term.



<ul>
<li> [[Main Page/PHYS 4210/How to Choose Experiments|How to Choose Experiments]] </li>
<li> [[Main Page/PHYS 4210/How to Write Reports|How to Write Reports]] </li>
<li> [[Main Page/PHYS 3220/Lab Safety|Lab Safety]] </li>
</ul>

<h2> List of Available Experiments </h2>
<table width=750>
<tr><td width=550>'''Experiment'''</td><td width=100>'''Location'''</td><td width=150> </td></tr>

<tr><td>[[Main Page/PHYS 3220/Radioactive Decays|Radioactive Decays]]</td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Muon Lifetime|Muon Lifetime]]</td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Waveguides|Waveguides*]] </td><td> 111 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Rutherford II|Rutherford Scattering II]]</td><td> 111 PSE </td><td></td></tr>
<tr><td> [[Main Page/PHYS 3220/Rutherford I|Rutherford Scattering I]] </td><td> 111 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Gamma Ray Spectroscopy|Gamma Ray Spectroscopy*]] </td><td> 111 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 3220/new Excitation Potentials|The Franck-Hertz Experiment - Excitation Potentials of Mercury and Neon]]</td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Zeeman Effect|Zeeman Effect*]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Electron Spin Resonance|Electron Spin Resonance]] </td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Coaxial Cable|Coaxial Cable]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Digital Oscilloscope|Digital Storage Oscilloscope]] </td><td> 126 PSE </td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/High-TC Superconductivity|High-TC Superconductivity*]]</td><td> 126 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/Fourier Optics|Fourier Optics*]] </td><td> 126 PSE</td><td></td></tr>

<tr><td>[[Main Page/PHYS 4210/Mass Spectrometer|Mass Spectrometer]]</td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 3220/Viscosity|Viscosity]] </td><td> 150 PSE</td><td></td></tr>
<tr><td>[[Main Page/PHYS 4210/He-Ne Lasers|He-Ne Lasers*]]</td><td> 150 PSE</td><td></td></tr>








<p> Experiments with an asterisk (*) are better suited for PHYS 4211 students. </p>





















<p> </p>