As gamma-rays move through a material, three interactions dominate the interchange of energy between the gamma-ray and the atoms of the host material. At the lower energies, typically in the lower kev ranges, the photoelectric effect dominates. In this case, the entire gamma energy is utilized to free an electron from orbit and to then accelerate that electron with the remaining energy. At energies above 1.022 Mev, a direct conversion of energy to matter can take place under the influence of the field of the nucleus. Two particles of equal mass and opposite sign, an electron and a positron, are formed and the excess gamma energy is applied to the acceleration of each.
Notice that both processes remove the gamma ray completely, leaving charged particles moving away from the interaction site. The third primary interaction mechanism does not completely remove the gamma-ray. Known as Compton Scattering after one of the early physicists who investigated this and other photon phenomena, the model can be easily described using a classical mechanics approach applied to a two-body collision.
The model assumes that the electron involved is unbound and at rest. This introduces no problem for that region of gamma-ray energies for which Compton Scattering is of interest. At low energies of the incident photon, of the same magnitude as the binding energy of the electron, the probability of Compton Scattering is many orders of magnitude lower than the probability of a photoelectric effect occurring.
The model employs a "before and after" consideration of both momentum and of energy, to assure that momentum and energy are both conserved across the interaction (i.e.. this is an elastic collision).
Before the collision, the gamma possesses all the energy and all the momentum in the system, while after collision the scattered gamma and the ejected electron each possess energy and momentum.
Note that we assume that we know the energy of the incident gamma-ray, and we must evaluate the energy and direction of the scattered gamma, the velocity and direction of the ejected electron, four unknowns in all. The equations available are only three: two momentum balances (X and Y direction) one energy balance. We can only frame the solution in terms of one of the unknowns.
The equations may be framed in either photon wavelength or photon energy notation: the latter will be used here. Proceeding, we find that:
where:
Note that alphao is the incident gamma energy expressed in electron rest mass units.
The above result yields information only about the relation of the energy and direction of the scattered gamma-ray. The probability of scattering in any one direction is not addressed.
The probability of a given gamma-ray scattering in a given direction was developed in 1934 utilizing relativistic quantum mechanics. The result, known as the Klein-Nishina Equation expresses the probability of a gamma ray scattering "off" a single electron through an angle theta into a differential solid angle about theta (cm2 / electron-steradian):
Note that this equation predicts the interaction between one gamma-ray and one electron: it is at the microscopic level, hence the units are in barns per steradian per electron! Integration of this expression over all solid anlgle yields the total cross-section for compton scattering.
The unit "steradian" is the basic measure of solid angle: for reference, a sphere subtends 4pi steradians about any point within the sphere.
The term re is the classical radius of the electron, or 2.82 x10-13 cm.
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In use, the above model is applied to the evaluation of the scattered component of the gamma-ray flux by first assessing the number of gammas scattered through an increment of scattering angle. Create an axis system at the electron which scatters the gamma (at the collision site): then to determine the number of gammas scattered into an angle (d theta) about theta, first define the belt shown on the spherical surface surrounding the collision point.
We have an expression (above) for the probability of the scattered gamma being in unit solid angle about the scatter angle theta. Now we have to determine the magnitude of the solid angle subtended by that belt. Using the Theorem of Pappas, we find the belt area:
where:
and:
The surface area of a sphere is:
Therefore, the solid angle subtended by the belt is:
Applying this result to a group of Ne electrons (as in some material) located at the center of the sphere, each electron having the differential cross-section, and subject to N gammas impinging on that group per second, the number of gammas scattered through the belt shown will be:
Turning now to a practical situation, consider a point source emitting S monoenergetic (Eo) gammas/s located a distance r1 from a small mass m of material having an atomic number Z. If we ignore self-shielding in the source and scatterer, the scattered component of the gamma flux arriving at point P, a distance r2 from the scatterer, can be calculated.
The gamma flux incident on the scatterer is:
The number of electrons in it is:
NA is Avagadro's number, and A is the atomic number of the scatterer.
If we imagine the radius of the sphere in the previous section to be r2, and divide the number of gammas going through the belt by the belt area, we have the scattered gamma flux at point P:
