Gamma Cross Sections

In order to prepare for the lab, you should read over the full theory section below (Sec. 2) and in particular pay attention to the following questions. Some of these can be answered based on what you find in the theory section, but others may require outside research online. Make sure you can answer the following before you arrive for Day 1:

  • 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.
  • 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?
  • How does a NaI+PMT detector work? What are the different features in a pulse height spectrum?
    • In addition to the information below, you can read about photomultiplier tubes (PMTs), sodium iodide (NaI) crystals, and pulse height spectra here.

1 Goals

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 ~100 keV to ~1.5 MeV as they pass through different elements. You will use these measurements to test how well the Thomson scattering model predicts the cross section, and to look for evidence of different or additional processes that may also be present in addition to Thompson scattering.

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 – as described below in Section 2.1 – 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) the model makes reasonably accurate predictions over a range of energies. We want you to take data precisely and with care so as to probe exactly how far we can push this model: where does it hold, where does it break, and what can we learn from how the data deviate when it does break.

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.

2 Theory

2.1 Interactions of photons with matter

Click here to see details on how photons interact 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]

2.2 Cross section

Click here for a description of what a cross section is.

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.

2.3 Sodium iodide detector and photomultiplier tube

Click here to see the content of this section.

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.

2.4 Spectrum features

Click here to see the content of this section.

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.

3 Experimental procedure

In a very general sense, the process of doing experimental physics can be broken down into the following three parts:

  1. Research: In this part, you conduct background research in order to understand what you are measuring, how you will go about making the measurement, and what factors are going to limit how well you can make the measurement. Mostly this is done by going to the literature and learning about the current state of activity in the field or about the tools available.
  2. Execution: This is the part where you assemble your experiment, collect and analyze data, and determine where improvements need to be made. This process is repeated – likely coupled with more research – until you have obtained a scientifically significant result and you are confident that you fully understand all aspects of the experiment.
  3. Conclusions: This is where you finalize the full analysis of the data (and any associated uncertainties) and where you make connections between your work and the work that has come before. This may involve comparing data to expectations or to the results of others, or it may mean finding explanations for what you found on your own. You may ultimately write up the results (e.g. for publication in a journal), prepare a presentation (e.g. for a colloquium or conference) or share the results more informally.

For this first experiment, we are going to concentrate mostly on the Execution phase of the process. You will have two 4-hour lab periods to learn how to use the apparatus, investigate how best to perform the necessary measurements, and understand the limitations of the data you collect. When you leave the lab at the end of the second period it is not sufficient to just have a bunch of data… you need to know the following:

  • …that you have sufficient data to ensure that you get good results. This means you should have done a quick analysis of the data while in lab in order to obtain preliminary results and make sanity checks. If you find out your data is incomplete or faulty at home, it is harder to fix it.
  • …what the important experimental uncertainties are and to understand how they impact your interpretation of the data. This means that you need to have identified all potential sources – statistical and systematic – and taken steps to mitigate their influence or collected necessary data to quantify their impact.
  • …that you understand what you did and how you did it well enough to justify whatever conclusions you draw from the data. Make sure you fully understand the equipment. Make sure you know how the data and model are related. Make sure you get answers to your questions.

Since this is your first experiment, we will walk you through much of the process, pointing out what the important considerations are along the way. In future experiments you will be expected to do more and more of this on your own.

3.1 Overview

The goal of the experiment is to make measurements of the total interaction cross section for gamma rays from 81 keV to about 1.27 MeV for one absorbing material – either aluminum (Al) or iron (Fe). Doing this essentially boils down to measuring 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.

In order to perform this experiment you are provided with the following:

  • Radioactive sources which emit gammas at known energies.
  • A set of absorbers of different thicknesses.
  • A PMT+NaI detector which produces an electrical pulse when struck by a gamma.
  • A pulse height analyzer (PHA) which collects the pulses from the PMT+NaI detector and displays a histogram of pulse sizes.
  • A caliper to measure absorber thickness.

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

Note you are NOT “measuring gamma rays”! Gammas which strike the detector produce pulses, and it is these pulses which you are actually counting. You will need to make a determination of which pulses in the PHA spectrum correspond to gamma rays of a particular energy, and then you will have to determine the best way to estimate the number of these pulses. You also have to know how long the apparatus was counting (in order to get a rate of pulses detected) and you will have to measure the thickness of the absorber.


For each measurement, we have three basic quantities which we are measuring:

  • The thickness of the absorber.
  • How much time was spent counting.
  • The number of pulses corresponding to a particular gamma energy.

