Penelope Model
Introduction
The class G4PenelopeBremsstrahlung calculates the continuous energy loss due to soft \(\gamma\) emission and simulates the photon production by electrons and positrons. As usual, the gamma production threshold \(T_{c}\) for a given material is used to separate the continuous and the discrete parts of the process.
Electrons
The total cross sections are calculated from the data [STPerkins89], as described in Generic Calculation of Total Cross Sections and Livermore Model. The energy distribution \(\frac{d\sigma}{dW}(E)\), i.e. the probability of the emission of a photon with energy \(W\) given an incident electron of kinetic energy \(E\), is generated according to the formula
The functions \(F(\kappa)\) describing the energy spectra of the outgoing photons are taken from Ref.[SB86]. For each element \(Z\) from 1 to 92, 32 points in \(\kappa\), ranging from \(10^{-12}\) to 1, are used for the linear interpolation of this function. \(F(\kappa)\) is normalized using the condition \(F(10^{-12})=1\). The energy distribution of the emitted photons is available in the library [SB86] for 57 energies of the incident electron between 1 keV and 100 GeV. For other primary energies, logarithmic interpolation is used to obtain the values of the function \(F(\kappa)\). The direction of the emitted bremsstrahlung photon is determined by the polar angle \(\theta\) and the azimuthal angle \(\phi\). For isotropic media, with randomly oriented atoms, the bremsstrahlung differential cross section is independent of \(\phi\) and can be expressed as
Numerical values of the “shape function” \(p(Z,E,\kappa;\cos\theta)\), calculated by partial-wave methods, have been published in Ref. [KQP83] for the following benchmark cases: \(Z\)= 2, 8, 13, 47, 79 and 92; \(E\)= 1, 5, 10, 50, 100 and 500 keV; \(\kappa\)= 0, 0.6, 0.8 and 0.95. It was found in Ref. [eal01] that the benchmark partial-wave shape function of Ref. [KQP83] can be closely approximated by the analytical form (obtained in the Lorentz-dipole approximation)
with \(\beta' = \beta (1+B)\), if one considers \(A\) and \(B\) as adjustable parameters. The parameters \(A\) and \(B\) have been determined, by least squares fitting, for the 144 combinations of atomic numbers, electron energies and reduced photon energies corresponding to the benchmark shape functions tabulated in [KQP83]. The quantities \(\ln(AZ\beta)\) and \(B\beta\) vary smoothly with Z, \(\beta\) and \(\kappa\) and can be obtained by cubic spline interpolation of their values for the benchmark cases. This permits the fast evaluation of the shape function \(p(Z,E,\kappa;\cos\theta)\) for any combination of \(Z\), \(\beta\) and \(\kappa\). The stopping power \(dE/dx\) due to soft bremsstrahlung is calculated by interpolating in \(E\) and \(\kappa\) the numerical data of scaled cross sections of Ref. [BS82]. The energy and the direction of the outgoing electron are determined by using energy-momentum balance.
Positrons
The radiative differential cross section \(d\sigma^+(E)/dW\) for positrons reduces to that for electrons in the high-energy limit, but is smaller for intermediate and low energies. Owing to the lack of more accurate calculations, the differential cross section for positrons is obtained by multiplying the electron differential cross section \(d\sigma^-(E)/dW\) by a \(\kappa\)-independent factor, i.e.
The factor \(F_{p}(Z,E)\) is set equal to the ratio of the radiative stopping powers for positrons and electrons, which has been calculated in Ref.[eal86]. For the actual calculation, the following analytical approximation is used:
where
Because the factor \(F_{p}(Z,E)\) is independent on \(\kappa\), the energy distribution of the secondary \(\gamma\)’s has the same shape as electron bremsstrahlung. Similarly, owing to the lack of numerical data for positrons, it is assumed that the shape of the angular distribution \(p(Z,E,\kappa;\cos\theta)\) of the bremsstrahlung photons for positrons is the same as for the electrons. The energy and direction of the outgoing positron are determined from energy-momentum balance.
Bibliography
- BS82
M.J. Berger and S.M. Seltzer. Stopping power of electrons and positrons. Technical Report Report NBSIR 82-2550, National Bureau of Standards, 1982.
- 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.
- eal86
L.Kim et al. Ratio of positron to electron bremsstrahlung energy loss: an approximate scaling law. Phys. Rev. A, 33():2002, 1986.
- KQP83(1,2,3)
L. Kissel, C.A. Quarles, and R.H. Pratt. Shape functions for atomic-field bremsstrahlung from electrons of kinetic energy 1–500 keV on selected neutral atoms 1 ≤ z ≤ 92. Atomic Data and Nuclear Data Tables, 28(3):381–460, May 1983. URL: https://doi.org/10.1016/0092-640X(83)90001-3, doi:10.1016/0092-640x(83)90001-3.
- STPerkins89
S.M.Seltzer S.T.Perkins, D.E.Cullen. Tables and graphs of electron-interaction cross-sections from 10 ev to 100 gev derived from the llnl evaluated electron data library (eedl), z=1-100. Technical Report UCRL-50400 Vol.31, Lawrence Livermore National Laboratory, 1989.
- SB86(1,2)
S.M. Seltzer and M.J. Berger. Bremsstrahlung energy spectra from electrons with kinetic energy 1 kev - 100 gev incident on screened nuclei and orbital electrons of neutral atoms with z = 1–100. Atomic Data and Nuclear Data Tables, 35():345, 1986.