# Swift charged particles in a degenerate electron gas: An estimation for the Barkas effect in stopping

## TL;DR

This paper analytically investigates the stopping power of a degenerate electron gas for fast charged particles, emphasizing the charge-sign-dependent Barkas effect and providing a refined estimation using phase shifts and a Hulthén-type potential.

## Contribution

It introduces an analytical approach using scattering phase shifts to accurately characterize charge-sign effects in stopping power, improving upon conventional estimations.

## Key findings

- The exact leading phase shift captures both charge-sign-independent and dependent terms.
- The Barkas term is found to be smaller than traditional estimates.
- The Bloch term relates to differences between Born and exact treatments.

## Abstract

Applying a recently suggested new form [Phys. Rev. A {\bf 94}, 042704 (2016)] for the stopping power in terms of scattering phase shifts, here we show analytically that an exact leading phase shift may contain that information which is completely enough to characterize the asymptotic charge-sign-independent Bethe term ($\propto{Z_1^2}$) and a charge-sign-dependent Barkas term ($\propto{Z_1^3}$) in the stopping force of a degenerate electron gas for fast projectiles with charge $Z_1$. Our analytic implementation is based on a Hulth\'en-type potential with velocity-dependent screening. The next term in an asymptotic expansion, the Bloch term ($\propto{Z_1^4}$), measures the difference of the exact and first-order Born treatments with Coulomb potential. We found a reduced value for the Barkas term, in comparison with the conventional estimation which rests on the transport cross section.