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Introductory Lab: Gamma Cross Sections

For your first lab of this course, you will conduct a high energy particle counting experiment to test the limits of a model describing the interaction of gammas with electrons within a material.

Counting experiments are a common type of experiment in the fields of high energy particle and nuclear physics, and comparing between predicted values (given by models) and measured values (given by precision experiments) helps scientists determine which models remain plausible explanations for the real world and which have to be abandoned.

For example, the experiment which detected and measured the mass of the Higgs Boson was a test of the Standard Model of particle physics. That model predicted a particle that had certain known characteristics (e.g. the interactions that could produce it) and certain unknown characteristics (e.g. its exact mass), and scientists “discovered” the particle when they were able to show that experimental data was consistent with the predictions.

Now, scientists continue to investigate the particle and measure its mass to ever increasing degrees of precision because even a small deviation from the predictions of the Standard Model could be evidence of new physics.

In this experiment, you will use a PMT+NaI detector to make measurements of the total interaction cross section, $\sigma$, of gamma particles (high energy photons) in an energy range from ~50 keV to ~1.5 MeV as they pass through different elements. You will use these measurements to first test how well the Thomson scattering model predicts the cross section, and then look for evidence of different or additional processes that may also be present in addition to Thompson scattering.

The purpose of this lab is not just to show that the Thomson model is incomplete; that is something which we already know. Rather, the goal is to learn how to perform an experiment to test a model to a high degree of precision and to gain insight into the considerations and complications which go into making such a measurement.

Thomson scattering is a classical (i.e. non-quantum mechanical) model for photons scattering from electrons. We now consider Compton scattering to be a more complete and more accurate model than Thomson scattering, and we also know that in certain energy ranges processes like the photoelectric effect and pair production also contribute to the cross section.

For the purposes of this lab, we have chosen to test the Thomson model because (despite its incompleteness) it was the model that early particle physicists had in mind when these types of experiments were first performed. Measurements (like the type you will do here) helped to build up evidence for other interaction mechanisms, and in turn better models.

Learning goals

By the end of this experiment, students are expected to be able to do the following:

  • explain how a NaI+PMT gamma detector works;
  • be able to interpret the features of a gamma emission spectrum and estimate the relative energies of such features;
  • estimate statistical uncertainties from raw detector counts;
  • identify full energy peaks on spectra and determine total count rates and uncertainties in those peaks;
  • explain (and apply) the functional relationship between count rate and absorber thickness for gammas and explain the physics behind the terms “cross section” and “linear attenuation coefficient“;
  • explain contributions to the total gamma absorption cross section, and understand which contributions dominate in which energy regimes;
  • propose and explore possible systematic contributions to uncertainty in particle detection experiments;
  • fit the peak of a PHA spectra to a Gaussian plus appropriate background, evaluate the goodness of fit, and iterate (as needed);
  • fit data with error bars to an exponential function, evaluate the goodness of fit, and iterate (as needed); and
  • quantitatively compare results to predicted models and evaluate agreement.

Before you come to lab...

Prelab assignment (5 points)

To prepare for each lab this year, you will be asked to read over some preliminary information (theory, background, motivation, etc.) and answer some questions. This prelab assignment will be included on all experiments, but the questions and readings will be tailored to each experiment to make sure you are prepared to start working as soon as you get to lab.

The prelab assignment is due before lab begins and should be submitted to Canvas. There are no exceptions. For example, if your lab begins at 1:30 pm and you submit at 1:31 pm, you will receive zero credit. (Likewise, if your lab begins at 2:00 pm and you submit at 2:01 pm, you will receive no credit.)

For this experiment, read over the full theory section below, and answer the following questions (in a separate document). Some of these can be answered based on what you find in the theory section, but others may require outside research online.

  • What is an interaction cross section? You do not need to know how to derive it, but you need to know what it is conceptually and understand the relationship between a cross section and a linear attenuation coefficient. We give a brief description below, but search the internet for more information.
  • What is Thomson scattering? This will form the model against which we will test data in this lab. What does Thomson scattering predict for the cross section of gammas interacting with electrons? What are the limits of this model?
  • Similarly, what are Compton scattering, the photoelectric effect, and pair production? At what gamma energies (roughly) do these interactions with electrons occur?
  • How does a sodium iodide (NaI) detector work? What are the different features in a pulse height spectrum? You can read about photomultiplier tubes (PMTs), sodium iodide (NaI) crystals, and pulse height spectra here.

Python tutorial (10 points)

In parallel with your first lab, you also will need to complete the Python Tutorial assignment. Completion of the assignment will count for 10 points towards your first lab.

This assignment will be due by Friday, October 18 @ 5:00 pm, but we encourage you to get started on it as soon as possible. There will be many opportunities to use Python in the first lab, so the earlier you feel comfortable with it, the earlier you can start using it as an analysis tool!

LaTeX tutorial (optional)

We also provide a (completely optional) LaTeX tutorial here.

You may write-up your PHYS 211 assignments in any program you like (e.g. Microsoft Word or Google Docs), but we strongly suggest that you take this as an opportunity to learn professional scientific typesetting with LaTeX

Theory

Interactions of photons with matter

Photons interact with matter in the following three ways:

  • photoelectric effect: a photon is fully absorbed by an atom, liberating an electron from the shell in the process; probable only at low energies where the incident photon energy is comparable to electron binding energies.
  • Compton scattering:  a photon electromagnetically “collides” with an electron, giving up some energy which the electron takes as kinetic energy; possible at all photon energies.
  • pair production: a photon spontaneously splits into an electron-positron pair; possible only when the photon possesses more energy that twice the rest mass of the electron, $E \ge 2m_e c^2$ .

In Fig. 1, the linear attenuation coefficient is plotted for aluminum, as are the various components which contribute to the total. (A higher resolution PDF version – as well as PDFs for other materials – is available on the Linear Attenuation Coefficient Plots page.) Notice that this quantity is highest at low energies, but decreases by about four orders of magnitude as the energy changes from 10 keV to 10 MeV.

A figure showing the linear attenuation coefficients for aluminum. There are a total of four plots in the figure. One is attenuation due to the photoelectric effect, dominating at low energies. The second is due to Compton scattering, which is most relevant at intermediate (greater than 20 keV) energies. The third is for pair production, which only begins when the energy is twice that of the rest mass of the electron, but grows quickly. Finally the total attenuation coefficient is plotted, which is a sum of the other three factors.
Figure 1: The linear attenuation coefficient for aluminum is plotted with individual contributions to the total shown. To read the logarithmic axes, you must interpret the values of the grid lines between numbered tick marks appropriately. Each grid line represents one tenth of the value of the numbered tick mark preceding it – and all gridlines between numbered marks represent the same numerical change even though they gradually become more closely spaced. For example, starting at the 0.2 mark on the y-axis, the next grid lines correspond to 0.22, 0.24, 0.26, … , 0.48, and finally 0.50 where there is a new numbered mark. The next set of gridlines will then be worth 0.05 (one tenth of 0.5) and the values continue as 0.50, 0.55, 0.60, … [Source: Harshaw Radiation Detectors Catalog]
Cross section

As a photon passes through a material, there is a probability of interacting with an electron (through one of the above interactions) which increases with thickness. After such an interaction, the photon either disappears (e.g. it is absorbed in the photoelectric effect or is converted to mass in pair production) or is scattered into a different direction (as in Compton scattering). Either way, we don't observe the actual process, but instead see only the consequence: for a steady number of photons incident on a material, the number of photons which pass through it without interaction decreases as the thickness of the material increases.

To see why this is the case, consider the following model, illustrated in Fig. 2.

A figure illustrating the origin of attenuation increasing with thickness. The material has several circular regions, representing electrons, each with an effective area of sigma. As the material becomes thicker (dx increases), there is a higher likelyhood of passing through a region containing an electron.
Figure 2: A slab of material with a face of area A and thickness dx has an average number of electrons $NAdx$, each with “effective area” (or cross section) $\sigma$

Suppose we have a a small volume of material with a face of area $A$ and thickness $dx$. If this material has an electron density (that is, a number of electrons per unit volume) of $N$, and each electron has an “effective area” $\sigma$, then the fraction of the area which is covered in electrons is $(NAdx)\sigma /A = N\sigma dx$. (We know that electrons are point-like, but they can interact at a distance through the Coulomb interaction… hence the “effective” area.)

If a beam of photons with rate of intensity $R$ (number of photons per unit time) is incident on our slab, then the intensity will be reduced by an amount

$dR = -N\sigma dxR$. (1)

Rearranging and integrating, we find

$R = R_0e^{-N\sigma x}$. (2)

The rate of photons decreases exponentially as the thickness of the material increases. In this context, we call $\sigma$ the total interaction cross section. The standard unit for the cross section is the barn, where 1 barn = $10^{-24}\textrm{cm}^2$. As mentioned above, the cross section is energy-dependent.

The electron density can be computed from the material's mass density by

$N = \rho Z/A$. (3)

where $\rho$ is the mass density (mass per unit volume), $Z$ is the atomic number (number of electrons per atom) and $A$ is the atomic mass (average mass of one atom). It is common to rewrite this equation not in terms of the cross section, but in terms of the linear attenuation coefficient, $\lambda = N\sigma$ such that

$R = R_0e^{-\lambda x}$. (4)

The standard unit for the linear attenuation coefficient is the inverse centimeter, cm${}^{-1}$.

