https://physwiki.apps01.yorku.ca//api.php?action=feedcontributions&feedformat=atom&user=TaylorwPhysics Wiki - User contributions [en]2026-05-11T13:30:22ZUser contributionsMediaWiki 1.34.4https://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62056Main Page/PHYS 3220/Radioactive Decays2013-12-17T21:23:12Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Three types of radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p><br />
<br />
<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles and Fiesta plates. Original Coleman mantles used radioactive elements until 1990; the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. Radioactive elements were also used in glazing for bathroom tiles and Fiesta plates (no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the χ<sup>2</sup> obtained, and quote the decay rate, with its standard error. Include histograms of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outline on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis'', University Science Books, 1997.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62055Main Page/PHYS 3220/Radioactive Decays2013-12-17T21:22:01Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p><br />
<br />
<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles and Fiesta plates. Original Coleman mantles used radioactive elements until 1990; the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. Radioactive elements were also used in glazing for bathroom tiles and Fiesta plates (no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the χ<sup>2</sup> obtained, and quote the decay rate, with its standard error. Include histograms of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outline on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis'', University Science Books, 1997.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62054Main Page/PHYS 3220/Radioactive Decays2013-12-17T20:07:30Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p><br />
<br />
<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles and Fiesta plates. Original Coleman mantles used radioactive elements until 1990; the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. Radioactive elements were also used in glazing for bathroom tiles and Fiesta plates (no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the χ<sup>2</sup> obtained, and quote the decay rate, with its standard error. Include histograms of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outline on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis'', University Science Books, 1997.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62053Main Page/PHYS 3220/Radioactive Decays2013-12-17T20:06:03Z<p>Taylorw: Added learning outcomes. Minor edits.</p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p><br />
<br />
<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles and Fiesta plates. Original Coleman mantles used radioactive elements until 1990; the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. Radioactive elements were also used in glazing for bathroom tiles and Fiesta plates (no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the χ<sup>2</sup> obtained, and quote the decay rate, with its standard error. Include histograms of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outline on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62052Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:58:08Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p><br />
<br />
<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program "Particle Tracking.vi" performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (ch. 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (ch. 12 in ref. 5; an example is given on pg. 235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62051Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:49:57Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (refs. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g., refs. 1,2,6), and include a concise description in your own words in your report.</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped as it passes through matter. This fact is exploited both in shielding and in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and doubly charged, therefore, they give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped; γ-rays have the best penetration characteristics, i.e., are the most difficult to shield. α particles, which have typical energies of 5 MeV, are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through a specially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is exploited, e.g., by having fast particles penetrate healthy tissue with limited damage but sufficient slow-down such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity it was not understood whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution can be approximated by a Gaussian distribution. </p><br />
<br />
<p>Rutherford performed experiments that showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62050Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:34:41Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Radiation detection technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (ref. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g. ref. 1,2,6), and provide a concise description inyour own words with your report</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped by passage through matter. This is used both in shielding, as well as in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and, therefore, give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped. Gamma rays have the best penetration characteristics, i.e., are hardest to shield. α particles, which have typical energies of 5 MeV are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through an especially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is used, e.g., by having fast particles penetrating healthy tissue with limited damage but sufficient slow-down, such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity the question had to be resolved whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution goes over into a Gaussian. </p><br />
