Table of Contents
Compton Scattering
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 Cs-137 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 second experiment of the course (Introductory Lab: Gamma Cross Sections), you measured the interaction cross section for gammas 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 and the angle through which it scattered.
Specifically, you will do the following:
- perform a precise measurement to rigorously test a model;
- develop an informed data collection strategy; and
- identify and account for sources of systematic bias in your data.
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 prelab exercises.
Theory
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
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.)
Prelab exercises (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 as you collect it.
Do the following:
- Make a plot of scattered photon energy, $E^{\prime}$, as a function of scattering angle, $\theta$. This plot should be done in Python and include proper axis labels.
- Answer the following questions:
- Where is the scattered energy the greatest? Where is it the smallest?
- Is there any symmetry in the plot? (Should there be symmetry? Why or why not?)
- Does the scattered energy ever go to zero? (Should the energy ever go to zero? Why or why not?)
Day 1: Understanding the apparatus // Quick scan
Detector setup
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.
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.
Verify that everything is functioning properly by doing the following:
- Place the Cs-137 calibration source on the lip of the aperture of the detector housing. (Don't let it fall in!)
- Apply high voltage 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. (You may also want to use the scope transfer program to capture a screenshot and/or the raw data.)
- NOTE: This PMT requires positive high voltage. A value of +900 V is a good starting point. Do not exceed +1200 V.
- Disconnect the signal from the scope and attach it to the PHA. Look for a Cs-137 spectrum.
- You may need to adjust either the high voltage or the gain (coarse and/or fine) to bring features onto screen.
In this experiment you will be measuring the energy of 662 keV gammas after they have scattered from an aluminum rod. What is the highest energy gamma you expect to have to measure?
To take advantage of the full dynamic range of the PHA, you should use the software gain and the PMT high voltage 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.
Do the following:
- Further adjust the high voltage and gain so that this “highest energy” is towards the upper end of the spectrum.
- At this scale, sketch this spectrum in your lab notebook (and save the raw data). Identify the full energy peak and the Compton edge in the spectrum.
- Take a quick spectrum with any known source (anything from the selection of calibration sources will do), and verify that the energy peak falls (roughly) where you would expect.
Calibration
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 calibrate 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.
Do the following:
- 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.
- For each spectrum, remember to save your pulse height spectrum files (in both *.spu and text format).
- Produce a full calibration plot of energy versus channel (or vice versa) which includes a best fit function and uncertainties.
Quick scan
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 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 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, and 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.
- NOTE: 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.
Day 1 Analysis (20 points)
For your Day 1 analysis, you will present your energy calibration and your initial interpretation of the quick scan. These tasks serve as the foundation upon which the rest of the experiment will build.
Do the following:
Describe how you performed the calibration. (8 points)
- Describe your calibration procedure.
- For each radioactive source, present a plot of the full spectrum (which can be from either your initial or your final run) and annotate the relevant features.
- Using the nuclear decay schemes, identify which energies you expect from each source and identify the corresponding full energy peaks on your spectra.
- For both the initial and the final data sets, measure the peak channel positions and estimate uncertainties.
- Describe how you estimate the positions and uncertainties.
- Provide a table of values showing the literature value for the energy, the position (with uncertainty) for the initial calibration run and the position (with uncertainty) for the final calibration run.
Show a plot and fit of the calibration data. (7 points)
- Plot your data for both the initial and final runs (either on one figure or as separate figures).
- Fit the data to an appropriate fit function, and present the final fit values (with uncertainties).
- Discuss the reduced chi-square value of the fits.
- Did the calibration drift between your initial and final data collection runs? (What constitutes significant drift versus normal statistical fluctuation?)
Show and discuss your quick scan of scattered energy. (5 points)
- Plot your data and the predicted theory line.
- Qualitatively, does your data agree with the prediction?
- Are your data statistically scattered around the prediction or does there appear to be some systematic bias?
- Are there specific regions (e.g. large angle, small angle, or negative angle) where the data seems to be less in agreement than others?
- Propose one or more tests that can be performed to study statistical bias or to improve the quality of the data collected.
- Do not just propose “more/better data”. Be concrete and specific. For each proposal, consider whether you are seeking to improve statistical uncertainty, study a systematic effect, or eliminate a bias in your measurement.
Your Day 1 analysis is due by 11:59 pm the day before your next day in lab.
Days 2 and 3: Assessment of potential systematic biases // Data collection
NOTE: On any day where you make any energy measurements, you will need to calibrate again (and monitor the calibration for drift.)
For the remainder of the experiment, you will be working on two tasks:
- You must propose (and discuss with an instructor or TA) one or more experiments to study potential systematic bias.
- Your must collect a full set of scattered energy versus angle data, and quantitatively compare it to prediction.
It is up you in which order you tackle these tasks (or whether to interweave the data collection and systematic studies); the nature of the questions you are asking about systematics will influence the order.
