Before coming to the lab on the first day, read through the Theory section below.

In addition, consider the following:

In this experiment, you will use a PMT+NaI detector to measure the energy lost when photons Compton scatter off of electrons in an aluminum rod as diagramed in the Fig. A.

As you saw in the gamma cross sections experiment, the probability that a gamma passing through matter interacts (by scattering or being absorbed) is related to the linear attenuation coefficient. For our aluminum rod, some incident gammas will Compton scatter while others will pass through without interacting at all. From the chart for aluminum, estimate the linear attenuation coefficient and use this value to calculate the fraction of incident gammas that will pass through the scatterer without scattering. Assume the rod has a diameter of 1 cm.

Next, assume that we make two measurements – one at large scattering angle (Detector Position 1 in Fig. A) and another at a small (but non-zero) scattering angle (Detector Position 2). If we are trying to measure only the energies of the Compton scattered photons, what are the potential consequences of unscattered gammas passing through the rod for the two positions? Considering the fraction of gammas which do not scatter in the rod, do we need to worry about the effect of the unscattered gammas at each position?

Figure A: Schematic drawing of Compton Scattering

The more advanced theory dealing with the cross-section and absolute intensity of Compton scattering will not be addressed here. For more information, see the Klein-Nishina formula.

Consider the scattering of a gamma (photon) from a free electron as shown in Fig. 1.

Figure 1: An incident gamma of energy E “collides” with an electron and scatters with energy E' at angle θ relative to the initial trajectory.

The energy of a gamma scattered by a free electron, $E'$, depends on the scattering angle, $\theta$, and the energy of the incident gamma, $E$. It can easily be derived from the conservation of energy and momentum as

where $mc^2 = 511 \mathrm{keV}$ is the rest energy of the electron. This is the relationship which you will test in the lab.

Figure 2: The Compton scattering apparatus.

The experimental apparatus is shown schematically in Fig. 2.

A collimated beam of 662 keV $\gamma$ -rays produced in the decay of cesium-137 is incident on a cylindrical aluminum rod. Some of the 662 keV photons will Compton scatter from electrons in the rod and be detected by a NaI(Tl) detector coupled to a photomultiplier tube (PMT). Each gamma which strikes the NaI crystal will produce an output pulse from the PMT. The total charge contained in the pulse is proportional to the energy of the gamma which struck the NaI crystal. The pulses from the PMT+NaI detector are sent to a pulse height analyzer (PHA), in this case the Spectrum Techniques UCS-30. The PHA measures the size of each PMT pulse and displays a histogram of recorded PMT pulse sizes. The histogram of pulse height sizes is referred to as a pulse height spectrum.

By positioning the PMT+NaI detector at various angles about the scatterer, the energy of the Compton-scattered gammas can be determined from the resulting spectrum at each angle. To do this you will need to calibrate the pulse height axis of the spectrum using gammas of known energies.

A PMT coupled to a NaI crystal is a common detector used to measure the energy of gammas. A monoenergetic beam of gammas incident on a PMT+NaI detector will produce a characteristic spectrum of pulse heights. Understanding the details of this pulse height spectrum is necessary for analyzing your data. A detailed description of the pulse height spectrum produced by a PMT+NaI detector is given on the NaI Detector Physics and Pulse Height Spectra page. Make sure that you understand this description before the end of the first day of the experiment.

A pair of ${}^{137}\textrm{Cs}$ sources produce 662 keV gammas. These sources sit at the center of a lead pig to shield you from the radiation. The radiation emerges from the pig in a collimated beam aimed at the scatterer in the middle of the table.

The “source” is actually two sources having strengths as follows:

• 32.5 millicuries (mCi), produced 5/19/69
• 30.0 mCi, produced 7/11/69

These activities are nominal values only, as the activity will decay with time. (Cesium-137 has a half-life of 30.17 years.) When not in use, the pig is “closed” by a tungsten rod inserted into the exit aperture of the pig. A locking brass door holds the plug in place.