Consider the things that limit how well you can measure each of these quantities. For example…

  • You will use a caliper to measure the thickness of the absorber. Assuming the caliper is properly calibrated, your uncertainty in these measurements could be due to how well you can interpolate between tick marks on the scale or it could be due to variations in the actual thickness of the absorber due to irregularities in its faces.
  • The counting time is reported by the PHA. Assuming that it is properly calibrated, how well do you know that value?
  • You are counting pulses produced by gammas from a radioactive source, and you know that they are emitted at random but with a well-defined rate determined by the lifetime of the source. The number of gammas counted in a given interval of time is described by the Poisson distribution and the uncertainty in the number of counts measured is $\sqrt{(N)}$ where $N$ is the total number of counts recorded. (It may be helpful to explore our page on Poisson statistics.)
  • Real data is messy. In this experiment, there will be unwanted background pulses mixed in with the pulses you want to count. These background pulses may come from other sources of radiation in the room (including sources you may not be able to remove, such as natural radiation in the building materials or radiation from cosmic sources), thermal noise in the PMT, or even from the specific radioactive source you are use (e.g. in the case where the Compton edge or tail of a photopeak from one energy gamma overlaps with the photopeak of another gamma). You will have to decide how you will determine how many of the pulses you count are due to background and work that “correction” into your calculations. Incomplete or improper estimation of the background could be a source of systematic uncertainty (meaning that the uncertainty is not governed by random statistics, but instead is a bias that applies to all measured data in the same way). Quantifying systematic uncertainties are often more difficult than quantifying statistical ones, and so you may have to conduct some short experiments (or perform different exploratory analyses) in order to get an idea of how to treat them.

You will also have to propagate uncertainties through calculations. While you were introduced to this in your undergraduate labs, you may find it useful to refresh your memory here.

Parameters determined by the user

In addition to the measured quantities, there are a number of other important decisions you need to make about how you will conduct the experiment.

  • Where to place the source and the detector. Placing them closer together will give higher counting rates, but you need to space to fit absorbers in between.
  • Where exactly will you place the absorbers? Next to the source? Next the detector? At the mid point between the two? Each option has consequences for your measurements.
  • How many different thicknesses should you measure and what is the thickest absorber you need? There are no right answers here, but instead is a matter of trade-offs. Using more absorbers and collecting for longer times will give you more data points to work with and smaller uncertainties on the number of counts. However, you have a finite amount of time in which to collect all the data you need, so you need to decide on where to strike the balance. Two hints:
    • Don't blindly go through all the absorbers in sequence; this amounts to relying on luck to get good data. Instead, start with the thinnest and thickest absorbers you have, and make these the two end points of your data set. In that way, you will have some idea how long you need to count for the thicker absorbers, and can then make an informed decision on how long you should count on the ones in between.
    • Ten is not the magic number of data points to take. Plot your data as you take it, point by point. Keep in mind that you are expecting to see an exponential and that you eye can guide you at determining when you have enough data points to define the shape of the curve well.

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.

It may be helpful for you to know that this is not just an academic exercise in keeping a lab notebook for the purpose of getting a good grade. You will in fact have to rely on the contents of your lab notebook in later quarters. You will do at least one experiment where you revisit an experiment you have already done in order to extend the measurement. You will have to rely on your notebook to guide you. You will also have to write up a full journal article on one of the experiments which you did, and again you will rely heavily on the information in your lab notebook in order to recall all of the details of what you did months ago. So… you have a vested interest in keeping good records of what you do in lab.

3.2 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.

NOTEBOOK: Sketch the apparatus and connections. Note the approximate distance from source to absorber and from source to detector.

3.3 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 The icon for the UCS 30 spectrometer software 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.

Looking at signals on the oscilloscope

When the PMT is powered, photons incident on the detector will produce output signals. Before using the PHA to analyze these signals, we want to look at them on the oscilloscope.

Plug the output of the PMT (the smaller BNC cable) into one of the channels of the oscilloscope. (Make sure to use a 50$\Omega$ terminator to prevent reflections.) Place a radioactive source close to the face of the detector and find a signal on the scope.

Pulse height analyzer (PHA)

Now disconnect the PMT output from the scope and plug it into “Input” jack on the back of the PHA. Note that the 50 Ω terminator should not be used with the PHA; it should remain attached to the oscilloscope. In the USX program under the menu option “Mode”, make sure that the software is set to “PHA - Pre-Amp”.

When the software tells the PHA to collect (by hitting the green diamond, “Go”), each incoming analog signal is measured, digitized and counted, producing a histogram of these counts based on the voltage of each pulse. Again, as the pulse voltage is proportional to the energy deposited in the crystal, the channels along the x-axis are proportional to gamma energy. Though we will not need to do so for this experiment, the axis can be calibrated from channel into true energy. 

In “Pre-Amp” mode, the PHA can amplify the signal before binning and display. The amplification factor is called the gain and be set in software, with the coarse gain (x1, x2, x4, x8, x16, x32, or x64) and/or the fine gain (continuous from x1 to x2.5) options.

