Table of Contents
Compton Scattering - Winter 2024
Arthur Holly Compton was awarded the Nobel Prize in 1927 for his work (published in 1923) of careful spectroscopic measurements of x-rays scattered at various angles by light elements. He found that x-rays scattered at larger angles had systematically larger wavelengths, and he was able to explain these observations by considering the scattering as a collision between a single photon and a single electron in which energy and momentum are conserved; this effect now bears his name. The Compton effect demonstrates the essential duality of waves and particles in an especially clear way: Modeled as a particle (localized, having energy and momentum) one can apply conservation of energy and momentum to predict the relation between scattered x-ray energy and scattering angle. On the other hand, modeled as a wave, one can understand x-ray interference and diffraction phenomena.
Do not attempt to open the Cs137 source until you have been properly instructed on the appropriate safety measures by a member of the lab staff or a TA.
Some applications
Compton scattering is not just the footnote of quantum mechanics history; it is an experimental technique that still has practical use today. The following are some examples of the use of Compton scattering in modern research:
- Compton scattering is being used to model the distribution of x-rays produced when a black hole disturbs a star.
- Compton scattering based tomography can be used to map the electron density of a material
- Compton scattering is a critical theoretical component of investigating the substructure of nucleons (e.g. protons)
- Wooden materials can be non-destructively probed via Compton scattering, which can give insight into the density distribution of materials and possible defects present.
Teaching Points
In the first experiment of the course (Introductory Lab: Gamma Cross Sections), you measured the interaction cross section for gamma rays in aluminum. You most likely found that the purely classical Thomson Scattering model was not always in good agreement with you data. It was postulated in that experiment that Compton Scattering is a more complete description of how photons scatter off of free electrons.
A purely classical wave model of scattering allows for no change in wavelength at a boundary (due to continuity constraints), and thus would predict that scattered gammas would have the same energy as the source. In this experiment, you will test the Compton scattering model's prediction (which was foundational in establishing modern quantum mechanics) for the relationship between the energy of the scattered gamma ray and the angle through which it scattered.
- Perform a precise measurement to rigorously test a model.
- Identify and account for sources of systematic bias in your data.
- Develop an informed data collection strategy.
A Note About Uncertainties
This is an example of an experiment where it is easy to collect data for long enough that your statistical error bars become so small that some other, non-statistical effect becomes the dominant factor in setting your final uncertainty. These non-statistical effects are what we refer to as systematic uncertainties. This is a common and important theme in experimental physics, particularly in the fields of nuclear and particle physics. Learning how to identify systematic biases in your data, and then how to investigate and quantify these uncertainties is a significant part of this particular lab.
Before you come to lab...
In order to prepare for the lab, you should read over the full theory and apparatus sections below. After that, complete the short pre-lab exercise of plotting the model you plan to test.
Theory
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For a more rigorous description of Compton scattering you can turn to any modern physics or quantum mechanics textbook. Wikipedia also has a good discussion here: https://en.wikipedia.org/wiki/Compton_scattering. In this section, we provide just a brief overview.
Consider the scattering of a gamma (photon) from a free electron as shown in Fig. 1.
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| 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 be derived from the conservation of energy and momentum as
| $E' = \dfrac{E}{1+\frac{E}{mc^2}(1-\cos\theta)}$, | (1) |
where $mc^2 = 511\;\mathrm{keV}$ is the rest energy of the electron. This is the model which you will test.
Experimental apparatus
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The experimental apparatus is shown schematically in Fig. 2.
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| Figure 2: The Compton scattering apparatus. |
A collimated beam of 662 keV gammas produced in the decay of Cs-137 is incident on a cylindrical aluminum rod. A PMT+NaI detector which has been magnetically shielded and housed in a lead container is attached to a goniometer allowing it to be rotated about the scattering rod. Pulses from the PMT+NaI detector are sent to a UCS-30 pulse height analyzer (PHA). (See Spectrum Techniques Spectrometers and Software for more details.)
Radioactive source
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.
CAUTION: Do not place any part of your body in front of the open port of this source for an extended time. This source is on the order of 1000 times stronger than the plastic button sources used in other labs. (The activity is of the order of milli cuires rather than micro curies).
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.
Calibration sources
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.)
1. Plot the Compton scattering model (5 points)
When testing a theoretical model, it is always helpful to know what you expect to see in your data before you begin the experiment. Without this context, it is difficult to tell whether the data you are collecting is appropriate or not as you collect it.
Before coming to lab, make a plot of scattered photon energy, $E'$, versus scattering angle, $\theta$. This plot should be done in Python and include proper axis labels.
Exercises
Before you begin to collect the bulk of the data, you will complete a number of specific tasks, each of which is focused on a skill or technique which you need to understand in order to complete the experiment. Successfully completing these tasks, as determined by the instructors during the lab, will count for a total of 25% of the grade of this lab (including the 5% allotted to the pre-lab activity above).
