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. |

3 Experimental procedure

3.1 Turn on and set the PMT high voltage

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. The pulse height-to-energy calibration can change significantly during the first half hour after the high voltage (HV) is first turned on.

• 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.

3.2 View pulses on the scope

This is a good time to take a look at the PMT pulses on a scope. Before we send the signal through the PHA (which can be a bit of a black-box), it's good to understand the characteristics of the raw signal and check the behavior of the apparatus.

* Move the detector to zero degrees – directly in line with the source – and remove the aluminum rod scatterer.

• Open the door to the lead pig and use the tongs to remove the steel pin.

The collimated beam from the source is now directly incident on the detector and we should have a very strong signal. Adjust the scope until you can observe the pulses.

NOTEBOOK:  Sketch a single pulse. Measure the 'typical' rise time, fall time, duration and amplitude of the pulses. Record these values along with your sketch.

If you are using an analog scope (the ones with the glowing green screen), then the screen will display not only the most recent trigger event, but many of the previous events which have not yet faded away. Since we can see many pulses at once, we can actually learn a bit about probabilities by comparing the intensity of different pulses.

NOTEBOOK: Look at the range of pulses you see. You should see that some bands of pulse height are brighter (i.e. more frequent) than others. What physical reason gives rise to the different intensities for different pulse heights? Are there any bands where the intensity gets very dim or close to zero? Is there a maximum pulse height?

3.3 Adjust the high voltage and gain

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. 

• Return the pin to the pig and close the door.
• Reset the aluminum rod, and turn the detector away from zero. (The exact angle doesn't matter.)
• Disconnect the PMT output cable from the scope and connect it to the Input on the back of the UCS-30. Do not use the 50 Ω terminator on the PHA; 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 [Math Processing Error]γ { $\gamma$ -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.

<blockquote> 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. </HTML> 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.*

</blockquote></HTML> *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.

WARNING: Once you have set the high voltage and gain, DO NOT adjust either for the remainder of the experiment.

3.4 Energy calibration

3.4.1 In-software calibration

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.

3.4.2 Post-experiment calibration

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.

NOTE: It does not hurt to do the two-point calibration described above even when you plan to do a better calibration later. When exporting the data in the *.tsv format, both the channel number and calibrated energy values are saved. In this way, you may use the rough calibration values from the software as a guide while in lab, but do the proper calibration when producing final plots and analysis.

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.

NOTE: The energy calibration is not necessarily constant over time. In fact, it is likely to shift a small amount, even under ideal circumstances. It is good practice to establish the stability of the calibration by repeating some measurements multiple times over the course of performing the experiment; in this way, you can quantify any changes over time. At a minimum, you should take calibration data at the beginning of your data collection and at the end on each day. By looking at how the calibration changes over the course of the experiment, you gain insight into an important source of systematic uncertainty in the data.

Some factors which can cause the calibration of the detector to change over time include the following:

• The gain of the PMT can take 30 to 60 minutes to settle down to its equilibrium value. After about an hour, the performance of the PMT typically becomes more stable.
• Too high of a rate of  { $\gamma$ -ray's striking the detector can cause a reduction in the size of the pulses from the PMT. This phenomena is known as pulse height sagging and is a result of too much current flowing through the PMT. * Small changes in the high voltage (for example due to fluctuations in the wall line voltage) will affect the gain.

3.5 Collecting scattering data

• 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 { $d\mu = \frac{\sigma}{\sqrt N} = \frac{\left(\frac{\Gamma}{2\sqrt{2\ln 2}}\right)}{\sqrt{N}}$ , where [Math Processing Error]Γ { $\Gamma$  is the full width at half maximum (FWHM) and [Math Processing Error]N { $N$  is the net count for the peak.

NOTE: Choosing the centroid and uncertainty by moving your cursor around may arbitrary way to assign an uncertainty to a measured quantity, but uncertainties really are nothing more than the experimenter's judgement of how well they have measured a quantity; this is the simplest method.

Another legitimate way to estimate the uncertainty in this measurement would be to estimate the location of the full energy peak with the cursor multiple times (say 10 or 20), average those values and look at the standard deviation of the average. One could also fit the full energy peak to a function like a Gaussian sitting on a background and obtain the value of the centroid and its uncertainty more quantitatively.

For this experiment, fitting the peaks may mean a lot of extra work, but with only a small gain in the accuracy of the measurement. Using the cursor to estimate by eye gives an answer that is “good enough” because it turns out that other factors out of your control may dominate over the uncertainty in the peak locations. Use your judgement and justify your choices in the report.

3.5.1 Considerations for data taking

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 [Math Processing Error]γ { $\gamma$ -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.

CAUTION: Do NOT take data over night. Doing so is unnecessary in terms of getting an adequate number of counts, and the change in the calibration can be significant.

3.6 Preliminary analysis ("Day 2 Question")

In completing the following task between Day 1 and Day 2, you may completely ignore uncertainties.

Using your Day 1 calibration data, obtain a preliminary calibration of the detector channel-energy relationship. Then, select one of your Compton scattering spectra taken at any one fixed angle on Day 1, extract the channel location of the photopeak in the spectrum – you may do this approximately; we just want a rough estimate – and convert it to an energy for the scattered gamma using your preliminary calibration. Assuming that we know the incident gamma ray energy for the Cs-137 source, use the energy of the scattered gamma that you just found and the known scattering angle for selected spectrum to extract a rough value for the electron mass. Is it close to the literature value? Again, you do not need a statistically rigorous comparison here; a percentage difference is fine.

A TA will check with you to discuss this calculation at the start of Day 2.

4 Analysis

4.1 Energy calibration of pulse hight spectrum

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.

4.2 Scattering relationship

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.

4.3 Angular acceptance

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 report, 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.

Rubric

When writing your report, consult the rubric and notes below for the appropriate quarter.

Autumn quarter

Winter quarter

Analysis Rubric

In writing your report it should be assumed that the reader is fully familiar with the contents of the wiki.  Note that complete omission of a section will result in 0 points.