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.
Photoelectric effect cross section
This phenomenon describes the situation in which a gamma
photon interacts with an electron located in the
atomic structure. This results the ejection of that
electron from the
atom. The
photoelectric effect is the dominant energy transfer mechanism for
X-ray and gamma ray photons with energies below 50
keV. It is much less important at higher energies, but still needs to be taken into consideration.
Usually, the cross section of the photoeffect can be approximated by the simplified equation of[1][2]
For higher energies (>0.5
MeV) the cross section of the photoelectric effect is very small because other effects (especially
Compton scattering) dominates. 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]
where all input parameters are presented in the Table below.
n
an
bn
cn
pn
1
1.6268∙10−9
-2.683∙10−12
4.173∙10−2
1
2
1.5274∙10−9
-5.110∙10−13
1.027∙10−2
2
3
1.1330∙10−9
-2.177∙10−12
2.013∙10−2
3.5
4
-9.12∙10−11
0
0
4
Compton scattering cross section
Compton scattering (or Compton effect) is an interaction in which an incident gamma
photon interact 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.
for energies higher than 100 keV (k>0.2). For lower energies, however, this equation shall be substituted by:[6]
which is proportional to the absorber's
atomic number, Z.
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]
for low energies (k<4),
where
.
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<k<4.6:
For 4.6<k<6:
For 6<k<18:
For k>14 Haug proposed to use a shorter form of Borsellino equation:[9][10]
Total cross section
One can present the total cross section per atom as a simple sum of each effects:[2]
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:
where ai,j parameters are presented in Table below. 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<k<20,000) and absorber's atomic numbers (from Z=1 to 100).
ai,j
i=0
i=1
i=2
i=3
i=4
i=5
i=6
j=0
0.0830899
-0.08717743
0.02610534
-2.74655∙10−3
4.39504∙10−5
9.05605∙10−6
-3.97621∙10−7
j=1
0.265283
-0.10167009
0.00701793
2.371288∙10−3
-5.020251∙10−4
3.6531∙10−5
-9.46044∙10−7
j=2
2.18838∙10−3
-2.914205∙10−3
1.26639∙10−3
-7.6598∙10−5
-1.58882∙10−5
2.18716∙10−6
-7.49728∙10−8
j=3
-4.48746∙10−5
4.75329∙10−5
-1.43471∙10−5
1.19661∙10−6
5.7891∙10−8
-1.2617∙10−8
4.633∙10−10
j=4
6.29882∙10−7
-6.72311∙10−7
2.61963∙10−7
-5.1862∙10−8
5.692∙10−9
-3.29∙10−10
7.7∙10−12
For lower energy region (<1 MeV) the Fornalski equation is more complicated due to the larger function variability of different
elements. Therefore, the modified equation[11]
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. Analogically, the equation is valid for all Z from 1 to 100.
ai,j
j=0
j=1
j=2
j=3
j=4
j=5
j=6
i=0
-1.539137959563277
0.3722271606115605
-0.018918894979230043
5.304673816064956∙10−4
-7.901251450214221∙10−6
5.9124040925689876∙10−8
-1.7450439521037788∙10−10
i=1
-0.49013771295901015
7.366301806437177∙10−4
-8.898417420107425∙10−5
3.294237085781055∙10−6
-8.450746169984143∙10−8
7.640266479340313∙10−10
-2.282367050913894∙10−12
i=2
-0.05705460622256227
0.001957234615764126
-6.187107799669643∙10−5
2.1901234933548505∙10−6
1.9412437622425253∙10−8
-5.851534943255455∙10−10
2.7073481839614158∙10−12
i=3
0.001395861376531693
-7.137867469026608∙10−4
2.462958782088413∙10−4
-9.660290609660555∙10−6
1.295493742164346∙10−7
-6.538025860945927∙10−10
8.763097742806648∙10−13
i=4
5.105805426257604∙10−5
0.0011420827759804927
-8.177273886356552∙10−5
4.564725445290536∙10−6
-9.707786695822055∙10−8
8.351662725636947∙10−10
-2.545941852995417∙10−12
i=5
-5.416099245465933∙10−4
5.65398317844477∙10−4
-5.294089702089374∙10−5
5.437298837558547∙10−7
1.4824427385312707∙10−8
-2.8079293400520423∙10−10
1.247192025425616∙10−12
i=6
3.6322794450615036∙10−4
-2.186723664102979∙10−4
1.739236692381265∙10−5
-3.7341071277534563∙10−7
1.1585158108088033∙10−9
3.1805366711255584∙10−11
-2.0806866173605604∙10−13
XCOM Database of cross sections
The
USNational 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.
^Davisson, C.M. (1965). Interaction of gamma-radiation with matter. In: Alpha-, Beta- and Gamma-ray Spectroscopy: Volume 1. Edited by Kai Siegbahn. Amsterdam: North-Holland Publishing Company.
^Attix F.H. 1986. Introduction to radiological physics and radiation dosimetry. John Wiley & Sons
^Maximon L.C. 1968. Simple analytic expressions for the total Born approximation cross section for pair production in a Coulomb field. J. Res. Nat. Bur. Stand., vol. 72B (Math. Sci.), no. 1, pp. 79-88
[1]
^
abHaug E. 1981. Simple analytic expressions for the total cross section for γ-e pair production. Zeitschrift für Naturforschung, vol. 36a, pp. 413-414
^Haug E. 1975. Bremsstrahlung and pair production in the field of free electrons. Zeitschrift für Naturforschung, vol. 30a, pp. 1099-1113
^Berger, M.J., Hubbell, J.H., Seltzer, S.M., Chang, J., Coursey, J.S., Sukumar, R., Zucker, D.S., and Olsen, K., 2010. XCOM: Photon Cross Section Database (version 1.5), National Institute of Standards and Technology, Gaithersburg, MD, USA, DOI: 10.18434/T48G6X
[2]