Inverting the linear attenuation coefficient gives us the radiation length, $X_0 = 1/\lambda$. The intensity of a beam of photons through a material will decrease by a factor of $1/e$ in one radiation length. Therefore, a material will make an effective “shield” against gamma radiation of a particular energy if it is at least several radiation lengths thick.

Sodium iodide detector and photomultiplier tube

Think back to Sec. 2.1. There, we spoke of how photons interact with a material; in each possible process, the photon gives up some or all of its energy to the material as it passed through. If we had a way to measure the energy which was deposited in the material, we could tell when a photon was scattered or absorbed and how much energy it left in the process.

For this lab, we use a crystal of sodium iodide doped with thallium – NaI(Tl) – as our detector. Iodine is a high Z material (i.e. it has lots of electrons), so there is a large cross section for interaction. When a high-energy photon scatters in the crystal, electrons carry away the deposited energy and zip through the solid. These high energy electrons knock into other electrons and create many smaller energy events. In turn, the thallium dopant is excited, and when it quickly de-excites, new photons are emitted. Though the whole process is complicated, it is relatively fast (a few hundred nanoseconds) and the amount of light given off by the thallium dopant is proportional to the energy left by the initial higher-energy photon. This process is called scintillation.

In order to make use of this new scintillation light, the crystal is optically coupled to a photomultiplier tube (PMT). This tube is a series of plates (called dynodes), each held at a successively higher voltage potential. The scintillation photons released by the thallium are absorbed in the first dynode of the PMT and kick out electrons via the photoelectric effect. These electrons are accelerated toward the next dynode where they produce more electrons. At each stage the number of electrons kicked out exceeds the number of electrons coming in, so a small input signal gets “multiplied” into a larger output signal at the final plate. The multiplication factor of the PMT (and therefore the size of the final signal) is determined by the number of dynodes and the applied high voltage.

Summarizing, the Na(Tl) crystal coupled to the PMT reliably produces an output voltage which is proportional to the amount of energy deposited in the crystal by the incoming photon.

An animation of the processes occurring in a sodium iodide detector coupled to a photomultiplier tube. I have no idea how to better describe this than the preceding text in the section.

For more information, see NaI Detector Physics and Pulse Height Spectra.

Spectrum features

The output signal from the PMT can be examined on an oscilloscope, but the main way we will study this signal is by processing it with a pulse-height analyzer (PHA). Such a device measures the voltage (or current) of the PMT output signal and places those pulses into channel bins in a histogram according to their amplitude (or total integrated charge). Since the pulse voltage and current are proportional to the amount of energy deposited in the crystal, we can interpret channel number as a proxy for photon energy.

If we collect these pulses over a period of time, we build up an energy spectrum. A typical gamma spectrum has several features, as illustrated in Fig. 3:

  • If a photon manages to deposit all of its energy in the detector, the PMT signal will be proportional to the full photon energy and we will find a Gaussian peak centered on the true photon energy. This is called the full energy peak or photopeak.
  • If the photon Compton scatters once before escaping from the detector, it will deposit only some of its energy in the detector. There is a range of possible energies which the photon can give up, and such events form a continuous region called the Compton plateau or Compton shelf. This plateau, however, does not extend all the way to the full energy peak; there is, instead, a maximum energy transfer which occurs when the photon is scattered through an angle of $180^\circ$. This maximum defines the Compton edge.
  • Some photons initially travel away from the detector before scattering and changing direction back toward the detector. Such photons are called backscatter photons. In principle, backscatters can occur at many different angles leaving photons with many different energies. However, some geometries are such that scatters through particular angles are more likely to scatter back into the detector than others. For the vertical geometry used here, we are susceptible to 180-degree backscatters – photons which initially travel away from the detector, scatter off the table top, then pass back through the source into the detector.
  • There is a nice relationship between the energies of the full energy peak $E_\gamma$, the Compton edge $E_{CE}$ and the 180-degree backscatter peak $E_{BS}$ : $E_\gamma = E_{CE} + E_{BS}$.
A plot of the gamma spectrum produced by Cesium 137. There is a prominent Gaussian peak to the right (at relatively high energies). Following the spectrum left (to lower energies), there is then a small area of relatively low count rate activity, preceding the Compton edge. The radiation intensity increases rapidly at the Compton edge until it levels off to a nearly stable Compton plateau. However, at yet lower energies more features appear, such as a prominent gaussian backscatter peak overlaying the Compton shelf. Finally, at the lowest energies, there is a very large peak of low energy, background gamma radiation that is commin in most atomic spectra.
Figure 3: A pulse-height spectrum produced by Cs-137 with the prominent features labeled. The energy of the main photopeak is 662 keV.