<p>Rutherford performed experiments which showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62049Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:33:21Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Geiger-Müller detector technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understand previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (ref. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g. ref. 1,2,6), and provide a concise description inyour own words with your report</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped by passage through matter. This is used both in shielding, as well as in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and, therefore, give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped. Gamma rays have the best penetration characteristics, i.e., are hardest to shield. α particles, which have typical energies of 5 MeV are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through an especially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is used, e.g., by having fast particles penetrating healthy tissue with limited damage but sufficient slow-down, such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity the question had to be resolved whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution goes over into a Gaussian. </p><br />
<p>Rutherford performed experiments which showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62048Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:32:40Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Geiger-Müller detector technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with energies higher than X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understood previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (ref. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g. ref. 1,2,6), and provide a concise description inyour own words with your report</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped by passage through matter. This is used both in shielding, as well as in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and, therefore, give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped. Gamma rays have the best penetration characteristics, i.e., are hardest to shield. α particles, which have typical energies of 5 MeV are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through an especially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is used, e.g., by having fast particles penetrating healthy tissue with limited damage but sufficient slow-down, such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity the question had to be resolved whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution goes over into a Gaussian. </p><br />
<p>Rutherford performed experiments which showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62047Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:30:26Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Geiger-Müller detector technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable, by emitting either an electron or a positron. In the case of neutron-rich nuclei, a neutron converts into a proton, electron and antineutrino. The fast electron is emitted from the nucleus, corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei, a proton is converted into a neutron, positron and a neutrino (β<sup>+</sup> decay). The fast positron emerges from the nucleus. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with higher energies than hard X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understood previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (ref. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g. ref. 1,2,6), and provide a concise description inyour own words with your report</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped by passage through matter. This is used both in shielding, as well as in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and, therefore, give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped. Gamma rays have the best penetration characteristics, i.e., are hardest to shield. α particles, which have typical energies of 5 MeV are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through an especially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is used, e.g., by having fast particles penetrating healthy tissue with limited damage but sufficient slow-down, such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity the question had to be resolved whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution goes over into a Gaussian. </p><br />
<p>Rutherford performed experiments which showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Radioactive_Decays&diff=62046Main Page/PHYS 3220/Radioactive Decays2013-12-17T19:24:15Z<p>Taylorw: </p>
<hr />
<div><h1>Radioactive Decays</h1><br />
<br />
<h1>Learning Outcomes</h1><br />
<ol><br />
<li>Radioactivity<br />
<li>Poisson statistics<br />
<li>Geiger-Müller detector technology<br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<br />