Begin collecting a spectrum to make sure that the equipment is working and to familiarize yourself with the software. You may stop the collection by hitting the red button and erase (without stopping collection) by hitting the “X”.

3.4 Collecting and recording data

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 approcimately 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.

Counting procedure

In this section, we outline the procedure for collecting count rate as a function of absorber thickness. Begin with either the Cs-137 or Na-22 source, and repeat this procedure for each additional source.

When beginning with a new source, do the following:

  • Take a quick spectrum (with no absorber) to identify features.
    • Use the energies listed above to know what peaks to look for. See Nuclear Decay Schemes to understand the origin of these gamma energies.
  • Adjust the gain settings in the software so that all features are visible, but spread over the full channel range.
    • Do not change the high voltage unless instructed to do so by a TA or staff member. Doing so will have a similar effect as changing the gain, but pushing the voltage too low can affect overall count rate, while pushing the voltage too high can damage the detector.
    • There is no need to calibrate the x-axis; record peak positions in channel and infer energy based on the spectrum features.
NOTEBOOK: Record the high voltage and gain settings for this sample. Draw a quick sketch of the spectrum (to scale) with features identified (by channel and energy, if appropriate).
  • For each peak on the spectrum, set a Region of Interest (ROI) around the peak.
    • When running, the software will compute both gross counts (all) and net counts (gross counts minus a linear background formed by drawing a line from the left edge to the right edge and subtracting off points below that line) within that ROI.
    • Keep in mind how the background is set when choosing the ROI. Include enough of the peak so that significant counts are not lost, but do not go too far beyond the peak since we need a background that can be reasonably modeled as linear.
    • Once set, KEEP THE ROI FIXED for all absorber thicknesses.
NOTEBOOK: Record the ROI settings for each peak. (This is essential in case you ever need to reset ROIs after a software crash or in case they are otherwise lost.)
  • Now, clear your sample spectrum and collect fresh spectra over a range of absorber thicknesses (including zero thickness).
    • For each thickness, run long enough to get “good statistics”.
      • If the total number of counts under a peak is $N$, then the statistical uncertainty is $dN = \sqrt{N}$).
        • Note that this assumes that there is no background or that the background is removed completely with no uncertainty. This is rarely the case. You may need to think a bit about how to account for additional uncertainty due to the background.
      • “Good statistics” means that the fractional uncertainty, $dN/N$, is few percent. In some cases this can be achieved in a few seconds. In other cases, this will require an inordinately long time and you will have to settle for less.
    • At the end of each run, save the spectrum in both “spectrum” (*.spu) and “tab-separated” (*.tsv) formats.
      • A *.spu file can be read only by the SpecTech software. This file is needed if you want to look at the spectrum again while still in lab of if you ever need to make adjustments to ROIs after a run.
      • A *.tsv can be opened as a text file or read in by a program like python or Excel for analysis. This file is needed to reproduce the plot at home.


IMPORTANT: Choose absorber thicknesses intelligently. A good strategy is to take a zero thickness measurement followed by a very thick measurement. If the count rate has fallen sufficiently (e.g., by a factor of 5-10) at this second point, you now have two endpoints; fill in the intermediate thicknesses appropriately.

  • Do not necessarily proceed in fixed thickness steps.
  • Do not necessarily proceed thinking that each provided absorber thickness represents a data point. (You may stack more than one absorber together to get a greater thickness and you need not use all absorbers on all sources.)
  • Choose enough points to map out the whole range; 5-8 should be sufficient if your endpoints are appropriate.
  • You do not have to use the same set of thickness for each source.
  • You may move the absorber holder up or down to maintain “good geometry” and keep the absorber centered between source and detector. However, do not change the total distance between the source or detector once you begin collecting data.
NOTEBOOK:  In a data table, record the thickness, live time, gross counts and net counts for each peak. Compute the rate at each thickness as you go. You may also want to keep track of other quantities such as uncertainties; use your own judgement.

Record the filename for each spectrum (or, alternately, describe your naming scheme somewhere in your notes and stick to it.) Also, note any anomalous behavior you see as you collect data and record other stray observations. Remember that your notebook is a record of all the work you do; more is better!

4 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.

4.1 Spectra

For you submitted analysis, we want to see one of your spectra fully discussed and annotated.

Choose one spectrum (for a source of your choice) and replot the data (from the *.tsv file) using Python. Identify all the features of the plot including full energy peaks, Compton edges and backscatter peaks (if visible). This can be done by adding annotation text on the plot, by adding annotations by hand after the plot is made, or by making careful notes in the figure caption.

