Compton Scattering (PHYS 334)

Arthur Holly Compton was awarded the Nobel Prize in 1927 for his work, published in 1923, of careful spectroscopic measurements of x-rays scattered at various angles by light elements. He found that x-rays scattered at larger angles had systematically larger wavelengths. He discovered that the observations were accounted for by considering the scattering as a collision between a single photon and a single electron in which energy and momentum are conserved. This effect now bears his name. The Compton effect demonstrates the essential duality of waves and particles in an especially clear way: Modeled as a particle (localized, having energy and momentum) one can apply conservation of energy and momentum to predict the relation between scattered X-ray energy and scattering angle. On the other hand, modeled as a wave, one can understand x-ray interference and diffraction phenomena.

Do not attempt to open the Cs137 source until you have been properly instructed on the appropriate safety measures by a member of the lab staff.

A. A. Bartlett, Am J. Phys. 32, 120 (1964) This paper is a historical review of the experiments that were later explained by Compton's discovery of the Compton effect.

A. H. Compton, Am. J. Phys. 29, 817 (1961) Compton reviews the experimental evidence and the theoretical considerations that led to the discovery and interpretation of x-rays acting as particles.

Some Applications

  1. Compton scattering is being used to model the distribution of x-rays produced when a black hole disturbs a star.
  2. Compton scattering based tomography can be used to map the electron density of a material
  3. Compton scattering is a critical theoretical component of investigating the substructure of nucleons (e.g. protons)
  4. Wooden materials can be non-destructively probed via Compton scattering, which can give insight into the density distribution of materials and possible defects present.

Experimental Goal

The goal of this experiment is to measure the differential cross section for photons scattering off of electrons in an aluminum target. You will compare your results with the predictions of the Klein-Nashina formula. The experiment will be conducted in two parts.

In the first part of the experiment you will spend a few of days measuring the energy of 662 keV photons which have undergone Compton scattering in an aluminum target. The purpose of this exercise is to familiarize yourself with the properties of the detector system, to explore relevant data analysis techniques, and to understand the systematic and statistical uncertainties associated with the measurements. You are provided with details of the theory and experimental techniques required for this measurement. After completing this first part of the experiment, you will arrange a meeting with the course instructors to go over your results. Once the instructors are satisfied with your understanding of the Compton scattering experiment, you will proceed to the second part of the experiment.

In the second part of the experiment, you are not provided with step-by-step instructions on how to make the measurements. Instead, it is up to you to use the knowledge you have gained from the Compton scattering experiment to make the best possible measurement of the differential cross section using the equipment given.

To prepare, consider the following prompts:

What will you be measuring?

  • What is Compton scattering?
  • What quantities are you trying to measure.
  • What do you expect the relationship between these quantities to look like?
  • What is the apparatus at your disposal capable of measuring?

What measurements will you be making in the lab and how do they relate to the physical quantities you need? (In other words, what will your raw data look like and what do you have to do to it to get real-world quantities like photon energies, does anything need to be calibrated?)

  • How does your detector (a NaI crystal coupled to a photomultiplier and read out by a pulse-height analyzer) work? You have worked with these detectors and you seen pulse height spectra before in the Gamma Cross Sections experiment.
  • What will the pulse height spectra (PHS) from a monoenergetic gamma source look like, and how do you expect it to change as a function of scattering angle?
  • What will you measure from the PHS.

What factors will limit how well you are able to measure the relevant quantities?

  • Are there likely to be statistical uncertainties in measured quantities? How can you determine what an acceptable statistical uncertainty is? (Note, you really cannot answer this question unless you understand the previous questions related to what measurement will you be making.)
  • What systematic effects do you expect to be present? How will they bias your results and how might you deal with them?

Figure A: Schematic drawing of Compton Scattering

Theory and Apparatus

Compton Scattering

Consider the scattering of a gamma (photon) from a free electron as shown in Fig. 1.

Figure 1: An incident gamma of energy E “collides” with an electron and scatters with energy E' at angle θ relative to the initial trajectory.

The energy of a gamma scattered by a free electron, $E'$, depends on the scattering angle, $\theta$, and the energy of the incident gamma, $E$. It can easily be derived from the conservation of energy and momentum as

$E' = \dfrac{E}{1+\frac{E}{mc^2}(1-\cos\theta)}$ (1)

where $mc^2 = 511 \mathrm{keV}$ is the rest energy of the electron. This is the model which you will test.

Experimental Apparatus


Figure 2: The Compton scattering apparatus.

The experimental apparatus is shown schematically in Fig. 2.

A collimated beam of 662 keV $\gamma$ -rays produced in the decay of cesium-137 is incident on a cylindrical aluminum rod. A PMT+NaI detector which has been magnetically shielded and housed in a Lead PIG is attached to a goinometer allowing it to be rotated about the scattering rod. Pulses from the PMT+NaI detector are sent to a UCS-30 pulse height analyzer (PHA) Spectrum Techniques UCS-30.