<p>In this experiment a Geiger-Müller counter with a computer interface is used to detect the radiation coming from the natural background, as well as from some weak sources. The statistics of the decays is investigated to confirm the independence of the decay mechanism. The dependence of the count rate on the distance from the source is also investigated. Also, the Geiger-Müller method for detection of radioactivity will be investigated.</p><br />
<br />
<h2>Radioactive Decays</h2><br />
<p>Radioactive nuclear decays can be classified according to their decay mechanism: </p><br />
<br />
<ol style="list-style-type:lower-roman"><br />
<li><p><b>α-decay:</b> heavy radionucleides often decay via the emission of a cluster composed of 2 protons and 2 neutrons, i.e., a <sub>2</sub>He<sup>4</sup> nucleus.</p></li><br />
<li><p><b>β-decay:</b> nuclei away from the line of stability N = Z, where N is the total number of neutrons, and Z the total number of protons, can lower their energy, and hence become more stable by emitting either an electron or a positron. In the case of neutron-rich nuclei a fast electron is emitted from the nucleus, thus converting a neutron into a proton (and an electron + antineutrino) - corresponding to the β<sup>-</sup> decay of free neutrons (half-life 10.6 min.). For proton-rich nuclei a fast positron emerges from the nucleus (β<sup>+</sup> decay) whereby a proton is converted into a neutron and a neutrino. This latter process may seem counterintuitive as it cannot occur for free protons (why?). The rest of the nuclear system supplies the energy necessary for the reaction to take place.</p></li><br />
<br />
<li><p><b>γ-decay:</b> the emission of photons with higher energies than hard X-rays (MeV-range) is the result of a nuclear transition from an excited to a lower state in complete analogy with photon emission from excited atoms (eV to keV-range). This decay almost always accompanies α- and β-decays, since these processes usually leave the daughter nucleus in an excited state.</p></li><br />
<br />
<li><p><b>spontaneous fission:</b>the emission of nuclear clusters bigger than α-particles is a rare process that has been studied recently in a systematic way at heavy ion facilities. It represents an alternative but rare decay mechanism, which provides insight into the nature of nuclear forces.</p></li><br />
</ol><br />
<br />
<p>All modern physics texts contain a chapter that describes nuclear phenomenology as well as a table of isotopes. Understand the basic principles (there will be no need to understood previous chapters of the book for this!). See, e.g., refs. 1-3. </p><br />
<br />
<h2>Detection of radiation</h2><br />
<br />
<p>The detection of nuclear radiation relies on the property that it ionizes the surrounding matter through which it passes. This statement is obvious for the charged α, and β particles. For γ particles the ionization arises through the photoeffect and Compton scattering (ref. 1-3). This ionization can be detected through the electric spark induced between condenser plates that are biased with a high voltage, resulting in a short burst of current. This is the principle of a Geiger-Müller (GM) tube. The efficiency of detection depends on the voltage applied to the gas-filled tube (why can’t one use a vacuum tube?). It is important to realize that the detector has a finite efficiency, i.e., it does not detect every single α, β, or γ particle entering the detector. In particular, the efficiency depends on the voltage applied with a threshold behaviour (around 900 V) followed by saturation. In small hand-held radiation counters the high voltage is produced by a DC-DC converter as used in electronic flashlights. Read the description of GM counters available in many texts (e.g. ref. 1,2,6), and provide a concise description inyour own words with your report</p><br />
<p>Other detection mechanisms used for monitoring are: (i) exposure blackening of photographic film, e.g., in personal total dose monitors; (ii) scintillator counters; (iii) triggering of semiconductor devices; etc. </p><br />
<br />
<h2>Absorption of radiation</h2><br />
<p>Radiation is slowed down and eventually stopped by passage through matter. This is used both in shielding, as well as in applications of radiation for energy deposition (e.g., burning of cancer cells in radiation medicine). The absorption of the three different forms of radiation by matter is very different: α particles are heavy and, therefore, give up their energy readily in collisions with the nuclei of the surrounding matter; β particles are lighter and faster (as they emerge from the decay), and therefore pass more readily through matter until they are stopped. Gamma rays have the best penetration characteristics, i.e., are hardest to shield. α particles, which have typical energies of 5 MeV are stopped by a few centimeters of air, since they are doubly charged and slow compared to β particles. They are detected by GM counters only if they enter through an especially designed opening (transparent to them provided they are fast enough). </p><br />
<br />
<p>The stopping power and energy deposition is also a function that depends strongly on the kinetic energy of the ionizing particles. In radiation medicine this is used, e.g., by having fast particles penetrating healthy tissue with limited damage but sufficient slow-down, such that energy deposition becomes efficient when the tissue to be destroyed is reached. Usually physicists with nuclear medicine training are in charge of designing a radiation plan for each patient depending on the location of the tissue to be destroyed, vicinity of vital organs, etc. This is a non-trivial process, since secondary radiation (e.g., production of electrons) contributes to the energy deposition and may diffuse the flux of radiation. Read the detailed write-up for the “Absorption of Radiation” experiment.</p><br />
