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When using radioactive materials in experiments, medical procedures, and other applications, shielding is used to protect equipment from unwanted background noise in the form of high-energy photons (known as “gamma rays”) [1]. Nuclei which are more likely to interact with and absorb the energy of incident photons are more effective shielding materials. Accordingly, the overall effectiveness of a shield depends its thickness as well as on the probability for a material to interact with a photon. The effective area around a particle, within which a photon must pass in order to interact, is known as the “cross-section” of the particle; this cross-section can be used to determine the probability that an incident photon interacts with a given material [2].
A material’s photon cross-section is energy-dependent, with different interactions occurring as the energy of the incident gamma increases (Figure 1):
Introduction
Figure 1: Different photon interactions become relevant for different incident photon energies. For gammas, pair production becomes the dominant absorption process [3].
There are three interactions by which a photon loses energy in a material: the photoelectric effect, Compton scattering, and pair production [2]. Photoelectric interactions are negligible in gamma rays, which possess an energy of at least 1 MeV. Compton scattering is relevant for gammas, but this interaction only reduces the energy of the gamma rather than terminates it entirely. This means that pair production is the primary process by which high energy gammas are absorbed in a material [2].
Pair production occurs when an incident gamma interacts with a virtual photon created by the electric field of a heavy nucleus in a solid [4]. If energy of the incident gamma ray is greater than 1.022 MeV, the interaction creates an electron and a positron, each with rest mass 511 keV and some kinetic energy (
) [5]. Once created, the positron will “thermalize,” quickly losing most of its kinetic energy as it interacts with charged particles in its environment [4]. After the positron has thermalized, it will annihilate with one of the many electrons in its environment (
). This process produces two real 511 keV photons, each traveling with velocities approximately equal in magnitude but opposite in direction as to conserve momentum (Figure 2).
Figure 2: A schematic of a pair production event. The positron annihilates to produce two 511 keV photons; the electron produced via pair production will not be detected [5].
Since pair production is the dominant process by which high energy photons are absorbed in materials, one must measure the pair production cross-section of a material in order to effectively design a shield which attenuates gammas. This property can be measured by subjecting a slab of material to a source of gamma rays and observing the resulting rate of pair production in the slab [6]. One way to observe this rate is to measure rate of emission of the two 511 keV photons emitted due to pair production. However, since the 511 keV photons also interact with the slab, some of them may lose their energy and become absorbed in the slab. This effect acts to reduce the rate of 511 keV emission for thicker slabs, making it nontrivial to disentangle the pair production cross section from the observed rate of 511 keV emission. In this study, we developed a model for the thickness-dependence of the rate of 511 keV emission of a slab of material, and compared this model to measurements of a lead slab of varying thickness when subjected to a 60Co gamma ray source.
where μ is the attenuation length of the material, which is related to the interaction cross section σ of the material by the equation:
Theory
Pair production requires two components: a beam of incident gamma rays and nuclei with which the gamma rays will interact. Following the theory development of [6], a gamma ray beam of initial intensity I0 will attenuate to an intensity I after passing through a slab of material of thickness L according to:
where ρ is the mass density of the material, NA is Avogadro’s number, and A is the mass number of the material [6]. The intensity here is given as the number of photons per unit area per unit time, so the number of photons absorbed
in the slab per unit area per unit time is given by:
Equations 1-3 refer to the overall attenuation length of a material at the energy corresponding to the incident gammas. This attenuation length receives contributions from all possible photon interactions in the material. Therefore, in
order to determine the expected rate of pair production in the material, one must multiply the overall rate of photon absorption by the probability that a single absorption interaction was due to pair production:
This photon flux, N511, represents the number of 511 keV pairs which will escape the slab of material. The final number of detected photon pairs will depend on the effective solid angle of the detectors used in the measurement, as
well as on the quantum efficiency of these detectors. This relationship can be expressed simply as:
where σp is the pair production cross-section, and the probability that a single absorption was due to pair production is just a ratio of the pair production cross-section and the total photon cross-section of the material [6].
