Penelope Model

Total cross section

The total cross section of the \(\gamma\) conversion process is determined from the data [BH87], as described in Generic Calculation of Total Cross Sections.

Sampling of the final state

The energies \(E_{-}\) and \(E_{+}\) of the secondary electron and positron are sampled using the Bethe-Heitler cross section with the Coulomb correction, using the semiempirical model of Ref. [eal94]. If

\[\epsilon = \frac{E_{-}+m_{e}c^{2}}{E}\]

is the fraction of the \(\gamma\) energy \(E\) which is taken away from the electron,

\[\kappa = \frac{E}{m_{e}c^{2}} \quad \textrm{and} \quad a = \alpha Z,\]

the differential cross section, which includes a low-energy correction and a high-energy radiative correction, is

(14)\[\frac{d\sigma}{d\epsilon} = r_{e}^{2} a (Z+\eta) C_{r} \frac{2}{3} \Big[ 2 \Big( \frac{1}{2} - \epsilon \Big)^{2}\phi_{1}(\epsilon)+ \phi_{2}(\epsilon) \Big],\]

where:

\[\begin{split}\phi_{1}(\epsilon) &= \frac{7}{3} - 2 \ln (1+b^{2}) -6b\arctan (b^{-1}) \\ &- b^{2}[4-4b \arctan(b^{-1})-3 \ln(1+b^{-2})] \\ &+ 4\ln (R m_{e} c/\hbar) - 4f_{C}(Z) + F_{0}(\kappa,Z)\end{split}\]

and

\[\begin{split}\phi_{2}(\epsilon) &= \frac{11}{6} - 2 \ln (1+b^{2}) -3b\arctan (b^{-1}) \\ &+ \frac{1}{2}b^{2}[4-4b \arctan(b^{-1})-3 \ln(1+b^{-2})] \\ &+ 4\ln (R m_{e} c/\hbar) - 4f_{C}(Z) + F_{0}(\kappa,Z),\end{split}\]

with

\[b = \frac{Rm_{e}c}{\hbar} \frac{1}{2\kappa} \frac{1}{\epsilon(1-\epsilon)}.\]

In this case \(R\) is the screening radius for the atom \(Z\) (tabulated in [JHHOverbo80] for \(Z=1\) to 92) and \(\eta\) is the contribution of pair production in the electron field (rather than in the nuclear field). The parameter \(\eta\) is approximated as

\[\eta = \eta_{\infty}(1-e^{-v}),\]

where

\[\begin{split}v &= (0.2840-0.1909a)\ln(4/\kappa)+(0.1095+0.2206a)\ln^{2}(4/\kappa) \\ &+ (0.02888 - 0.04269a)\ln^{3}(4/\kappa) + (0.002527+0.002623)\ln^{4}(4/\kappa)\end{split}\]

and \(\eta_{\infty}\) is the contribution for the atom \(Z\) in the high-energy limit and is tabulated for \(Z=1\) to 92 in Ref. [JHHOverbo80]. In the Eq.(14), the function \(f_{C}(Z)\) is the high-energy Coulomb correction of Ref. [eal54], given by

\[\begin{split}f_{C}(Z) &= a^{2}[(1+a^{2})^{-1}+0.202059-0.03693a^{2}+0.00835a^{4} \\ &- 0.00201a^{6}+0.00049a^{8}-0.00012a^{10}+0.00003a^{12}];\end{split}\]

\(C_{r} = 1.0093\) is the high-energy limit of Mork and Olsen’s radiative correction (see Ref. [JHHOverbo80]); \(F_{0}(\kappa,Z)\) is a Coulomb-like correction function, which has been analytically approximated as [eal01]

\[\begin{split}F_{0}(\kappa,Z) &= (-0.1774 - 12.10a + 11.18a^{2})(2/\kappa)^{1/2} \\ &+ (8.523 + 73.26a - 44.41a^{2})(2/\kappa) \\ &- (13.52 + 121.1a - 96.41a^{2})(2/\kappa)^{3/2} \\ &+ (8.946 + 62.05a - 63.41a^{2})(2/\kappa)^{2}.\end{split}\]

The kinetic energy \(E_{+}\) of the secondary positron is obtained as

\[E_{+} = E - E_{-} - 2m_{e}c^{2}.\]

The polar angles \(\theta_{-}\) and \(\theta_{+}\) of the directions of movement of the electron and the positron, relative to the direction of the incident photon, are sampled from the leading term of the expression obtained from high-energy theory (see Ref.[eal69])

\[p(\cos\theta_{\pm}) = a(1-\beta_{\pm}\cos\theta_{\pm})^{-2},\]

where \(a\) is the a normalization constant and \(\beta_{\pm}\) is the particle velocity in units of the speed of light. As the directions of the produced particles and of the incident photon are not necessarily coplanar, the azimuthal angles \(\phi_{-}\) and \(\phi_{+}\) of the electron and of the positron are sampled independently and uniformly in the interval \((0, 2\pi)\).

Bibliography

BH87

M.J. Berger and J.H. Hubbel. Xcom: photom cross sections on a personal computer. Technical Report Report NBSIR 87-3597, National Bureau of Standards, 1987.

eal01

F. Salvat et al. Penelope - a code system for monte carlo simulation of electron and photon transport. Technical Report, Workshop Proceedings Issy-les-Moulineaux, France; AEN-NEA, 5-7 November 2001.

eal54

H. Davies et al. Theory of bremsstrahlung and pair production. ii. integral cross section for pair production. Phys. Rev., 93(4):788–795, 1954.

eal94

J.Baró et al. Analytical cross sections for monte carlo simulation of photon transport. Radiat. Phys. Chem., 44():531, 1994.

eal69

J.W. Motz et al. Pair production by photons. Rev. Mod. Phys, 41():581, 1969.

JHHOverbo80(1,2,3)

J.H. Hubbell, H.A. Gimm and I. Øverbø. Pair, Triplet, and Total Atomic Cross Sections (and Mass Attenuation Coefficients) for 1 MeV-100 GeV Photons in Elements Z=1 to 100. Journal of Physical and Chemical Reference Data, 9:1023–1148, October 1980. doi:10.1063/1.555629.