Completing these exercises will likely take most of the first one or two days of lab. Go slowly, and make sure you understand each step!
2. Verify that the detector is functioning properly (5 points)
You are working with the same PMT+NaI detector and UCS-30 pulse height analyzer as you used for the Gamma Cross Sections experiment; therefore, you should already be familiar with how these devices work.
Verify that everything is functioning properly by doing the following:
Apply HV to the PMT and find the signal on the scope. Sketch a typical PMT pulse in your notebook and note its rise time and fall time.
Note that this PMT requires positive high voltage. A value of +900 V is a good starting point.
Use the Cs-137 calibration sourse to collect a pulse height spectrum. Identify the full energy peak and the Compton edge in the spectrum. Sketch this spectrum in your lab notebook.
The calibration sources are kept in a small wooden box behind some lead shielding on the far side of the lab bench from the detector.
The size of the pulses from the PMT are proportional to both the energy of the incident gamma and the high voltage (HV) applied to the PMT. (Increasing the HV will increase the size of the PMT pulses.)
The PHA you are using measures and digitizes the size of each pulse from the PMT. The x-axis of the spectrum displayed by the USX software is the digitized “size” of the pulses recorded. In this experiment you will be measuring the energy of 662 keV gammas after they have scattered from an aluminum rod. Thus, you know what is the highest energy gamma you expect to have to measure (what is this energy?). To take advantage of the full dynamic range of the PHA, you should use the software gain and the PMT HV setting to make sure that the full energy peak of the highest energy gamma you need to record falls in a channel near the upper end of the pulse height range.
- Set the amplifier gain in the USX software, and the PMT HV to ensure that the full energy peak of the highest energy gamma you expect to measure is appropriately positioned.
NOTE: Do not exceed +1200 V on the PMT. If you have the PMT at this HV and still need to increase the pulse sizes, use the amplifier gain in the software.
Take a quick spectrum with any known source and verify that the energy peak falls (roughly) where you would expect.
3. Calibration (5 points)
When you did the Gamma Cross Sections experiment, you were interested in the number of gammas which were recorded. However, in this experiment you need to measure the energy of the scattered photons. In order to measure absolute energy, you will have to calibration the channel axis of the pulse height spectrum.
The position of features on the x-axis of the spectrum is proportional to the total energy deposited in the crystal by an incoming gamma. Accordingly, we can calibrate the x-axis in terms of incident gamma energy by placing calibration sources in front of the detector and measuring the pulse height spectrum channel corresponding to the different energy gammas emitted by those sources.
You are provided with a number of small sources which can be used for calibration. These sources provide gammas of known energy from 31 keV up to 1.27 keV. You can look up the primary emission energies for each of these sources on the Nuclear Decay Schemes page. From the plot you made before lab, you should know what range of scattered photon energies you expect to have to measure.
- Use the available calibration sources to calibrate pulse height channel to gamma energy. Use as many known gamma energies as you can to cover the full range of Compton scattered energies you expect to measure in your experiment.
- A full calibration will consists of a plot (with best fit function and uncertainties) showing energy versus channel (or vice versa).
4. Quick scan (5 points)
This is now a good time to collect a quick set of Compton scattering data in order to gain a sense of how the experimental apparatus as a whole is performing. Make measurements of the scattered gamma energies as a function of scattering angle, making sure to cover as much of the full range of angles as possible. You should consult with an instructor about how much time you should spend collecting this data based on when you get to this part, but 2 hours is a good nominal figure.
The goal here is not to think of this as your final data set. Instead you want to collect just enough data that you can go do a full analysis of it and obtain a preliminary result. As you go through this process you will run into things which do not seem to be working as you would expect, and which you will need to spend some more time in the lab investigating. The issues which you will uncover are may constitute systematic biases (which crop up in all experimental work), and identifying systematic biases (and understanding their impact on your experiment) is a critical part of doing experimental science.
Keep the following in mind as you collect this data:
- From your plot of the Compton scattering model you can make an informed decision of which scattering angles to test for this measurement. You want to choose these angles so that your data will efficiently cover the model.
- This is a quick measurement. You want to fit all of the data collection into a couple of hours so you cannot spend too much time collecting data for any given scattering angle; we are not yet concerned with optimizing our data collection strategy to minimize statistical uncertainty.
- Since this is intended to be a preliminary run through the whole data collection and analysis process, you can determine the centroid location of your full energy peaks using the functionality built into the USX software. There is no need – at this point – to fit peaks (though you may wish to do so in later runs).
- Save all of your pulse height spectra in both spectrum (*.spu) and text (either *.tsv or *.csv) formats.
Do the following:
- Save all pulse height spectra.
- Record measurements of the full energy peak centroid locations from each pulse height spectrum. (These should be tabulated neatly in your notebook including units and estimates of their uncertainties.)
- Convert channel locations to absolute energy using your calibration from earlier. (Again, data should be neatly tabulated in your notebook with units and uncertainties.)
Produce a plot of the scattered gamma energy versus scattering angle along with the prediction from the Compton scattering model.