• The source is “opened” by swinging the door away from the face of the pig and removing the plug using the long handled tongs so that your hands are not exposed to the beam.  
• When you are finished taking data, the tongs should be used to reinsert the plug and the door should be closed.

To calibrate the pulse height axis of the PHA, a set of small radioactive sources is provided. Sources include ${}^{241}$Am, ${}^{133}$Ba, ${}^{57}$Co, ${}^{137}$Cs, and ${}^{22}$Na, and should yield discernible gamma peaks with energies between 59.5 keV and 661.6 keV.  

You need not consider energies above 662 keV when doing your calibration.

Energies and relative intensities of the calibration sources are available from the nuclear decay schemes. Note that these sources all have low activity so as to not overwhelm the detector with counts and cause charge pileup (also known as voltage sag.)

NOTEBOOK: While the pig is closed you should sketch the layout of the experimental apparatus in your lab notebook, and important dimensions should be recorded. Of particular interest are the following:

• The distance from the source to the scatterer. (Assume that the radioactive sources are located in the middle of the pig.)
• The distance from the scatterer to the front face of the detector shielding.
• The dimensions of the opening in the detector shielding which determines the range of scattering angles which the detector sees.

Do not attempt to remove the NaI+PMT detector! Additional information about the apparatus geometry – including dimensions of several hard-to-measure quantities – is given in a separate PDF. You may consult this document, but be sure to verify measurements (and determine uncertainties) yourself, recording values in your notebook.

To minimize electronic drifts, the power supply and pulse height analyzer should be turned on early and left on for the duration of the experiment. If the voltage to the PMT changes, the pulse height-to-energy calibration will change as well. Let the PHA run for at least five minutes before taking data to ensure the high voltage has stabilized.

• Make sure that the high voltage connector on the back of the PMT is connected to the HV output of the UCS-30. Also check that the UCS-30 is connected to a USB port on the computer.
• Turn on the UCS-30 and then run the USX application on the computer.
• In the Settings menu, select High Voltage/Amp/ADC. Verify that the high voltage is set to positive polarity. (Most of the PMTs in the lab have the appropriate HV polarity marked on the back of the tube.)
• In the upper left corner of the USX software, type in a value of 1500 for the high voltage, and click the On/Off button to apply this voltage.

The size of the pulses from the PMT scale with the magnitude of the high voltage applied to the tube. Later, we will adjust this voltage (and the amplifier gain) in order to take full advantage of the dynamic range of the PHA, but for now we can do some preliminary tests.

Now we wish to adjust the HV so that the full energy peak of the highest energy $\gamma$-ray you expect to observe is near the right end of the x-axis of the spectrum displayed by the USX software. 

• Make sure the pin to the pig is inserted and the door is closed.
• Make sure the aluminum scattering rod is in place.
• Turn the detector away from zero. (The exact angle doesn't matter.)
• Make sure the PMT output cable is connected to the Input on the back of the UCS-30. Do not use the 50 Ω terminator on the PHA; if present, leave it attached to the scope.
• From the drop-down Mode menu in software, make sure that “Preamp” is selected.
• Select a calibration source which will provide $\gamma$-rays of the highest energy you expect to observe in the Compton scattered data. (HINT: consider the source of the $\gamma$-rays which will be scattered off of the aluminum rod and ask yourself what is the highest possible Compton scattered photon energy you could obtain.) * Place the appropriate calibration source in the entrance aperture of the detector, near the NaI crystal.
• Start the data collection on the PHA software. You should observe the normal features of a pulse height spectrum – a full energy peak, a Compton plateau and (maybe) a backscatter peak.
• Adjust the coarse and/or fine gain until the full energy peak is near the upper end of the pulse height range. Leave yourself a little bit of room… about 85% over is fine.
• If you need to set the gain very high or if you are unable to move the peak far enough, you may also raise the high voltage. DO NOT EXCEED +2000 V.

