Bethe stopping-power formula and its corrections

The classical and quantum theories leading to the asymptotic Bethe formula of the stopping power of matter for charged particles heavier than the electron are briefly reviewed. Models and approximations for the practical calculation of various corrections that extend the validity range of the formula are described. The asymptotic formula and the associated shell correction were determined previously from an extensive database of atomic generalized oscillator strengths, calculated for an independent-electron model with the Dirac-Hartree-Fock-Slater (DHFS) self-consistent potential, with due account for relativistic departures from the Bethe sum rule. The nonrelativistic Bloch correction is extended to the relativistic domain by means of the Lindhard-S\o{}rensen formulation, and an accurate parametrization for point projectiles with small charges is proposed. The density-effect correction and the Barkas correction are obtained from a semiempirical model of the optical oscillator strength (OOS), built from the calculated DHFS contributions of inner electron subshells plus the OOS of outer-shell electrons represented by an analytical expression, which is determined by the composition, mass density, and empirical mean excitation energy, or $I$ value of the material. Inclusion of the shell, density-effect, Lindhard-S\o{}rensen, and Barkas corrections into the asymptotic formula leads to the corrected Bethe formula. A general strategy is proposed to determine the stopping power in terms of only the $I$ value of the material. It is shown that, with the empirical $I$ values recommended in Report 37 of the International Commission on Radiation Units and Measurements, the stopping powers calculated numerically from the corrected formula are in close agreement with available measurements of the stopping power of elemental materials for protons and alpha particles with energies higher than 0.75 and 5 MeV, respectively.