Radioactive source

A pair of ${}^{137}\textrm{Cs}$ sources produce 662 keV gammas. These sources sit at the center of a lead pig to shield you from the radiation. The radiation emerges from the pig in a collimated beam aimed at the scatterer in the middle of the table.

CAUTION: Do not place any part of your body in front of the open port of this source for an extended time. This source is on the order of 1000 times stronger than the plastic button sources used in other labs. (The activity is of the order of milli cuires rather than micro curies).

The “source” is actually two sources having strengths as follows:

  • 32.5 millicuries (mCi), produced 5/19/69
  • 30.0 mCi, produced 7/11/69

These activities are nominal values only, as the activity will decay with time. (Cesium-137 has a half-life of 30.17 years.) When not in use, the pig is “closed” by a tungsten rod inserted into the exit aperture of the pig. A locking brass door holds the plug in place.

  • The source is “opened” by swinging the door away from the face of the pig and removing the plug using the long handled tongs so that your hands are not exposed to the beam.  
  • When you are finished taking data, the tongs should be used to reinsert the plug and the door should be closed.

Calibration sources

To calibrate the pulse height axis of the PHA, a set of small radioactive sources is provided. Sources include ${}^{241}$Am, ${}^{133}$Ba, ${}^{57}$Co, ${}^{137}$Cs, and ${}^{22}$Na, and should yield discernible gamma peaks with energies between 59.5 keV and 661.6 keV.  

You need not consider energies above 662 keV when doing your calibration.

Energies and relative intensities of the calibration sources are available from the nuclear decay schemes. Note that these sources all have low activity so as to not overwhelm the detector with counts and cause charge pileup (also known as voltage sag.)

Part I: Compton scattering – Energy relation

PMT high voltage

  • The PMT you are using requires positive HV which should not exceed +1500V.
  • When the HV is first turned on it takes some time for the performance of the PMT to come to equilibrium. During this time the noise and gain characteristics of the PMT may change by a small amount.
    • Note you do not have to guess when the PMT has come to equilibrium. You can make measurements which will tell you what you need to know.
    • Also note that you have four hours in the lab period to work with the apparatus so you cannot wait hours before beginning data collection. You need to balance the need to collect a minimum set of data in a finite amount of time vs. optimizing the detector performance.

Tips on data collection

You have limited time to take data. Therefore, it is prudent to develop a strategy to optimize data taking. Important questions to ask are the following:

  • At which angles should data be taken?
    • If the function under test varies smoothly, then a simple rule is: Take more data where the expected slope is greatest.
  • At how many angles should data be taken?
    • If the function under test varies rapidly, then taking data at more angles is appropriate
  • How much data should be collected at each angle?
    • In a range where a function under test varies rapidly, then acquiring more counts in that range is appropriate.
  • How do we minimize the effects of electronic drift on the calibration?
    • The correspondence between PHA channel number and gamma energy may be affected by electronic warm-up, room temperature changes, or other factors. To minimize these effects:
      • Wait at least 30 minutes after turning on the PMT high voltage before taking data.
      • Take at least two sets of calibration data, one at the start and another at the end of a lab session. A third calibration in the middle of the day may even be useful.
      • Avoid very long (overnight) data runs.

The purpose of this measurement is to gain familiarity with the apparatus and experimental techniques which will be used to measure the differential cross section. Keep in mind that you will need to finish this part of the experiment in only a few days. Take data for a number of appropriately chosen scattering angles and use this experience to understand your measurements, their limitations, and the analysis of the data.

Part II: Testing the Klein-Nishina relation

Part II of the experiment is intentionally left open-ended. You may need to consult outside resources for more theory information or equipment manuals for details about the apparatus. Given your experience collecting data in Part I, use your judgement to determine effective collection and analysis strategies, and budget your remaining time in lab appropriately.

The information given below is only a suggestion for how to proceed. Based on your interests and the quality of your work after Part I, you and the faculty instructor may discuss alternate goals. You are encouraged to bring ideas and propose ideas at your meeting, but you should also be prepared to defend approaches which stray far from the outline below.

Klein-Nishina relation for differential scattering cross section

The Klein-Nishina formula predicts the relationship between the gamma scattering cross section and the gamma energy. Use the same apparatus from the first part of the experiment to determine the total scattering cross section as a function of energy (angle).

Some things you will need to calculate, simulate and/or measure are the following:

  • the total flux of photons which would strike the detector in the absence of a scatterer;
  • the efficiency of the detector as a function of gamma energy;
  • how to account for the angular acceptance of the detector; and
  • different sources of background gammas striking the detector.

Detector photopeak efficiency

For information on how to measure the efficiency of the detector as a function of gamma energy, see this scintillator manufacturer handout. Optionally, data from Harshaw is available here.

If you really want to dig into the weeds, the original naval research labs report with calculations and tables is also available: calculated_efficiences_of_nai_crystals.pdf