NOTEBOOK: Once you have finished setting the high voltage, save the resulting spectrum to the hard drive. It is recommended that you save all the spectra you collect in this experiment in both *.spu and *.tsv formats. It is good practice to record the filenames you use in the appropriate part of your lab notebook, along with enough information to know exactly what the spectra in the files represent.

NOTEBOOK:

• Record the final high voltage, coarse gain and fine gain values in your lab notebook.
• Sketch the pulse height spectrum to scale and label all axes.
• Identify and label the full energy peak, Compton shelf and Compton edge in the spectrum
• Use the Compton scattering formula and the location of the full energy peak to calculate where the Compton edge feature should be. Note whether or not the calculated location for the Compton edge is consistent with the location in your spectrum.

The purpose of the above exercise is to confirm that you have correctly identified the full energy peak associated with the 662 keV gammas. Mistakes are possible due to the presence of multiple gammas with overlapping features or because a high voltage or gain set was set too high or too low. In experimental work it is important to understand your data well enough to verify that things make sense. Otherwise you can waste time on something which is not what you are trying to study. If your calculated and measured values for the location of the Compton edge are consistent with one another then you can proceed with the experiment with increased confidence that you are on the correct path. If however these values are inconsistent, you should identify the problem before proceeding. This exercise is intended to be a 'quick & dirty' check; we are not looking for precise agreement.

Use the calibration sources provided to determine the relationship between gamma energy and pulse height channel on the PHA. The pig containing the two ${}^{137}$Cs sources should remained closed whenever you take calibration data, and ideally the PMT will have already been powered for 30 minutes or more so that the voltage is stable.

The PHA software has a two- or three-point calibration feature which allows the x-axis of the display to change from channel number to energy. To calibrate in this way, one needs to know the true energy and the corresponding channel location for two (or three) gamma spectrum features.

NOTEBOOK: Collect a PHA spectrum for each of the calibration sources provided and sketch each in your notebook. In addition, save each in *spu and *tsv format so you can access them again later if needed. For each photon, Identify the full-energy peak and record the peak centroid (with uncertainty). While the software cannot incorporate the peak uncertainties into its calibration, you may wish to comment on these when discussing the calibration in your analysis. For each feature, identify the corresponding known photon energy in the appropriate decay scheme, and record this value in your notebook as well, keeping as many digits as provided. 

Once the peak positions are known, select Settings: Energy Calibrate: 2 point or Settings: Energy Calibrate: 3 point from the drop down menu. Follow the software instructions and make sure to enter values of energies (in keV) with the full number of digits available from the decay schemes.

The in-software calibration method described above has several limitations.

• It does not incorporate the uncertainties on the measured peak channel positions.
• It does not explicitly provide the fit function and cannot produce uncertainties on the fit parameters determined.
• One can only use two or three calibration points.

Therefore, it is preferable to collect data in raw channel number and then, at home, fit the channel versus energy data to a function to find the conversion formula.

It is suggested that you do a quick plot of the $\gamma$ -ray energy vs. pulse height channel in-lab as a sanity check. You may use any software you like for this purpose (e.g. Microsoft Excel), but you should detail the results (including a print-out or sketch) in your notebook. The purpose of plotting this data as you take it is to catch any problems such as misidentified peaks while you have time to correct the problem. Your calibration data should form a pretty good straight line. If any data point is visibly out of place (fitting is not necessary here), this is a sign that you may have misidentified the full energy peak or some other problem.   

NOTEBOOK: Include a preliminary plot of your calibration data and determine (roughly) the slope of the line.

• To minimize interference, when not in use the calibration sources should be kept behind the lead bricks at the far end of the bench from the detector.
• With the aluminum scatterer in place, open the pig using the long handled tongs provided.
• Position the detector at a scattering angle of interest. Use the angle scale to measure the angle of the detector and estimate the uncertainty of the angle.
• Collect data at each angle for a long enough period of time to be able to accurately determine the location of the full energy peak by eye using the cursor function of the UCS-30 software.
  • Alternately, you can use the “centroid” reading provided by the USX software, which calculates a weighted average.
• To estimate the uncertainty in your determination of the full energy peak, move the cursor left or right until it just becomes obvious that the cursor is no longer at the center of the peak. The number of channels which you moved over is your uncertainty in the measurement of the peak position.
  • Alternately, you can estimate the uncertainty in the peak center as $d\mu = \dfrac{\sigma}{\sqrt N} = \dfrac{\left(\frac{\Gamma}{2\sqrt{2\ln 2}}\right)}{\sqrt{N}}$, where $\Gamma$ is the full width at half maximum (FWHM) and $N$ is the net count for the peak.