<br />
<h2>Lifetimes of radioactive sources</h2><br />
<p>A proper understanding of nuclear decays on the basis of a nuclear shell model (in analogy to atomic structure of electronic energy levels) enables one to predict the energies of the emitted particles as well as the half-lifes. The lifetime is related to the broadening in energy of the decaying state and can be understood from Heisenberg's uncertainty principle. (As a function of time the number of decaying particles is described by an exponential decay law.) </p><br />
<br />
<p>The radioactive sources that we use in this experiment do not permit a measurement of the decay law, since they have long lifetimes (tens to thousands of years), i.e., it is impossible to observe the decrease in radioactivity over a reasonable time span. However, sources with a short lifetime can be produced by exposure of a sample to a high-flux source, e.g., a reactor, which results in the conversion of stable nuclei into unstable ones.</p><br />
<br />
<h2>Statistics of nuclear counting</h2><br />
<p>In the early studies of radioactivity the question had to be resolved whether radioactivity was a purely random process, or whether the emission of one particle might effect the emission of others. One can prove that the observation of the number of independent decays per time interval (count rate) as a function of time should result in a Poissonian distribution (ref. 5). In the limit of high count rates the Poissonian distribution goes over into a Gaussian. </p><br />
<p>Rutherford performed experiments which showed that the probability, ''P(n)'', of observing ''n'' counts in a fixed time interval followed the Poisson formula</p><br />
<br />
<table width=400 align=center><br />
<tr><td><br />
<p align=justify>[[File:Rd-eqn1.png|150px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<p>where the ''average'' number of counts per interval is calculated as </p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn2.png|280px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>For all the measurements in this experiment that are performed with computerized data acquisition and data analysis, the Poissonian character of the statistical distribution of decay events are to be investigated and verified. Since the computer program ''Radiation Counter'' <b>***CHANGE THIS PROGRAM REFERENCE****</b> performs the statistical analysis automatically, it is crucial that you think through the steps involved in obtaining the histogram (cf.. also chapter 11 in ref. 5).</p><br />
<br />
<p>To illustrate how one explicitly analyzes the data we include an example for your convenience.</p><br />
<br />
<ol><br />
<li>Let us say that you record the number of counts heard during 100 five-second intervals by entering a mark in the column appropriate for that number of counts (col. 2 in the table below).<br />
<table width=420 align=center><br />
<tr><td width=120><b>Number of Counts in interval (n)</b></td><td width=120><b>Number of times Count occurs</b></td><td width=100><b>''P(n)''</b></td><td width=100><b>Total Counts</b></td></tr><br />
<tr><td>0</td> <td>I(1)</td> <td>0.01</td> <td>0x1=0</td></tr><br />
<tr><td>1</td> <td>II(2)</td> <td>0.03</td> <td>1x3=3</td></tr><br />
<tr><td>2</td> <td>IIII I(5)</td> <td>0.05</td> <td>2x5=10</td></tr><br />
<tr><td>etc..</td><td></td><td></td><td></td></tr><br />
</table><br />
</li><br />
<br />
<li><p>Now construct a bar graph for the results, showing ''P(n)'' vs ''n'', where ''P(n)'' is the probability for finding n counts:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn3.png|260px|center]]<br />
</p><br />
</td></table><br />
<p>Then, using the Poisson distribution (Eq. 1) evaluate ''P(n)'' and graph the theoretical distribution over the same range of values. To do this, you require the value of n-bar; this should be the mean number of counts in your measurement:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn4.png|110px|center]]<br />
</p><br />
</td></table><br />
<p>Thus, your theoretical distribution and your experimental results will have the same mean.</p><br />
</li><br />
<li><p>Now calculate the standard deviation of your data:</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn5.png|180px|center]]<br />
</p><br />
</td></table><br />
</li><br />
<br />
<li><p>Compare this with the expected standard deviation from the theoretical probability distribution, which is (for a Poisson distribution):</p><br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-eqn6.png|140px|center]]<br />
</p><br />
</td></table><br />
<p>Note that this simple relation between the '''mean''' and the standard deviation is not a property of all distributions.</p><br />
</li><br />
<br />
<li><p>To see if the numbers of counts obey Poisson statistics in a quantitative way, we use the Chi-squared (χ<sup>2</sup>) test (cf.. chpt. 12 in ref. 5, an example is given on p.235). From the reduced χ<sup>2</sup> value one infers the agreement.</p><br />
</li><br />
<br />
</ol><br />
<br />
<h1>Experimental Procedure</h1><br />
<br />
<p>Familiarize yourself with the computer-interfaced GM counter and associated computer software.</p><br />
<br />
<h2>Required Components</h2><br />
<ol><br />
<li>[[Media:Radioactive-ACratemeter.JPG|AC Powered Table-Top GM Counter]]</li><br />
<li>[[Media:RDHandHeldGM.JPG|Hand-held GM Counter]]</li><br />
<li>[[Media:RDBeigeFiesta.JPG|Beige 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDOrangeFiesta.JPG|Orange 'Fiesta' Ceramic Dish]]</li><br />
