SBETHE: Stopping powers of materials for swift charged particles from the corrected Bethe formula (new version announcement)

Abstract

A new version of the Fortran program sbethe is presented. This program 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–Sørensen 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 radiative contribution to the stopping power of muons is obtained from interpolation of tables given by Groom et al. (2001) [5]. The program yields reliable stopping powers and particle ranges for arbitrary materials and projectiles with kinetic energy larger than a certain cutoff value $E_{\mathrm{cut}}$, 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.

New version program summary

Program Title: sbethe

CPC Library link to program files: https://doi.org/10.17632/7zw25f428t.2

Licensing provisions:: CC By NC 3.0

Programming language:: Fortran 90

Journal reference of previous version:: Comput. Phys. Commun. 287 (2023) 108697

Does the new version supersede the previous version?:: Yes

Reasons for the new version:: The present version extends the original Fortran code by implementing a more realistic extension formula for low-energy protons and alphas in various materials. The program now accounts for radiative effects for high-energy muons. It also produces additional output files with relevant data.

Summary of revisions:: The present program differs from the one published by Salvat and Andreo [1] in the following aspects: 1) The program uses a fitted extension formula for low-energy protons and alphas in various materials for which enough measured stopping-power data are available. 2) Sbethe generates the output file named PENstp.dat with values of the stopping power tabulated at the energy grids used by the Monte Carlo simulation codes penelope and penhan [2,3]. 3) The modified program accounts for radiative effects for high-energy muons. 4) The manual of the code has been expanded to describe the new features covered by the program.

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 $E_{\mathrm{cut}}$, 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)
  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–Sørensen correction, which generalizes the Bloch correction for relativistic projectiles, and
• 4)
  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.

An extrapolation formula is used to extend the calculated electronic stopping power to energies less than $E_{\mathrm{cut}}$ 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 [4], while radiative stopping powers for muons are obtained from tables derived by cubic spline interpolation from tables in Groom et al. [5].

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 Reports 37 and 49 [6], 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 by electrons, tables of radiative stopping powers of muons) that are used in the calculations.

References

• [1]
  F. Salvat, P. Andreo, SBETHE: Stopping powers of materials for swift charged particles from the corrected Bethe formula, Comput. Phys. Commun. 287 (2023) 108697.
• [2]
  F. Salvat, penelope-2024: A Code System for Monte Carlo Simulation of Electron and Photon Transport, OECD Nuclear Energy Agency, document NEA/MBDAV/R(2024)1, OECD Publishing, Paris, 2025, https://doi.org/10.82155/1vk5-0513.
• [3]
  F. Salvat, C. Heredia, Electromagnetic interaction models for Monte Carlo simulation of protons and alpha particles, Nucl. Instrum. Meth. B 546 (2023) 165157.
• [4]
  S.M. Seltzer, M.J. Berger, Bremsstrahlung spectra from electron interactions with screened atomic nuclei and orbital electrons, Nucl. Instrum. Meth. B 12 (1985) 95–134; S.M. Seltzer, M.J. Berger, Bremsstrahlung energy spectra from electrons with kinetic energy 1 keV–10 GeV incident on screened nuclei and orbital electrons of neutral atoms with Z= 1–100, At. Data Nucl. Data Tables 35 (1986) 345–418.
• [5]
  D.E. Groom, N.V. Mokhov, S.I. Striganov, Muon Stopping power and range tables 10 MeV-100 TeV, At. Data Nucl. Data Tables 78 (2001) 183–356.
• [6]
  ICRU Report 37, Stopping Powers for Electrons and Positrons (ICRU, Bethesda, MD, 1984); ICRU Report 49, Stopping Powers and Ranges for Protons and Alpha Particles (ICRU", Bethesda, MD, 1993).