In this experiment you need to decide how many and at what angles to take data, as well as how best to determine the location of the center of the full energy peaks from the spectrum. When it comes to deciding on a data collecting strategy, you are faced with balancing trade-offs associated with the fact that you have a finite amount of time to do the experiment. How many angles should you make measurements at? Which angles should you use? How long do you spend collecting data at each angle? Learning how to reason your way to an appropriate answer to these questions is a big part of this course.

Here are some things to consider as you plan your data collecting strategy.

• What angles to take data at: Looking at the formula for Compton scattering, it should be readily apparent that there are well defined largest and smallest angles. To determine which angles in between the two extremes to use, it is helpful to plot the functional for Compton scattering. If the function is fairly smoothly changing over its full range, you can take evenly spaced data. However if there are regions where the function changes rapidly and others where it changes much more slowly, then it might be better to take more closely spaced data in the rapidly changing region at the expense of fewer data points where things are changing more slowly.
• How many angles to take data at and how long to count at each angle: These two factors are obviously related. The longer you count at each angle, the fewer angles you can make measurements at. There is no strictly “right” or “wrong” answer. The plot of the functional form of the Compton scattering formula can give you some idea of how many angles you should take data at to obtain good coverage. How long to count is a little more subtle. You might be tempted to use $\sqrt{N}/N$ as a guide. However, since you will be determining peak locations by eye, (using the cursor in the UCS-30 software), you need to collect data only long enough that the peak is smooth enough for you to feel confident in your ability to estimate its location. Ultimately, how long to spend collecting data at a given angle is another judgement call on your part. Note that for some angles the rate of scattered $\gamma$ -rays is higher than others and so you don't need to spend as long on those measurements.
• Minimize the effects of drift in the calibration:  As mentioned earlier, the calibration from pulse height channel to  $\gamma$ -ray energy changes over time. The change is most noticeable during the first half hour after turning on the high voltage to the PMT, and over time periods of a day or more. To minimize these effects, consider the following:
  • Try to wait 30 minutes after turning on the high voltage to the PMT before beginning data collection.
  • Take at least two sets of calibration data, one before and one after collecting your scattering data for the day. A third calibration in the middle of the day may even be useful.

Determine the energy calibration to be used for your analysis of the scattering data. You should have taken multiple sets of calibration data. Examine how stable the calibration was over the course of your experiment. Depending on how many calibration data sets you have, and how much the calibration changed over time, you have a couple of options.

• If your calibration was fairly stable, you could use one set of calibration data or an average of multiple calibrations.
  • If an average of multiple calibrations runs is used, you will need to take into account both the uncertainty in the fit parameters and the deviation in the average. If one of these two sources of uncertainty dominates, the other can be ignored. If they are both of the same order of magnitude then they should be appropriately combined to arrive at the overall uncertainty in the calibration. 
• If your calibration changed significantly, you could choose to apply different calibrations to the scattering data based on when the data were taken.

However you decide to handle the calibration, you should justify your method. A table, plot and fit of the pulse height to energy calibration values you extract from the different calibration runs should be included.

Include a discussion of the uncertainties in the calibration. If uncertainties had to be propagated through calculations, the formula for the propagation should be included.

Use your calibration(s) to determine the scattered $\gamma$ -ray energies measured for the different scattering angles. Account for the uncertainties in both the estimation of the full energy peak locations from the spectrum and the energy calibration. If one of the uncertainties dominates you can ignore the smaller. Otherwise, propagate the uncertainties. Justify how you decide to treat these uncertainties and, if necessary, show how they were propagated. Plot and fit your scattering data to the Compton scattering formula, Eq. (1). Extract the rest mass of the electron from your fit to the data and compare with literature value.