<li>[[Media:RDMantles.JPG|α,γ Source: <sub>90</sub>Th<sup>232</sup>, Lantern Mantles]]</li><br />
</ol><br />
<br />
<h2>Hardware instructions:</h2><br />
<p>The hand-held GM counter can be operated independent of the computer interface. You should use it in range I (up to 2000? counts per minute - cpm), and turn on the audio monitoring. The background rate should be in the range of up to a few counts per second. For sources we use a bag containing Coleman-type naphta lantern mantles - original Coleman mantles used radioactive elements until 1990 the clones still use a <sub>90</sub>Th<sup>232</sup> α emitter to enhance fluorescence. (Radioactive elements were used in glazing for bathroom tiles and Fiesta plates - no longer on the market). Make sure that the sources are some distance away from the GM counter when measuring the background radiation.</p><br />
<br />
<h2>Computer Instruction</h2><br />
<br />
<p>Data will be collected using a program called "Particle Tracking.vi" located on the desktop.<br />
This program uses the microphone input of the computer to monitor the counts from the "Radiation Alert- Monitor 4" detector. The operation of the program is is described below</p><br />
<br />
<table width=400 align=center><br />
<td><br />
<p align=justify>[[File:Rd-vi.png|800px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>''Note that the program displays a histogram of the results for you to see, but only the raw data of the counts is written to the output file.''</p><br />
<br />
<br />
<h2>Required Data</h2><br />
<ol><br />
<li>Test the statistics of nuclear background radiation. Note the direction in which the GM counter is pointing. Make sure that it is aiming at free space, and not at a potential radioactive source. Take at least two runs, one of which should be with a larger amount of data to observe an improvement in the fit to a Poissonian distribution. Comment on the chi-squared obtained, and quote the decay rate, with its standard error. Include print-outs of the distributions. Repeat the longer run with the GM counter pointing in an orthogonal direction. Are the data consistent with the previous run? Should they be? What are some sources of background radiation? Save the data points for one of the long runs to a data file. Perform the Poisson statistics analysis explicitly as described in the example in the previous section. How do these results compare to the results from the computer program?</li><br />
<br />
<li>Perform measurements similar to (1) while bringing the bag with lantern mantles (<sub>90</sub>Th<sup>232</sup> α,γ source) close to the opening of the GM counter. Comment on the obtained distribution. Use a detailed table of isotopes (with decay schemes) to identify the radionuclide of the thorium family (ref. 6). </li><br />
<br />
<li>Place the orange 'Fiesta' ceramic dish plate on the table. Mount the GM counter centered above the plate using a retort stand. Measure average count rates as a function of distance, e.g., 0.5 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm. Has the count rate at 25 cm reached the background count rate within errors? Plot the count rates after subtraction of the background rate as a function of distance. What functional behaviour do you find? Can you explain why the Geiger counter is responding when exposed to the Fiesta plate? Show relevant decay chain diagrams.</li><br />
<br />
<li>Turn on the AC powered table-top GM counter. Set the knob to HV and dial up an operating voltage not exceeding 1200 Volts. Set the knob to display count rate X1 (in counts per minute) and note the background radiation. Place the beige Fiesta dish close to the exposed GM tube (the aluminium shield can be rotated such that an opening appears). You may need to reduce the sensitivity of the meter by setting the knob to the X10 range. Then measure the count rate as a function of the operating voltage.</li><br />
</ol><br />
<br />
<p>Incorporate in your report an outlilne on the three nuclear decay mechanisms. The function of the GM counter should also be explained briefly in the report.</p><br />
<br />
<h1>References</h1><br />
<br />
<ol><br />
<li>Knoll, G.F., ''Radiation Detection and Measurement'', 2nd ed.</li><br />
<li>Tsoulfanidis, N., ''Measurement and Detection of Radiation''.</li><br />
<li>Rohlf, J.W., ''Modern Physics from α to Z<sup>0</sup>'', Wiley 1994</li><br />
<li>Brehm J.J., Mullin, W.J. ''Modern Physics'', Wiley 1989</li><br />
<li>Taylor, J.R., ''An Introduction to Error Analysis''.</li><br />
<li>Cork, J.M., ''Radioactivity and Nuclear Physics'', D. van Nostrand 195</li><br />
</ol></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62045Main Page/PHYS 3220/Cavendish2013-12-17T19:19:33Z<p>Taylorw: Added learning outcomes. Minor edits.</p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1> Learning Outcomes</h1><br />
<ol><br />
<li>Rotational motion and torque<br />
<li>Damped harmonic oscillations<br />
<li>Error analysis<br />
<li>Computational curve fitting<br />
<li>Exposure to LabView <br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m is the mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you through the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers - one is the centre pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of large spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclosure</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has been illuminated by the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62044Main Page/PHYS 3220/Cavendish2013-12-17T19:17:01Z<p>Taylorw: </p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1> Learning Outcomes</h1><br />