For more information, see NaI Detector Physics and Pulse Height Spectra.

Using the PHA Software

PHA setup

For more information, see SpecTech Spectrometers and Software

Starting the software The software we will use for this experiment is called USX. It is designed to control and communicate with the SpecTech UCS-30 pulse-height analyzers. A link to the program can be found on the computer desktop. Before starting the software, makes sure the PHA is on and that the USB cable is connectedto the computer. If the program initializes correctly, a box saying “Loading hex files” should briefly appear and the device name should appear along the title bar. If you see the warning “No device connected” or if some buttons on the software appear greyed out or unresponsive, the program may not have loaded properly. Try turning the PHA off then on again and restarting the program. If that does not work, restart both the PHA and the computer. Sometimes, usually after a system update, Windows will need to reload the drivers for the UCS PHA. If the loading drivers message comes up when the PHA is first turned on, allow the process to finish. There are two drivers which need to be loaded and it typically takes a minute or so.

Setting the high voltage

The PMT high voltage (HV) for this experiment will be supplied by the PHA and the value is controlled in software. Make sure the high voltage cable from the PMT (the cable coming from the red output port labeled “HV”, not the BNC signal cable from the output labeled “sig”) is plugged into the positive voltage output on the PHA and turn the voltage on by clicking the “OFF” button in the upper left corner. It should now read “ON” and be red. (Yes… this is confusing notation. We can't change it.) The light on the front of the PHA indicating HV should change from green (off) to red (on).

A screenshot of the high voltage controls, illustrating what the setup looks like with the voltage off. A screenshot of the high voltage controls, illustrating what the setup looks like with the voltage on.
High voltage when turned off High voltage when turned on

Set the high voltage to +1000 V. (Note that the voltage setting box does not display units… only numbers. The hidden units are in fact “volts”.) You may adjust the voltage down if needed later, but do not exceed +1000 V.

“Good” geometry

In order to perform an accurate count measurement, it is necessary to choose appropriate geometry. If we assume a point source of gammas and a detector with a circular face toward the source, then the gammas from the source which arrive at the detector all fall within the cone shown in Fig. 4.

A figure depicting the orientation of a photomultiplier tube (PMT), the absorber, and radioactive source relative to one another. A shaded cone between the source and PMT illustrates the possible path of detected gammas. The absorber is then placed such that any gamma that would be detected must pass through its entire thickness.
Figure 4: An illustration of “good geometry”. Note that the absorber is of a slightly larger diameter than the cone formed by the gammas emitted by the source and that the material is placed approximately equidistant between source and detector face.

Clearly, placing a scatterer between the source and the detector will reduce the number of gammas reaching the detector. However, if the scatter is smaller than the cone, some gammas will miss the scatterer and still be detected. On the other hand, if the scatterer is larger than the cone, some gammas which would have missed the detector will nonetheless scatter into it, artificially increasing the number of detections. A good compromise is to place the scatterer halfway between the source and the detector, with its diameter just slightly larger than the cone size at that position. We will call this setup good geometry.

As you change absorber thickness during the experiment, you may need to adjust the absorber holder position to maintain the absorber bulk at approximately the midpoint. Do not, however, change the overall source to detector distance once you have begun collecting data on a particular source.

Radioactive sources

We will use several different gamma-emitting sources for this experiment. Each of these sources are low-strength (on the order of microcuries) and are housed in small plastic containers called buttons. The sources we will use (and their most prominent photon energies) are the following:

A disc source (also known as a button source) containing a sample of radioactive Sodium 22. It consists of an inch-wide plastic disc that is approximately an eighth of an inch thick. In the middle is a small sample of radioactive material, sealed in with epoxy. The front face of the disc contains a sticker with information about the isotope, the date it was created, and the manufacturer. A disc source for Barium 133 A disc source for Cesium 137
Na-22 (511 keV and 1.27 MeV) Ba-133 (31 keV, 81 keV, 356 keV, and others)Cs-137 (32 keV, 662 keV)

You should collect a full range of data using all three sources, though we suggest you start with either Cs-137 or Na-22. Note that Ba-133 does emit more than three decay gammas, albeit at low intensities. See the barium decay scheme for details.

Experimental procedure

Overview

The goal of the experiment is to make measurements of the total interaction cross section for gammas from about 30 keV up to about 1.25 MeV for one absorbing material – either aluminum (Al) or iron (Fe). To do this, you will need to measure the fraction of gammas of a particular energy that are able to pass through absorbers of different thickness. Plotting this fractional absorption versus absorber thickness should yield an exponential relationship whose decay constant is related to the total cross section for the material.

