On the new and accurate (Goudsmit-Saunderson) model for describing e-/e+ multiple Coulomb scattering (Geant4 Technical Note)

TL;DR

The note presents a comprehensive GS-based MSC framework for e-/e+ in Geant4-10.4 that leverages Kawrakow’s EGSnrc formulation and incorporates energy-loss, spin-relativistic, screening, and delta-ray corrections. It implements an efficient run-time sampling scheme for MSC deflections by pre-computing smooth transformed PDFs $q^{2+}(u)$ on a grid and sampling via a rational inverse transform, enabling accurate condensed-history stepping. The approach uses a hybrid zero/single/multi-scattering decomposition to maintain computational efficiency while preserving key angular moments, with corrections to align first transport cross sections with more accurate ELSEPA DCS. The resulting model replaces the previous GS MSC implementation in Geant4 and aims to deliver artefact-free, high-precision electron transport suitable for a wide range of materials and energies, including mixtures and delta-ray considerations. Overall, this work provides a practical, high-accuracy MSC description for Geant4 that improves agreement with reference DCS calculations and supports robust electron tracking in complex geometries.

Abstract

A new model, for the accurate simulation of multiple Coulomb scattering (MSC) of e-/e+, has been implemented in Geant4 recently and made available with version Geant4-10.4. The model is based on Goudsmit-Saunderson (GS) angular distributions computed by utilising the screen Rutherford (SR) DCS and follows very closely the formulation developed by Kawrakow [1, 2] and utilised in the EGSnrc toolkit [3]. Corrections, for taking into accountenergy loss [2] neglected by the GS theory, spin-relativistic effects [3] not included in the SR but might be accounted on the basis of Mott DCS as well as the so-called scattering power correction [4], i.e. appropriately incorporating deflections due to sub-threshold delta ray productions, are all included similarly to the EGSnrc model [3]. Furthermore, an accurate electron-step algorithm [5, 6, 2] is utilised for path length correction, i.e. for calculating the post-step position in each condensed history simulation steps such that the corresponding single-scattering longitudinal and lateral (post step point) distributions are very well reproduced. An e-/e+ stepping algorithm, including the simulation step-limit due to the MSC and boundary crossing [2]), free from step-size artefacts, makes the model complete. Details on this new model, including all the above-mentioned components and corrections, are provided in this Geant4 technical note. It must be noted, that a Goudsmit-Saunderson model for MSC was available before Geant4-10.4., documented in [7], that has been completely replaced by the model described in this technical note (keeping only the G4GoudsmitSaundersonMscModel name of the C++ class from that previous version)

Figures (4)

• Figure 1: $q^{2+}(s,\lambda,a,A;u)$ (left) and $q^{2+}(s,\lambda,\tilde{a},A;u)$ transformed PDFs that correspond to the parameter values shown in Table \ref{['tb:kaw_table1']}-\ref{['tb:kaw_table4']}.
• Figure 2: The $R_{\text{MR}}$ Mott-to-Rutherford elastic DCS ratio, that shows the effect of the spin to the DCS, for $Au$ and $Al$ target atoms, electron and position projectiles at various projectile kinetic energy. See APPENDIX \ref{['app:mott-dcs']} for the details of the computation.
• Figure 3: Some $\kappa=A/A^{(M)}$ screening correction factors calculated by solving Eq.(\ref{['eq:first-trans-correction']}) numerically for the given projectile (left $e^-$; right $e^+$), target atom and kinetic energy to obtain the corrected $\kappa A^{(M)}$ screening parameter that satisfies Eq.(\ref{['eq:first-trans-correction']}) when used in the SR part of the SR$\times\text{R}_{\text{MR}}$ DCS. Similar correction is used in EGSnrc (see Figure 13. in kawrakow2000egsnrc or Figure 1. in kawrakow2000cross) and the reported correction factors are in a good agreement with the presented values (though different PWA DCS were used in kawrakow2000egsnrckawrakow2000cross as reference which explains the small observable differences).
• Figure 4: Differential Cross Sections (DCS-s) for Coulomb scattering of electrons on $Au$ and $Al$ target atoms at different projectile kinetic energies. See Table \ref{['tb:DCS-PLOT']} and the text.