Chose one of the full energy peaks and perform a fit to a Gaussian with a linear background,

$f(x) = \dfrac{N}{\sigma \sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} + Ax + B$. (5)

where $N$ is the number of net counts in the peak, $\sigma$ is the standard deviation, $\mu$ is the centroid, and $A$ and $B$ are the coefficients of the terms in the linear background. Report the values and uncertainties of each fit parameter. These values may be given as an annotation in the plot, or separately as a table of data. Be careful to include units and report an appropriate number of significant figures!

4.2 Linear attenuation

Next, you need to determine the linear attenuation (and then, by conversion, the total cross section).

For each energy (i.e. each peak you monitored during lab), plot the count rate as a function of absorber thickness. (You will have 4-7 energies total, depending on which sources you used and which peaks you were able to monitor.) Include error bars on each data point and use these uncertainties when computing a best fit to the function of the form

$R(x) = R_0 e^{-\lambda x} + B$. (6)
ANALYSIS: What is the purpose of including the constant B?

Each energy plot will yield one linear attenuation coefficient. Using the appropriate electron density for your absorber material (see Sec. 2.2), convert each linear attenuation coefficient to a cross section.

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.

 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: A plot 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.

ANALYSIS: Compile into a single table the following values and uncertainties at each energy:

  • measured linear attenuation coefficient (from fit),
  • literature value for the linear attenuation coefficient (from NIST data), and
  • measured cross section (from calculation using measured linear attenuation coefficient).

Comment on the agreement between your measurements and the literature.

4.3 Comparison with the Thomson scattering model

As mentioned at the start of the experiment, one of the earliest models for how a gamma interacts with an electron is the Thomson scattering model. We may now look at individual energies and determine how well the value of the cross section predicted by the Thomson model compares to the measured values.

ANALYSIS: For each energy studied, compare the Thomson cross section to the measured cross section. Are they in agreement? If not, is the measured value higher or lower (by a statistically significant amount) than the Thomson prediction. If the measured value is higher, what does this mean? If it is lower, what does that mean? Can you determine which process (photoelectric effect, Compton scattering or pair production) dominates at each energy explored in this experiment?


Normally, we will not provide such an explicit guide for the report, but for this first experiment, you can use the rubric below to make sure that you are covering all the topics you need to.

Item Good Adequate Needs Improvement Inadequate
Data handling Shows the PHA spectrum for one source with relevant features clearly identified.

Describes the process for extracting measured quantities from the raw data.

Includes rationale for method of choosing each region of interest.

Includes a Gaussian fit of one full energy peak with background along with all resultant fit parameters and uncertainties.
Presentation of how the raw data were handled is mostly clear, but one or more elements is missing or confusing.

Plots are not properly labeled, annotated, scaled, etc.

Some aspect of how data was handled demonstrates misunderstanding of experimental technique.

Gaussian fit is attempted, but flawed.
Overall description of data handling is confusing or important aspects are omitted which make it unclear how the raw data were processed.

Relevant parameters are incorrectly interpreted or the wrong parameters are measured.

Gaussian fit is missing.
Attempt made, but is incorrect or missing appropriate discussion.
Uncertainties Provides uncertainties in measured count rates.

Identifies and justifies dominant uncertainties.

Provides adequate calculations and description to make clear how uncertainties are propagated through calculations.
How the uncertainties were handled is made mostly clear, but one or more elements are missing or poorly presented.

All elements are present, but discussion lacks clarity.

Dominant uncertainty is not justified.
Uncertainties are presented without explanation or are incorrectly justified.

Propagation formulas are missing or misused.
Treatment of uncertainties is inadequate or missing important information.

Statements are made which are contradicted by the data.
Linear attenuation coefficient For each gamma energy, provides an appropriately labeled and annotated plot of the data and fit.

Clearly presents linear attenuation coefficients and associated uncertainties.

Correctly interprets goodness of fit parameters.

Identifies and adaquately discusses any anomalies in the the data.
All elements are addressed, but one or more elements are poorly presented.

Plots are not properly labeled or annotated, or are confusing to read.

Minor anomalies in data are present but not pointed out and discussed.
Goodness of fit parameters are incorrectly interpreted.

Major anomalies in data are present and clearly affect the final result or suggest that there were important mistakes made in collecting and analyzing the data without being accounted for.
Linear attenuation values are given, but no plots or fits are shown.

Attempt made, but is incorrect or missing appropriate discussion.
Identify dominant absorption mechanism Uses appropriate graphs or tables to compare measured linear attenuation coefficients with literature values.

Identifies dominant absorption mechanisms for each gamma energy measured.

Compares data to Thomson model and provides some justification for the why the measured value may be higher or lower than predicted.
Data is not presented in a clear fashion.

Discussion leaves out significant details of how dominant absorption mechanisms were identified.
Dominant absorption mechanisms misidentified.

No uncertainties given for measured linear attenuation coefficients.

Statements are made which are not supported by the data.
Attempt made, but is incorrect or missing appropriate discussion.

Statements are made which are contradicted by the data.