  • How will you determine the cross section (or, relatedly, the linear attenuation coefficient) for a given energy?
  • What are the raw data you are collecting? What are the limits of how well you can measure each quantity?

Lab notebook

You are expected to keep a record of your work in a permanent lab notebook. There is no exact formula for what should go into a lab notebook. A good rule of thumb is to record everything which you would need to refer back to if you wanted to exactly reproduce your experiment at a later time, or that you might need to know when writing a paper on your results or explaining to a colleague what you did and how you did it.

For this experiment, things like the distances between the source, absorbers and PMT should be recorded. Likewise, operating parameters like the high voltage setting for the PMT or any amplifier gain applied in the PHA should be recorded. You will be collecting digitized pulse height distributions on the computer which can be saved and used for analysis both in and out of the lab. Your lab notebook should have a record of each measurement including saved file name, source used, absorber used, date, etc., so that you know everything which went into collecting that pulse height spectra. After you leave the lab and sit down to do your analysis, you likely will not remember all of these details, so it it critical to record them.

We recommend (highly!) that while you are in the lab waiting for your apparatus to collect data you should be doing calculations and making quick plots of the data in order to evaluate how things are going and ascertain any corrections you may need to make. These calculations and plots should all go into your lab notebook.

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 40% of the grade of this lab.

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!

1. Verifying detector operation and identifying spectrum features (5 points)

No scientist does an experiment just by turning equipment on and taking data straight away. A significant amount of effort goes into making sure that you understand how your apparatus works and ensuring that everything is operating correctly. You will spend plenty of time tracking down why your experiment is not working as you expect, and this is also part of doing science.

For this task, you will familiarize yourself with the detector operation and look at pulses on the oscilloscope. You will take some sample data, and you will use this preliminary data to help you determine a good collection strategy.

To verify the operation of your detector system, do the following. (There is no need to estimate uncertainties for these tasks).

  • Connect the output of your photomultiplier tube (PMT) to the digital oscilloscope. PMTs produce fast pulses, so you will need to use the 50 $\Omega$ terminator provided. Be careful not to connect the HV supply of the UCS-30 to the input of the scope. (If you are unsure of which connections to make get assistance from an instructor or one of your fellow students.)
  • Apply +900V to the PMT and place a Cs-137 source up against the sodium iodide (NaI) scintillator crystal.
  • Find the PMT pulses on the scope. The output pulses will have negative polarity, a fast risetime (on the order of 10 ns) and a longer decay time (on the order 100s of ns). The amplitude of the pulses will vary based on both the high voltage applied and the energy of the gammas entering the NaI crystal, but for this PMT, typical pulses should be somewhere in the range of 10s or 100s of mV.
  • Obtain a stable trigger on the PMT pulses and use the cursor function of the scope to measure the amplitude, rise time, and decay time of a typical pulse. Make a sketch of the pulse in your lab notebook and indicate relevant features.

Though looking at the signals on the scope is useful as a diagnostic tool, it is a very difficult way to do counting measurements. Therefore, the main way in which you will collect data for this experiment is through pulse height spectra. You will need to understand the features of these spectra in order to properly select regions to be analyzed.

Disconnect the signal from the scope and connect it to the PHA. (Make sure to leave the 50 $\Omega$ terminator connected to the scope. Do not use the terminator on the PHA input.) Start data collection and adjust parameters as necessary to see the spectrum clearly.

Do the following:

  • Sketch the pulse height spectrum you see in your lab notebook, and identify/annotate the features which you observe. Use the cursor function in the USX software to measure the channel (or channel range) in which each feature occurs.
  • Use your knowledge of how the different features are formed and the known energies of the two gammas to verify the features you have identified. (Note that the pulse height channel on the $x$-axis of the spectrum is proportional to the energy deposited in the NaI crystal. For example, if one feature has twice the energy of another feature, it should appear at twice the channel as the first feature.)
  • Save the spectrum as both a *.spu file (which can be reopened using the USX software) and as a text file (either a comma-separated file, *.csv, or a tab-separated file, *.tsv) file. The text file can used to replot the data using Python later (or can be read into Excel or some other spreadsheet program to manipulate or plot).

Complete all relevant calculations in your lab notebook. Be prepared to discuss the spectrum with instructors.

Help, my .spu file is damaged!