• The energy of the incident gamma from decaying cesium can, in principle, be found from independent experiments; therefore, one would expect its value to be fixed in the fit, rather than allowed to float as a free parameter. However, your determination of absolute energies is dependent on the apparatus you are using and on your calibration; therefore, one might want to allow its best value to be determined by the fit in order to account for any issues arising from (even slight) mis-calibration. You may use your judgement (and justify) on whether to fit with only rest mass as the free parameter or both rest mass and incident gamma energy as the free parameters.

After displaying the fit, discuss how well your data agree with the Compton scattering formula and evaluate – in depth – the results.

Using the sketch and measurements of the geometry of your apparatus collected in Sec. 2.5, compute the angular acceptance of the detector (i.e. the full opening angle over which gammas scattered from the rod could enter the detector). For the analysis, discuss how this extended angular acceptance compares to your uncertainty in the angle position of the detector and discuss what affect a non-zero angular acceptance has on the data you collect. Note that this angular acceptance is not necessarily an explicit contribution to the uncertainty, but you should discuss its impact on how you collect data and what you measure.

Your written analysis that you submit to be graded should be built around your final conclusions. Everything in your analysis should support your final result and conclusions. For this experiment your final result(s) may end up being a comparison of your scattered photon energies to the predictions of the Compton model. Your conclusions would be your evaluation of how well your measured values did or did not match a theoretical prediction and a discussion of anything you may have discovered about how the results depend on different experimentat factors.

You need to make clear things you did, decisions you made in the lab which are important to understanding how you arrived at your results and conclusions. This might include:

• It should be made clear how any measured quantities were arrived at. For example how you measured your photopeak locations on the pulse height spectra should be made clear. Did you use the cursor to estimate peak locations, or create an ROI and use the centroid calculated by the USX software, or maybe you fit the spectra to gaussian plus background.
• How you performed your energy calibration.
• How you assessed, quantified and propagated uncertainties should be discussed and made clear including examples of how calculations were done where appropriate.
• Any effects you discovered such as drift in your calibration, or pulse height shifts at certain scattering angles should all be addressed.

The above list is not intended to be complete, nor should it be treated as a checklist of what should go into your written analysis. Your analysis needs to make clear to the reader what your results and conclusions are, show how your data support those conclusions, demonstrate how you processed the data, etc.

For this quarter we are focusing on developing your skills in data analysis and drawing appropriate conclusions from your data. Your analysis should focus on these things. You should not include sections on the apparatus, background theory, historical significance, and things like this. This is not to say that these things are unimportant, they are just not part of a report on your analysis and results.

Your analysis will be evaluated based on the following rubric. The rubric is not a format for your analysis, you are not expected to have a specific section on Data Handling or Presentation of Data. Elements of the different rubric categories will appear at different points through out your analysis writeup. For example you will be presenting data in your discussion of the calibration, your discussion of determining peak locations, and likely in your final results. Your writeup of your analysis should be structured in a way that is clear and readable, there should be a logic to the flow of it.

Each item below is graded on a 0-4 point scale:

• 4 – Good (A): completes all listed tasks and provides appropriate context; thinks carefully about data and analysis; addresses all concerns raised by the results (where appropriate).
• 3 – Adequate (B): misses one or more minor element or lacks appropriate context; leaves a problem or ambiguity unaddressed; does not present analysis clearly enough.
• 2 – Needs improvement (C): omits or mishandles one or more item which renders the analysis fundamentally incorrect or incomplete; presents results in an incorrect or unclear way.
• 1 – Inadequate (D): omits or mishandles multiple items or treats them at an insufficient level; presentation is overall muddled or inaccurate; flaws in logic or process.
• 0 – Missing (F): omits all elements or makes no meaningful attempt.

All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.