After pair production occurs, electron-positron annihilation takes place somewhere inside the slab. The 511 keV photons produced by this annihilation must then travel through the slab before they may be detected. If the annihilation occurred at a depth, x, in the slab, one photon of the 511 keV pair must travel a distance x to escape the slab, while the other must travel a distance L-x. This means the resulting 511 keV photons will attenuate as if they passed through the entire slab of thickness L and attenuation length at 511 keV given by μ511 :
In principle, this factor KI0 can be measured if the source activity, experimental geometry, and PMT quantum efficiency is known precisely. However, we will use it as a fitting parameter to confirm the exponential behavior of the pair production rate as slab thickness, L, increases.
Apparatus
A 60Co radioactive source was used to provide high energy photons for pair production. The source and a lead slab are positioned between 2 PMTs, which can be seen in the image below.
Figure 3: Our 2 PMTs, with the lead slab and 60Co source. The source is attached to the rod on the left side of the slab. The detector on the left serves as an enable signal for the detector on the right.
In order to count annihilation events, the PMTs were configured to detect coincident signals due to the incident 511 keV gammas. This coincidence circuit is shown below. First, the detection signal from detector 1 is amplified using the built in pre-amplifier. Signals from detector 1 are passed through a single channel analyzer which, serves as a voltage filter with a narrow band around the voltage produced by 511 keV gammas. If the signal is within this range, it is passed to the full width gate, which converts it into a pulse, which is sent to the multichannel analyzer (MCA). When the MCA gets an "enable" signal, it passes the data from detector 2 to the computer.. The data is then recorded using Maestro software.
Figure 4: Our coincidence circuit.
Results
Our results for the detected pair production rates for slab thicknesses ranging from 1 mm to 15.4 mm are shown below. Each data point was collected over a period of 12 hours.
Figure 5: Pair production rate data. The aside from the outliers at 5.4 mm and 7.2 mm, the data resembles the trend suggested by the predicted rate of pair production detection as derived earlier. Original figure.
Systematic errors introduced into the data had an impact on the reliability of the data, and as a result cast doubt on any conclusions that may be drawn from this measurement. These systematics were caused by an unexpected change in position of the 60Co sometime during the data collection process. We corrected for this effect by shifting the affected points by an amount which compensated for the resulting change in the geometrical factor, K. After performing this shift and removing outliers, a fit (shown in red above) of the data to theory yielded a value of χν2 =1.38.
Conclusion
Although significant systematic errors were present in this experiment, the model’s primary qualitative predictions seemed to have been upheld: the detected pair production rates increased initially, leveled off at some maximum value, then gradually decreased back to zero. Even with several outliers and overall offsets needed to correct this data, the results generally agreed with theory, yielding a χν2 value of 1.38. However, the level of systematic error present in this experiment significantly undermines any conclusion which may have been drawn from these results. Future iterations of this experiment must be executed using a stronger gamma ray source and a more methodical approach to assuring the geometric stability of the experiment while it collects data.
References
[1] Chabot, George E. “Shielding of Gamma Radiation.” Health Physics Society.
https://hps.org/documents/shielding_of_gamma_radiation.pdf.
[2] Hirayama, Hideo. “Lecture Note on Photon Interactions and Cross Sections.” KEK, High Energy Accelerator Research Organization. Nov 2000. http://rcwww.kek.jp/research/shield/photon_r.pdf
[3] “Radiation Interactions with Tissue.” Radiology Key. Jan 8 2016. https://radiologykey.com/wp-content/uploads/2016/01/c4-fig-0004.jpg
[4] Egerton, R. “Pair Production and Annihilation.” Portland State University. http://web.pdx.edu/~egertonr/ph311-12/pair-p&a.htm
[5] “Atomic and Molecular Interactions – Pair production.” University of Virginia. 2013. https://www.med-ed.virginia.edu/courses/rad/radbiol/01physics/phys-03-05.html
[6] L. Peralta, “A simple electron-positron pair production experiment,” Am. J. Phys. 74, 457
(2006). http://dx.doi.org/10.1119/1.2174030.
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