If your .spu file is malformed and you didn't save a .csv file, there may still be hope. Put the .spu file in the same directory as the Python notebook linked below and change the line fname = “YOUR_FILE_HERE.spu” to include the name of your file. It should then save the file with a .csv suffix with your data enclosed.

interpret_spu.ipynb

Identifying and assessing uncertainties

When doing an experiment one of the most difficult but important things you need to understand is how well you know your measured quantities. If you are trying to show agreement or disagreement with a theoretical prediction or a model, or trying to measure a fundamental property of a particle like the mass of the Higgs Boson to a higher degree of precision, you must know how to assess the uncertainties in your measurements – both statistical and systematic.

Before we dive into things, think about the following general questions:

  • What are the different quantities that you need to measure in this experiment? (Remember… you don't measure gammas and you don't measure aluminum… be specific.)
  • For each of the quantities, think about what will limit how well you make that measurement. Consider both random measurement fluctuations and systematic biases.

Estimating the number of counts in a photopeak

The total number of counts in a photopeak ($N$) is the primary measurement you will be making in this experiment. Unlike measuring the time or thickness, estimating $N$ involves a lot of subtle questions and you have to make decisions about how best to accomplish this. It would be easy if all of the pulses in the spectrum represented a detection of the gamma of interest; however, this is not the case. Some of the pulses are due to random noise fluctuations in the PMT or background radiation in the room not related to the source you are measuring. Some radioactive sources emit more than one gamma energy, resulting in multiple overlapping spectrum features.

As the experimenter, it is up to you to make decisions about how to analyze the data. The purpose of this course is NOT to tell you some so-called “right” way to make a particular measurement so that you get the “right” result. The purpose is to teach you how to identify the important considerations and how to deal with them yourself (because in research there is no one to tell you the “right” way of doing it).

We will suggest two ways one can go about estimating the number of counts corresponding to a feature in a spectrum. The first way is to use the functionality built into the USX software to define a region of interest. The second way (which may or may not be better depending on a multitude of factors) is to save the data from the spectrum and fit the photopeak to a functional form which accurately represents both the signal of interest and the background component.

Using your Cs-137 spectrum (with no absorber present), explore the two methods as follows.

2. Determining count uncertainty: Using the USX software (5 points)

The USX software allows you to use the cursor to read off the number of counts in an individual channel. You can also click and drag the cursor to highlight a region of interest (ROI) in the data and the software will provide the following statistical data about that region: the gross counts, the net counts, the centroid, and the full-width half-maximum (FWHM) of the distribution. You can use the help function in the software to get more details on what the software is doing.

If you understand what the software is doing (and you understand the features in the spectrum that you are studying), the method of defining an appropriate ROI and using the net counts as your estimate of $N$ typically results in “good” results. Doing so, however, does require making well-informed decisions about where to place the boundaries of the ROI.

Do the following:

  • Use the USX software to estimate the number of counts in the 662 keV photopeak, and determine the uncertainty in that estimate.
    • Note that the software begins by determining the gross counts which have a statistical uncertainty. It then estimates the number of counts in that region which are due to “background”, a number which also has an associated statistical uncertainty. Finally it subtracts the background counts from the gross counts to arrive at the estimate of for the net counts, which means you have to propagate the uncertainties through that calculation!.
  • Do this several times, erasing and reselecting the ROI each time to get a sense of how sensitive the resulting net count is to the ROI boundaries you choose. This has important ramifications for how you use this feature in the experiment.

To receive credit for this part of the task, instructors will want to see how you choose your ROIs – either through an accurate sketch in your lab notebook or a saved plot or screenshot. You need to record all of the measured values in your lab notebook and show the calculations and propagation of uncertainties.

3. Determining count uncertainty: Fitting the data (5 points)

Fitting your data can, if done properly, provide a better estimate for the number of counts in the photopeak after accounting for background. Fitting the data is more time intensive, however, and the results can be more challenging to assess. For this exercise, do the following (which can be done out of lab using the python scripts provided in the Python Tutorial, or the sample script below):

roi_fitter_basic_v2.ipynb

Note: There were some commands that did not work properly on a fresh Anaconda installation, Tuesday's students may want to re-download the file.

  • Import your saved Cs-137 spectrum and use the provided python scripts to perform a fit to the 662 keV photopeak. You will have to select a region of the data to fit, just as you did with defining the ROI in the USX software. You might have to try running the fit with different region selections in order to obtain the “best” fit you can.
  • The script gives you a simple model of the data to fit to – a Gaussian representing the photopeak and background consisting of a constant plus linear term. You are welcome to try fitting a more complicated functional form to the data if you decide you need to.

For this part of the exercise, instructors will want to see the final plot of your data with the best fit, the goodness of fit parameters, the best estimates for the functional parameters, and the estimated uncertainties on those fit parameters. Be prepared to discuss what these parameters tell you about your fit and why you think that the fit is good or bad.

