Livermore Model

Total Cross Section

The total cross section for the Compton scattering process is based on either EPDL97 or EPICS2017 data as described in Low Energy Livermore Model, and determined from the data as described in Generic Calculation of Total Cross Sections. To avoid sampling problems in the Compton process the cross section is set to zero at low-energy limit of cross section table, which is 100 eV in majority of EM Physics Lists.

Sampling of the Final State

For low energy incident photons, the simulation of the Compton scattering process is performed according to the same procedure used for the “standard” Compton scattering simulation, with the addition that Hubbel’s atomic form factor [Hub97] or scattering function, \(SF\), is taken into account. The angular and energy distribution of the incoherently scattered photon is then given by the product of the Klein-Nishina formula \(\Phi(\epsilon)\) and the scattering function, \(SF(q)\)  [Cul95]

\[P(\epsilon, q ) = \Phi( \epsilon ) \times SF(q) .\]

\(\epsilon\) is the ratio of the scattered photon energy \(E'\), and the incident photon energy \(E\). The momentum transfer is given by \(q = \frac{1}{\lambda} \sin(\theta/2)\), where \(\theta\) is the polar angle of the scattered photon with respect to the direction of the parent photon. \(\Phi(\epsilon)\) is given by

\[\Phi(\epsilon) \cong {\left[{1\over\epsilon} + \epsilon\right] \left[1-{\epsilon \over{1+\epsilon^2}} \sin^2\theta\right]} .\]

The effect of the scattering function becomes significant at low energies, especially in suppressing forward scattering [Cul95].

The sampling method of the final state is based on composition and rejection Monte Carlo methods [MC70, NHR85, BM60], with the \(SF\) function included in the rejection function

\[g(\epsilon) = \left[1-\frac{\epsilon}{1+\epsilon^2} \sin^2\theta \right] \times SF(q) ,\]

with \(0 < g(\epsilon) < Z\). Values of the scattering functions at each momentum transfer, \(q\), are obtained by interpolating the evaluated data for the corresponding atomic number, \(Z\).

The polar angle \(\theta\) is deduced from the sampled \(\epsilon\) value. In the azimuthal direction, the angular distributions of both the scattered photon and the recoil electron are considered to be isotropic [NSJC04, Ste03].

Since the incoherent scattering occurs mainly on the outermost electronic subshells, the binding energies can be neglected, as stated in reference [NSJC04, Ste03]. The momentum vector of the scattered photon, \(\overrightarrow{P'_{\gamma}}\), is transformed into the World coordinate system. The kinetic energy and momentum of the recoil electron are then

\[\begin{split}T_{el} &= E - E' \\ \overrightarrow{P_{el}} &= \overrightarrow{P_{\gamma}} - \overrightarrow{P'_{\gamma}} .\end{split}\]

Bibliography

BM60

J.C. Butcher and H. Messel. Nucl. Phys., 20(15):, 1960.

Cul95(1,2)

D.E. Cullen. A simple model of photon transport. Nucl. Instr. Meth. in Phys. Res. B, 101():499–510, 1995.

Hub97

J.H. Hubbell. Summary of existing information on the incoherent scattering of photons, particularly on the validity of the use of the incoherent scattering function. Radiation Physics and Chemistry, 50(1):113–124, jul 1997. URL: https://doi.org/10.1016/S0969-806X(97)00049-2, doi:10.1016/s0969-806x(97)00049-2.

MC70

H. Messel and D. Crawford. Electron-Photon shower distribution. Pergamon Press, 1970.

NSJC04(1,2)

C. Negreanu, J. Stepanek, O.P. Joneja, and R. Chawla. Validation of new electron and positron data libraries. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 213:55–59, jan 2004. URL: https://doi.org/10.1016/s0168-583x(03)01533-7, doi:10.1016/s0168-583x(03)01533-7.

NHR85

W.R. Nelson, H. Hirayama, and D.W.O. Rogers. EGS4 code system. SLAC, Dec 1985. SLAC-265, UC-32.

Ste03(1,2)

Jiri Stepanek. Electron and positron atomic elastic scattering cross sections. Radiation Physics and Chemistry, 66(2):99–116, feb 2003. URL: https://doi.org/10.1016/s0969-806x(02)00386-9, doi:10.1016/s0969-806x(02)00386-9.