SBETHE: Stopping powers of materials for swift charged particles from the corrected Bethe formula

The Fortran program SBETHE calculates the stopping power of materials for swift charged particles with small charges (electrons, muons, protons, their antiparticles, and alphas). The electronic stopping power is computed from the corrected Bethe formula, with the shell correction derived from numerical calculations with the plane-wave Born approximation (PWBA) for atoms, which were based on an independent-electron model with the Dirac-Hartree-Fock-Slater self-consistent potential for the ground -state configuration of the target atom. The density effect correction is evaluated from an empirical optical oscillator strength (OOS) model based on atomic subshell contributions obtained from PWBA calculations. For projectiles heavier than the electron, the Barkas correction is evaluated from the OOS model, and the Lindhard-Sorensen correction is estimated from an accurate parameterization of its numerical values. The calculated electronic stopping power is completely determined by a single empirical parameter, the mean excitation energy or I value of the material. The radiative stopping power for electrons, and positrons, is evaluated by means of Seltzer and Berger's cross section tables for bremsstrahlung emission. The program yields reliable stopping powers and particle ranges for arbitrary materials and projectiles with kinetic energy larger than a certain cutoff value Ecut, which is specific of each projectile kind. The program is accompanied by an extensive database that contains tables of relevant energy-dependent atomic quantities for all the elements from hydrogen to einsteinium. sbethe may be used to generate basic information for dosimetry calculations and Monte Carlo simulations of radiation transport, and as a pedagogical tool.Program summaryProgram title: SBETHECPC Library link to program files: https://doi .org /10 .17632 /7zw25f428t .1Licensing provisions: CC by NC 3.0 Programming language: Fortran 90/95 Nature of problem: The program calculates the stopping power of arbitrary materials for swift charged projectiles with small charges. The material is characterized by its chemical composition, mass density, and the empirical I value. The considered projectiles are electrons, positrons, negative muons, antimuons, protons, antiprotons, and alphas, which are described as point particles characterized by their mass and charge. If the actual I value of the material is known, the results from the program are expected to be reliable for projectiles with kinetic energy higher than a value Ecut, of the order of 1 keV for electrons and positrons, 150 keV for muons and antimuons, 0.75 MeV for protons and antiprotons, and 5 MeV for alpha particles.Solution method: The electronic stopping power is calculated by means of a corrected Bethe formula [1], which combines the conventional Bethe logarithm with the following corrections,1) the shell correction obtained from calculations based on the plane-wave Born approximation with the self-consistent Dirac-Hartree-Fock-Slater (DHFS) potential of neutral atoms in their ground-state configuration [2],2) the density effect correction, which accounts for the reduction of the stopping power caused by the dielectric polarization of the medium,3) a parameterization of the Lindhard-Sorensen correction, which generalizes the Bloch correction for relativistic projectiles, and4) the Barkas correction, which accounts for differences between the stopping powers of particles and their antiparticles. The density-effect and the Barkas corrections are calculated from a model of the optical oscillator strength (OOS) of the material, which combines the contributions of inner atomic subshells calculated with the DHFS potential, with a classical oscillator model for the contribution of valence electrons, A simple extrapolation formula is used to extend the calculated electronic stopping power to energies less than Ecut to allow the calculation of particle ranges. For electrons and positrons, the radiative stopping power is calculated from numerical tables prepared by Seltzer and Berger [3].Additional comments including restrictions and unusual features: The calculated stopping power is determined by a single parameter, the mean excitation energy or I value. The program assigns to each material a default I value, derived from the recommendations in the ICRU Report 37, which can be changed by the user. The distribution package includes text files with tables of atomic energy-dependent quantities (subshell optical oscillator strengths, shell corrections, scaled cross sections for bremsstrahlung emission) that are used in the calculations.References [1] F. Salvat, Phys. Rev. A 106 (2022) 032809. [2] F. Salvat, L. Barjuan, P. Andreo, Phys. Rev. A 105 (2022) 042813. [3] S.M. Seltzer, M.J. Berger, Nucl. Instrum. Methods B 12 (1985) 95-134.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by/4.0/).

Automated summary

The Fortran program SBETHE calculates the stopping power of materials for swift charged particles with small charges. It uses the corrected Bethe formula with shell correction derived from numerical calculations and the density effect correction based on an empirical optical oscillator strength model. The program is useful for dosimetry calculations, radiation transport simulations, and as a pedagogical tool.