Barkas effect in electronic stopping power: Rigorous evaluation for the harmonic oscillator

The Coulomb interaction between a heavy fast charged particle and an electron bound by a harmonic force has been evaluated accurately up to the second-order term in the Born series. The resulting mean energy transfer is given as a function of impact parameter, projectile velocity, and the resonance frequency of the oscillator. Integration over the impact parameter yields the stopping cross section up to third order in the projectile charge $e_1$ without introduction of an adjustable cutoff radius. Unlike previous treatments, the validity of the evaluation of the stopping cross section within the given physical model is not restricted to the high-speed limit. Numerical results cover the velocity range 0.1≤2${\mathrm{mv}}^2$/ħω≤20. We find a significant $e_1^3$ (or Barkas) correction in the mean energy transfer at all impact parameters. The classical Barkas correction, as given for the harmonic oscillator in the high-speed limit by Ashley, Ritchie, and Brandt [Phys. Rev. B 5, 2393 (1972)], approaches the quantal result for large impact parameters, but the approach is slow in the velocity range covered, and large discrepancies are observed in the range of impact parameters that determine the stopping cross section. Stopping cross sections for protons on atomic hydrogen and helium have been estimated roughly by assigning a single resonance frequency to the target atom. The results cover a velocity range down to somewhat below the stopping maximum where the Barkas correction turns out to be large. This indicates that a properly executed Barkas correction plays an important role in the prediction of the height and the position of the stopping maximum.