4. Investigating systematic biases (5 points)

Not all uncertainties in measurements are due to statistical fluctuations. In fact, the most important and challenging uncertainties are those which are related to the details of how you conduct your measurements, how your instruments perform, etc. These uncertainties are collectively referred to as “systematic effects”.

In order to get a better understanding of systematic effects, we want your group to choose one effect to study. It is up to you what you want to explore, but you must be able to clearly state the issue you are investigating, make a prediction about what effect that issue might cause, and design an experiment to explore that issue. You do not need to find a definitive answer – Sometime exploring a systematic effect is an even bigger task that making the measurement in the first place! – and you do not need to necessarily quantify the uncertainty related to the issue you investigate, but you do need to convince the instructors that you have explored this systematic effect in sufficient detail.

You do not have to study the effect right now. You may want to take data and come back to this exercise later after you've learned more about your apparatus and a better idea of what sorts of things could cause bias.

To receive credit for this task we expect to see a description, including clear diagrams, of the experiment you devise in your lab notebook along with the data you collect and calculations you perform. We will expect you to articulate what you are testing for, how your experiment will work and what your results are.

For this experiment, one possible systematic effect which could in principle introduce a bias to your measured rates is the relationship between the positions of radioactive source, the absorber and the detector. We will use this particular systematic effect to illustrate the process for assessing whether or not how you setup the source-absorber-detector impacts the data you collect.

Read the Theory section on Good Geometry. There, it is suggested that the geometry you use could bias your measurements. But how do you know if it this geometry has a meaningful effect? To answer that, you will need to devise a mini-experiment.

Do the following:

  • Explore the extreme cases of placing the absorber directly on top of the source and placing the absorber directly underneath the detector.
    • Based on your understanding of what you are attempting to measure, (i.e. the total interaction cross section), what undesirable effect might these two cases introduce in terms of the measurements you will be making? We will want you to articulate how you expect a potential bias to manifest itself in the data.
  • Once you have a clear idea of what you are testing for, conduct a simple experiment which would reveal such a bias if it is in fact present. This experiment can be as simple as three measurements, or as complicated as you wish to make it.
  • From the data you collect, do you observe any impact on the data? If so, what will you do to minimize or compensate for this potential bias?

5. Obtaining the linear attenuation coefficient at 662 keV (5 points)

For your final task, before going on to “do” the experiment, we want you to go through the whole process of data collection and analysis for one single energy. Doing this will reveal how well you understand all the steps and it will give you an opportunity to fix or figure out mistakes and missing steps.

Do the following:

  • For the 662 keV gamma of Cs-137, collect a full set of rate data at whichever absorber thicknesses you think are appropriate. For each rate, determine the total statistical uncertainty.
  • Plot rate versus thickness and fit the data to an exponential.
  • Extract the linear attenuation coefficient from the fit, with uncertainties.
  • Compare your measured value to the literature value.
  • Assess how well your measurement agrees with the literature value.

To receive credit for this task we will want to see the parameters of your experiment (like PMT high voltage, distances between source and absorber and detector) in your lab notebook. Your also needs to include your raw data with uncertainties, tabulations of your calculated rates with uncertainties, calculations related to extracting your linear attenuation coefficient from the fit to the data, and the final comparison of the data to the literature value.

We provide a Juypter notebook for you to use when plotting and fitting the data. Some of this work will likely be done outside of lab, but we want to see the results before you move on to collecting the rest of your data for the full experiment.

Note, the data you collect for this exercise will form part of your full data set for the experiment as a whole. There will be no need to retake this data, so it pays to take the time to do it right and understand what you are doing.

Completing the data taking

Once you are finished with the skill-building tasks above, use the rest of your remaining time to take a complete set of data for the remaining gamma energies.

  • From Cs-137, you should be able to observe gammas at 31 keV. (You may need to adjust the gain on your spectrum to bring the peak on screen, and you likely will need to use a very different set of absorber thicknesses than you used when studying 662 keV.)
  • From Na-22, you should be able to observe gammas at 511 keV and 1275 keV. (Again, adjust the gain to optimize the spectrum.)
  • If there is time, you may also use the Ba-133 source, which provides gammas at many energies, the most prominent of which are at 32 keV, 81 keV, and 356 keV.

As you progress, do not forget to continue recording relevant information in your lab notebook as you did for the exercises.

Final data analysis

Once you have collected all the data, the goal is to determine the total cross section (including the properly propagated uncertainty) at each gamma energy you tested. You will compare your measured values to the predictions of the Thomson model, and assess how well this model represents the results of your experiment.