Task 1: Assessment of potential systematic biases
Based on the results of your preliminary data collection and analysis, you and the instructors will identify one or more potential sources of systematic bias which may impact how you interpret your final results.
- Propose experiments for study and discuss them with lab instructors or TAs.
- Make sure that your proposals have been approved before starting data collection.
How to proceed is up to you. Keep in mind, however, that this exercise does need to be completed (and discussed with lab staff and/or TAs) before the end of Day 3.
Task 2: Data collection
You will want to collect a full set of Compton scattering data. You should allow yourself the equivalent of at least one full day in the lab to collect this data (though it can be spread over both days if you are concurrently doing other studies). 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 after the end of Day 3.
Remember to repeat calibration as needed during the data collection.
At home, you will need to plot and quantitatively compare this data to the prediction. As this constitutes the core of the experiment, expect your grader to be picky about the quality and trustworthiness of your data, and how your data collection influenced your study of systematic biases (or vice versa).
Final analysis (75 points)
Data handling and energy calibration (20 points)
Describe how you processed the data from raw pulse height spectra to final scattered energy measurements.
Do the following:
Show and describe how you chose to determine the pulse height channel corresponding to the full energy peak position from your PHA spectra. (10 points)
- Include at least one representative plot of the pulse height spectrum for a single scattering angle.
- Describe and show how you determined the channel position of the full energy peak in a spectrum. (If you performed a fit, include details of your procedure and example fits. If you used some other method, justify and detail your method and include examples.)
- Show how you determined the uncertainty in your measured quantities.
Provide the calibration plot(s) and fit function(s) used to determine energies for data included in your final plot. (5 points)
- 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).
- Discuss how you handled drift in the calibration (if present).
- You do not need to repeat the explanation of your calibration procedure provided in the Day 1 analysis or provide any energy spectra of the calibration sources.
Provide a table (properly labeled and organized) of your raw measured channel values and your calibrated energy values for your scattering data. (Include uncertainties, where appropriate.) (5 points)
- Show how you applied the energy calibration to your raw data, including how you propagated uncertainties though the calculations.
Systematic bias studies (30 points)
Describe the study (or studies) you conducted to investigate potential systematic biases and report the results (regardless of whether the results were conclusive or helped to mitigate any issues).
Do the following:
For each systematic bias study you conducted, do the following: (30 points total)
- Describe the possible source of systematic bias in your data which you identified and why you chose to study that source.
- Describe how you chose to investigate the bias effect(s).
- Include data, plots, fits or other measurements and observations from these investigations, as appropriate.
- Include plot(s) of the scattering data (including preliminary, partial, or final data, as appropriate) which demonstrate whether or not the effect under investigation contributed to a bias in your scattering data.
Final results and discussion (25 points)
Show your final results for the scattered energy as a function of scattering angle. Interpret and discuss the results.
Do the following:
Provide a plot of your final scattered energy data alongside the prediction from the Compton scattering model. (15 points)
- Show a fit of your data to the model along with a residual plot.
- The form of the fit – specifically which parameters are free and which are fixed – is up to you, but should be stated and justified.
- Provide values and uncertainties for all best 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 in the data or residuals).
- Show a quantitative comparison of your result to a known component of the model. (For example, if you leave the rest mass of the electron as a free parameter in your fit, compare your best fit value to the known rest mass for the electron. There are other ways this can be accomplished; this is simply one suggestion.)
Justify your results and discuss the context surrounding your measurements. (10 points)
- This may include asking yourself the following non-exhaustive list of questions:
- What do your results mean?
- E.g., for someone who is hearing about this experiment for the first time, why are your results important and why is this experiment the right way to make these measurements?
- Why should the reader believe your results and your estimates of uncertainties on those values?
- E.g., what about your technique and data collection or analysis strategy gives the reader confidence in your work?
- If you disagree with expectations, do you disagree in some systematic way or is it random?
- E.g., consider your results in the larger physics context, and address anything which isn't consistent with what you (and other experts) know. Separate statistical uncertainty from systematic bias.
- If you agree with expectation, what would be the next steps for increasing the precision of your measurement or for testing your expectations in new ways?
- E.g., are there improvements to the apparatus or procedure that would allow you to shrink error bars, or is there a different experiment than could be used to answer the questions of this experiment in a complementary way?
- Is your experiment complete, or are there holes in the picture?
- E.g., are there results that you cannot explain or disagreements that you cannot give plausible explanations for? Do you need more (or different data) to investigate the problem fully, or are there measurements that need to be retaken or rethought?
- Consider shortcomings in your work or things that you would improve if you were to continue this experiment (e.g. changes apparatus or technique, alternate analysis methods, or different models). Point towards future directions of study.
- Be specific! Do not just say “more data” or “better equipment”. Justify the suggestions you make.
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.