<ol><br />
<li>Rotational motion and torque<br />
<li>Damped harmonic oscillations<br />
<li>Error analysis<br />
<li>Computational curve fitting <br />
</ol><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m is the mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you through the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers - one is the centre pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of large spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclosure</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has been illuminated by the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62043Main Page/PHYS 3220/Cavendish2013-12-17T19:04:47Z<p>Taylorw: </p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m is the mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you through the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers - one is the centre pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of large spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclosure</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has been illuminated by the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62042Main Page/PHYS 3220/Cavendish2013-12-17T18:59:36Z<p>Taylorw: </p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m is the mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you through the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers - one is the centre pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of large spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclosure</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg. sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of the motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has has the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62041Main Page/PHYS 3220/Cavendish2013-12-17T18:57:23Z<p>Taylorw: </p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m is the mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you through the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers - one is the centre pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of lager spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclouse</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg. sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of the motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has has the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62040Main Page/PHYS 3220/Cavendish2013-12-17T18:55:40Z<p>Taylorw: </p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small masses rotating about their centre of mass). The external torque in our case is provided by the angular form of Hooke's law, as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which, however, is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
where C is the torque constant (cf. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m is the mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you though the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers- one is the center pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of lager spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclouse</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg. sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of the motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has has the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorwhttps://physwiki.apps01.yorku.ca//index.php?title=Main_Page/PHYS_3220/Cavendish&diff=62039Main Page/PHYS 3220/Cavendish2013-12-17T18:49:48Z<p>Taylorw: </p>
<hr />
<div><h1>Measurement of the Gravitational Constant ''G'' with a Torsion Balance: The Cavendish Experiment</h1><br />
<br />
<h1>Introduction</h1><br />
<p>Cavendish first performed the measurement of ''G'', one of the classic experiments in physics, in 1798. Since then, there have been many attempts <ref>J.W. Beams, <i>"Finding a Better Value for ''G''"</i>, [http://www.physicstoday.org/resource/1/phtoad/v24/i5/p34_s1 Physics Today, '''24''', 34 (1971)]</ref> to improve on this determination using variations of the same basic experiment. For example, see Gundlach<ref> J.H. Gundlach & S.M. Merkowtiz, <i>"Measurement of Newton's Constant Using a Torsion Balance with Angular Acceleration Feedback"</i>, [http://prl.aps.org/abstract/PRL/v85/i14/p2869_1 Phys. Rev. Lett., '''85''', 2869 (2000)]</ref> (2000) and Quinn<ref> T. Quinn, H. Parks, C. Speake & R. Davis, <i>"Improved Determination of ''G'' Using Two Methods"</i>, [http://prl.aps.org/abstract/PRL/v111/i10/e101102 Phys. Rev. Lett., '''111''', 101102 (2013)]</ref> (2013). The apparatus at your disposal is a modified form of that used by Cavendish and others. The high sensitivity of the apparatus demands patience, perseverance and care from the experimenter. This is a very delicate instrument so treat it gently.</p><br />
<p>A laser S (see Fig. 1) illuminates a small (effectively) massless mirror MM that is attached to a light horizontal rod holding two small lead balls of mass m at a separation of 10cm. The small balls and mirror are suspended from a 25cm bronze torsion wire (perpendicular to the page). The entire apparatus is enclosed in a rigid case that is mounted securely on a wall or table.</p><br />