You are welcome to talk with your lab partner(s) to discuss the analysis, the plotting and fitting, the calculations, and conclusions. However, everything you hand in must be your own work! You have to make all of your own plots, perform all of the fits to the data, and do all of the calculations yourself. You are not permitted to divide the work up with your partner (e.g. one student doing one part of the analysis while the other does another), and you are not permitted to use your partner's code. (However, you may talk to your partner and/or instructors about coding issues and you are welcome to use and modify the python scripts which are provided as part of the course.)

A good rule of thumb is that when you sign your name to your work you are attesting that everything in that body of work was done by you. Including work done by your lab partner (or anyone else) without attribution is plagiarism, which we something we take very seriously. (And even though including work done by your lab partner with attribution is not plagiarism, we cannot give you credit if you didn't do the work yourself.)

You have one week to perform a full and complete analysis of the data you collect in lab and submit the following assignments. You can score up to a maximum of 60 points total on these assignments, as divided into three categories (each worth 20 points).

Spectrum identification and fitting (20 points)

For each radioactive source, show the full spectrum (at zero thickness) and identify all visible features. In addition, show fits to each full energy peak (at zero thickness).

Do the following:

  • Provide a spectrum of each radioactive source (at zero thickness) with features labeled.
  • Show attempted fits to each full energy peak (in Na-22 and Cs-137, at least), with the following features:
    • full functional form described (including background),
    • range of data used in fit,
    • best fit parameters and uncertainties, and
    • goodness of fit parameters.
  • Provide a discussion of the quality of fit for each full energy peak fit.

Extracting linear attenuation coefficients (20 points)

For each gamma energy, show how you extract the linear attenuation coefficient and report your best estimate (with uncertainties).

Do the following:

  • Provide plots of rate vs. thickness for all energies that you study with fits and annotations.
  • Discuss how the raw data was collected for each data point (e.g. from software, from fit, etc.) and how it was processed for the plots.
  • Discuss how the uncertainties were determined for each data point.
  • Discuss the quality of fits and discuss your iterative process to obtain final results (if applicable).

Conclusions and comparison with literature (20 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 table of results and a plot showing your final linear attenuation coefficients vs. energy (on a log-log scale).
  • Provide a residual plot showing the difference between data and literature values for each point (with uncertainties included).
  • Discuss the agreement between the data and the literature (for individual points and/or overall, as appropriate).
  • Contextualize your results. This may include asking yourself the following non-exhaustive list of questions:
    • What do your results mean?
    • What processes dominate at which energies?
    • If you disagree with expectations, do you disagree in some systematic way or is it random?
  • 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.

Literature values

In order to compare your results to literature, we provide a table of data (in text document form) combined from two different National Institute of Standards(NIST) databases: X-Ray Form Factor, Attenuation and Scattering Tables (FFAST) and X-Ray Attenuation and Absorption for Materials of Dosimetric Interest (XAAMDI) . We converted the data density values to linear attenuation for you. This data is plotted in Fig. 6 for aluminum as an example, and is consistent with Fig. 1 above.

NOTES FOR US

NIST data is calculated, not experimental.

For XCOM data, one of the major data sources is the following paper by Storm and Israel https://doi.org/10.1016/S0092-640X(70)80017-1. They note that there is around a 3% spread between their values and experimentally measured values in the range of values we care about. There are no uncertainties given on the calculations.

For an example of this cropping up in research, see https://doi.org/10.1016/0378-4363(84)90037-8. The data presented is definitely off by 2-3% in some instances when compared to interpolated XCOM values.

That paper was found from the giant list of references in Bibliography of Photon Total Cross Section (Attenuation Coefficient) Measurements 10 eV to 13.5 GeV, 1907-1993 https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir5437.pdf

A little more modern discussion of gamma energies in biophysics can be found below: http://doi.org/10.1088/0031-9155/44/1/001. Also of note is that they discuss different individual contributions to attenuation, and they've got some plots comparing actual data to theory datasets.

That work was cited in http://dx.doi.org/10.1088/0031-9155/60/4/1641, wherein the authors state in the abstract that we can possibly put tighter uncertainties on the XCOM dataset than previously thought.

A log-log plot of linear attenuation coefficient versus incident gamma energy for aluminum. There are two distinct regions: a low-energy region where the attenuation decreases quickly with energy, and a higher energy region (starting around 1 keV) where the attenuation increases much more slowly with increasing energy.
Figure 6: Plots of the total linear attenuation coefficient data reported by NIST. Aluminum Values are consistent with Fig. 1.

Extract literature values from the file and add these (along with uncertainties) to the table of measured values. You may need to interpolate between given data points in order to find an estimate at certain gamma energies used in this experiment. For our purposes, you may assume a 3% uncertainty on all the data points in the file. (However, the true uncertainties are considerably more complicated, as detailed here.) You should use your own judgement when determining the uncertainty on your interpolated value.