- Since this is a quick measurement whose primary purpose is as a diagnostic tool, you can plot the data in any way that is convenient in the lab. You do not need to use Python for this exercise; a spreadsheet plot would be acceptable, or you may even plot by hand using pencil and paper.
Once you have your preliminary plot of the scattering data, find an instructor to go over the results. Using this data, the instructor will help you to identify potential sources of systematic bias and put together a plan for investigating the biases so that you can account for them in your final data analysis.
At this point you may be wondering why bother taking a preliminary set of data, as opposed to simply collecting all of the data you need now. This is a good question.
One reason to do this is because all experiments are subject to instrumental bias which can lead you to improperly interpret your data. Most systematic effects are not initially obvious and only show up when you start collecting and analyzing data. This is the point where you will start to see results which are not what you expect, at which point you may find out that you have to go back and retake some (or all) of the data. So, by collecting preliminary data and analyzing it to establish whether or not things are working as expected, you may potentially save yourself a lot of time and grief.
5. Assessment of potential systematic biases (5 points)
Based on the results of your preliminary data collection and analysis, you and the instructors will have identified one or more potential sources of systematic bias which may impact how you interpret your final results. (You should also have discussed how you can go about investigating these effects.)
Do the following:
- Conduct the proposed investigations and show your work and results to an instructor.
You do not have to do these investigations before moving on to collect a more complete data set to analyze. Depending on what you choose to study, these measurements might best be made before, after, or even during the primary Compton scattering data collection. How to proceed is up to you.
Keep in mind, however, that this exercise does need to be completed and evaluated by us before the end of your third day in the lab.
Final data collection
You are now prepared to collect a full set of Compton scattering data. You should have allow yourself at least one full day in the lab to collect this data. Use this time to acquire the best data you are able to as you will not have another opportunity to come into the lab to collect more.
Final data analysis
You have one week to perform a full and complete analysis of the data you collected in-lab and submit the following assignments. You can score up to a maximum of 50 points total on these assignments, but you should notice that the pool of points available is actually 60 points. This means that you don't need to do everything perfectly to get full credit (though you should still try to do everything as well as possible, because we will not consider make-up or extra credit). If you score more than the maximum of 50 points… “hooray”! (But you will still only get a score of 50; you don't get extra credit.)
All plots should be appropriately labeled and of publication quality.
Data handling and energy calibration (25 points)
Do the following:
- Show how you choose to determine the pulse height channel corresponding to the full energy peaks from your PHA spectra.
- Include one representative plot of the pulse height spectra for a single scattering angle, including the details of any fits you performed on the data.
- Show how you determined the uncertainty in your measured quantities.
- Provide a table (properly labeled and organized) of your measured quantities for your scattering data. Include uncertainties on all measured quantities.
- Show how you applied the energy calibration to your raw data, including how you propagated uncertainties though the calculations.
- Include the measured values used in your calibration (properly labeled and including uncertainties). The data should presented in an organized and easy to read form. (Tables are usually the right choice here.)
- Provide a plot of the data you used for determining the pulse height to gamma energy conversion.
- Show a fit of your data to the model and provide the associated uncertainties on the fit parameters.
- Evaluate the goodness of the fit (both in terms of the reduced chi-square value of the fit and in terms of any anomalies or trends you observe).
Systematic biases (25 points)
Do the following:
- Briefly describe the possible sources of systematic bias in your data which you identified and choose to investigate further.
- Briefly describe how you choose to investigate each of these effects.
- Include data from measurements which you made as part of these investigations.
- Include plots of the data which demonstrate whether or not the effect under investigation will bias your scattering data.
Conclusions and comparison with literature (25 points)
You will need to write a complete and persuasive conclusion that includes a comparison of your results to expectations/literature and puts the results in proper context. Even more than in the assignments above, this assignment leans heavily on your ability to write clearly and correctly. You may want to look over our page on Drawing Conclusions.
Do the following:
- Provide a plot of your final data and the predictions of the Compton scattering model.
- Show a fit of your data to the model and provide the associated uncertainties on the fit parameters.
- Evaluate the goodness of the fit (both in terms of the reduced chi-square value of the fit and in terms of any anomalies or trends you observe).
- Show a quantitative comparison of your results to a known component of the model. (For example in your fit you could leave the rest mass of the electron as a free parameter and then compare the best fit value from your data to the known rest mass for the electron. There are other ways this can be accomplished; this is simply one suggestion.)
- Discuss the final conclusions which you are able to draw based upon your data. Include a discussion of how you chose to deal with any sources of systematic bias you uncovered in your data.
References
A. A. Bartlett, Am J. Phys. 32, 120 (1964) This paper is a historical review of the experiments that were later explained by Compton's discovery of the Compton effect.
A. H. Compton, Am. J. Phys. 29, 817 (1961) Compton reviews the experimental evidence and the theoretical considerations that led to the discovery and interpretation of x-rays acting as particles.