The total cross section of gamma ray interactions is composed of several independent processes: photoelectric effect, Compton (incoherent) scattering, electron-positron pair production in the nucleus field and electron-positron pair production in the electron field (triplet production).
Other effects, like the photonuclear absorption, Thomson or Rayleigh (coherent) scattering can be omitted because of their nonsignificant contribution in the gamma ray range of energies.
The detailed equations for cross sections (barn/atom) of all mentioned effects connected with gamma ray interaction with matter are listed below.
The photoelectric effect phenomenon describes the interaction of a gamma photon with an electron located in the atomic structure.
The photoelectric effect is the dominant energy transfer mechanism for X-ray and gamma ray photons with energies below 50 keV.
Usually, the cross section of the photoeffect can be approximated by the simplified equation of[1][2]
where k = Eγ / Ee, and where Eγ = hν is the photon energy given in eV and Ee = me c2 ≈ 5,11∙105 eV is the electron rest mass energy, Z is an atomic number of the absorber's element, α = e2/(ħc) ≈ 1/137 is the fine structure constant, and re2 = e4/Ee2 ≈ 0.07941 b is the square of the classical electron radius in barns.
and EB is a binding energy of electron, and ϕ0 is a Thomson cross section (ϕ0 = 8πe4/(3Ee2) ≈ 0.66526 barn).
However, for precise calculations of the photoeffect cross section in high energy range, the Sauter equation shall be substituted by the Pratt-Scofield equation[4][5][6]
Compton scattering (or Compton effect) is an interaction in which an incident gamma photon interacts with an atomic electron to cause its ejection and scatter of the original photon with lower energy.
The probability of Compton scattering decreases with increasing photon energy.
Compton scattering is thought to be the principal absorption mechanism for gamma rays in the intermediate energy range 100 keV to 10 MeV.
The cross section of the Compton effect is described by the Klein-Nishina equation:
The additional cross section connected with the Compton effect can be calculated for the energy transfer coefficient only – the absorption of the photon energy by the electron:[7]
By interaction with the electric field of a nucleus, the energy of the incident photon is converted into the mass of an electron-positron (e−e+) pair.
The cross section for the pair production effect is usually described by the Maximon equation:[8][6]
However, for higher energies (k>4) the Maximon equation has a form of
The triplet production effect, where positron and electron is produced in the field of other electron, is similar to the pair production, with the threshold at k=4.
This effect, however, is much less probable than the pair production in the nucleus field.
The most popular form of the triplet cross section was formulated as Borsellino-Ghizzetti equation[6] where a=-2.4674 and b=-1.8031.
This equation is quite long, so Haug[9] proposed simpler analytical forms of triplet cross section.
Especially for the lowest energies 4
Next, using the Beer–Lambert–Bouguer law, one can calculate the linear attenuation coefficient for the photon interaction with an absorber of atomic density N:
The analytical calculation of the cross section of each specific phenomenon is rather difficult because appropriate equations are long and complicated.
Thus, the total cross section of gamma interaction can be presented in one phenomenological equation formulated by Fornalski,[11] which can be used instead:
This formula is an approximation of the total cross section of gamma rays interaction with matter, for different energies (from 1 MeV to 10 GeV, namely 2 For lower energy region (<1 MeV) the Fornalski equation is more complicated due to the larger function variability of different elements. is a good approximation for photon energies from 150 keV to 10 MeV, where the photon energy E is given in MeV, and ai,j parameters are presented in Table below with much better precision. The US National Institute of Standards and Technology published on-line[12] a complete and detailed database of cross section values of X-ray and gamma ray interactions with different materials in different energies. The database, called XCOM, contains also linear and mass attenuation coefficients, which are useful for practical applications.