<p>When two massive lead balls (each with mass M = 1.5kg) are placed asymmetrically as illustrated, in position AA, a small torque acts on the torsion balance twisting the torsion wire and causing the image of the light source to swing through a measurable distance Δ' along the opposite wall. The swing gradually decays until equilibrium is reached. The massive balls are next placed at diametrically opposite points, BB, for further measurement. After the oscillations have died away once more, a second equilibrium position is reached at a distance Δ' with respect to the other side of the zero position. We will use this pattern to evaluate a value for G, the gravitational constant.</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig1.png|500px|border|center]]<br />
<b>Figure 1 -</b> Experiment setup (Top View).<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>Method</h1><br />
<p><b>IMPORTANT:</b> Treat the equipment gently. For example, slamming the door or bumping into the table will considerably lengthen the experimental observation time. </p><br />
<p>In order for the balance to operate correctly, the torsion wire must be precisely vertical and must be able to move freely. The centres of the four balls must lie in one plane as well.</p><br />
<p>Both a manual recording technique and a computerized data collection method are used in this experiment. The position of the laser beam on the graph paper on the opposite wall is noted when the heavy balls are an equal distance from the small balls. The large balls are then rotated so that they almost touch the case. You should make this adjustment carefully to ensure that the heavy balls do not knock the case. If the case is accidentally knocked, the mirror will be set into a large amplitude oscillation that will take about 1 hour to decay. You can therefore save yourself a lot of time (and effort) by making this adjustment delicately in the first place. </p><br />
<p>If this step has been completed successfully the position of the laser spot on the opposite wall will change slowly with time. Note this position on the graph paper every 15 seconds for the first few minutes and every 30 sec. or 1 min. thereafter. The light spot will undergo damped oscillations about an equilibrium position Δ'. Determine the mean period of oscillation P with a stop watch and compare this with the value derived from a plot of displacement vs. time. Allow about 45 min – 60 min for these oscillations to damp out and measure the new equilibrium position x<sub>1</sub> (see Fig. 2).</p><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig2.png|400px|border|center]]<br />
<b>Figure 2 -</b> Graph of small-mass oscillations.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<p>Carefully reverse the position of the heavy balls and repeat the procedure recording position vs. time and the new position at equilibrium x<sub>2</sub>. Now repeat this twice more and obtain an average value for Δ. Measure the diameter of the heavy ball and the thickness of the case. Find the separation of the centres of gravity of the two balls. What approximation(s) does this involve?</p><br />
<p> In this particular experiment, data will be collected by hand, and concurrently automatically by the computer. Part of the laser beam is reflected onto a 128-segment linear photodiode array. The output of the photodiode array is monitored on an oscilloscope, and a computer connected to the oscilloscope downloads the data for processing. A control program written in Labview collects and processes the data, so you obtain data points which form a graph similar to the manually collected data.</p><br />
<br />
<h1>Theory</h1><br />
The torque generated by a force <b>F</b> acting on a mass ''m'' located at <b>r</b> is defined through the cross product <b>T</b> = '''r''' x '''F'''. For rotational motion in a plane described by an angle θ(t), angular velocity ω(t), and acceleration α(t) the torque has only a single non-zero component. It is perpendicular to the plane and is denoted by a scalar: T = T<sub>z</sub>. The combination of the definition of torque with Newton's law of motion, and the moment of inertia I results in the equation of motion T = I α, where T represents the sum of all torques acting on the system (in our case two small mass’ rotating about their center of mass). The external torque in our case is provided by the angular form of Hooke's law as applicable for the torsion of wires, and we ignore for the moment the internal friction in the wire (which however is crucial to obtain a steady-state solution after long intervals).<br />
<br />
If a torsion balance is twisted by a torque (couple) through a small angle θ (radians), the restoring torque is proportional to the angular displacement and is oppositely directed, <br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn1.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
i.e., Where C is the torque constant (cf.. spring constant) of the wire. (What are the units of C?) We combine this with the equation of motion to obtain<br />
<table width=200 align=center><br />
<tr><td><br />
<p align=justify>[[File:Cav-eqn2.png|110px|center]]<br />
</p><br />
</td><td> <b>(1)</b></td></tr></table><br />
<br />
<br />
Equation (1) is a homogeneous linear ordinary differential equation with constant coefficients. The friction responsible for the damping of the oscillations has been omitted here. Solve the equation and show that the period P is given by<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn3.png|110px|center]]<br />
</p><br />
</td><td> <b>(2)</b></td></tr></table><br />
<br />
<br />
Neglecting the material between the two small spheres, the moment of inertia I of the torsion balance about the axis of rotation is<br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn4.png|80px|center]]<br />
</p><br />
</td><td> <b>(3)</b></td></tr></table><br />
<br />
where m = mass of each ball and 2d is the distance between them.<br />
<br />
Thus by measuring the period P and calculating I, the torque constant C may be found from equation (2). If Δ' is the deflection of the light beam from the rest position upon rotation of the torsion wire through θ radians, (Fig. 1) then<br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn3b.png|80px|center]]<br />
</p><br />
</td></table><br />
<br />
<p>where ''D'' is the distance between the mirror and the recording medium.</p><br />
<p>Once the large spheres are moved to their alternate asymmetrical position, the total deflection Δ produced (after the oscillations have died out) is (see Fig. 2)</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn5.png|130px|center]]<br />
</p><br />
</td><td> <b>(4)</b></td></tr></table><br />
<br />
<p>The torque (couple) exerted by the large masses is</p><br />
<table width=200 align=center><tr><td><br />
<p align=justify>[[File:Cav-eqn6.png|160px|center]]<br />
</p><br />
</td><td> <b>(5)</b></td></tr></table><br />
<p>Here ''F'' is the magnitude of the gravitational force of attraction between the small and large masses, and is given by</p><br />
<table width=200 align=center><td><br />
<p align=justify>[[File:Cav-eqn7.png|90px|center]]<br />
</p><br />
</td></table><br />
<p>where ''b'' is the distance between the centres of the small (''m'') and large (''M'') masses at equilibrium.</p><br />
<br />
<br />
<h1>Procedure</h1><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig3new.png|500px|border|center]]<br />
<b>Figure 3 -</b> Experimental Setup.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h2>Data Collection</h2><br />
<ol><br />
<li>Run the program "Labview 8.2", and open the vi called '''"Cavendish v2.vi"'''.</li><br />
<li>Turn on the oscilloscope and power supply only using the power buttons. The power supply should be giving +5V, and the settings should not be touched.</li><br />
<li>To operate the laser, simply flip up the toggle switch of the laser power supply. The laser will appear a few seconds afterwards.</li><br />
<li>When the program runs, it leads you though the steps of turning on the laser and power supply. Follow the directions carefully, as the program needs to obtain a background light level with the laser off.</li><br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cavendishv2_vi.png|500px|border|center]]<br />
<b>Figure 4 -</b> The Labview control program.<br />
<br clear=right><br />
</p><br />
</td></table><br />
<li>The data will be saved as two lists of numbers- one is the center pixel number, and the other is the time. You need to convert centre pixel number to a displacement in order to calculate G.</li><br />
</ol><br />
<br />
<h2>Parameters of Apparatus</h2><br />
<table width=600><br />
<tr><td width=500> Diameter of lager spherical mass</td><td width=100> 6.386 cm</td></tr><br />
<tr><td> Diameter of small spherical mass</td><td> n.a. </td></tr><br />
<tr><td> Mass of large spherical mass</td><td>1500g</td></tr><br />
<tr><td> Mass of small spherical mass</td><td>20g</td></tr><br />
<tr><td> Thickness of Cavendish box enclouse</td><td>3.01cm</td></tr><br />
<tr><td> Separation between the centre of the mirror and the centre of the small spherical mass</td><td>5.0cm</td></tr><br />
<tr><td><b> Details of the Photodiode Array </b></td><td></td></tr><br />
<tr><td> Number of pixels</td><td>128</td></tr><br />
<tr><td> Active area length</td><td> 10.2cm</td></tr><br />
<tr><td> Pixel size</td><td> 0.8mm x 0.8mm</td></tr><br />
<tr><td> Distance from mirror to photodiode array</td><td> 59.2cm ± 0.5cm</td></tr><br />
</table><br />
<br />
<h2>Tasks</h2><br />
<ol><br />
<li>Derive ''G'' from the above equations in terms of the period of oscillation.</li><br />
<li>Discuss the effect of the attraction of the distant 1.5 kg. sphere for the small balls.</li><br />
<li>Enumerate and discuss other possible sources of error in the measurement of ''G'' using this apparatus.</li><br />
<li>When the beam of light oscillates about its final position, it slowly damps out. Assume that the damping force is proportional to the (angular) velocity to find the equation of the motion. From the data, find the damping constant. See textbook references on damped harmonic motion<ref>For a discussion on damping see: A.P. French, ''Vibration and Waves'', Norton, pp.62-70</ref>.</li><br />
<li>Find the expression for the damping force at any time and compare the frequency of the motion without damping to that with damping. Comment on the difference(s) between the two.</li><br />
<li>Why do you need to measure the period of oscillation?</li><br />
<li>This experiment makes efficient use of real-time data acquisition and analysis. This is accomplished using the programming language LabView. What logical steps is the program following to convert what it downloads from the oscilloscope into a reasonable estimation of the photodiode pixel number which has has the laser? You will need to look at the "block diagram" of the program. (You can inspect the block diagram while the program is running). You are only required to understand the general algorithm, not the details. You may find <ref>[http://www.ni.com/gettingstarted/labviewbasics/ Getting Started with LabView]</ref> a useful starting point for understanding LabView. </li><br />
</ol><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig4.png|400px|border|center]]<br />
<b>Figure 5 -</b> Cavendish Beam Schematic. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<table width=400 align=center><td><br />
<p align=justify>[[File:Cav-fig5.jpg|400px|border|center]]<br />
<b>Figure 6 -</b> Cavendish Beam. <br />
<br clear=right><br />
</p><br />
</td></table><br />
<br />
<h1>References</h1><br />
<references/></div>Taylorw