PreprintCC BY-NC-SA 4.0arXiv
Closed analytical formulas for the Ashley-Ritchie-Brandt and Jackson-McCarthy polarization functions
Abstract
The Z13 correction in the Bethe-Bloch formula that quantifies the electronic stopping power of matter for heavy charged particles is revisited. This correction accounts for the Barkas-Andersen effect. Ashley, Ritchie and Brandt (ARB) wrote it in terms of a non-relativistic polarization function, and Jackson and McCarthy (JM) obtained the relativistic counterpart. Both expressions involve intricate triple integrals. The present manuscript describes how the polarization functions of JM and ARB can be reduced to compact analytical formulas. The corresponding asymptotic behaviours are also found. The accuracy of two currently-used empirical fits to the ARB and JM polarization functions is assessed.
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BibTeX
@misc{fernandezvarea2026closed,
title = {{Closed analytical formulas for the Ashley-Ritchie-Brandt and Jackson-McCarthy polarization functions}},
author = {J. M. Fernández-Varea and A. Fernández-Mrosko},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08017},
eprint = {2611.08017},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08017},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Closed analytical formulas for the Ashley-Ritchie-Brandt and Jackson-McCarthy polarization functions
AU - J. M. Fernández-Varea
AU - A. Fernández-Mrosko
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08017
UR - https://arxiv.org/abs/2611.08017
AB - The Z_1^3 correction in the Bethe-Bloch formula that quantifies the electronic stopping power of matter for heavy charged particles is revisited. This correction accounts for the Barkas-Andersen effect. Ashley, Ritchie and Brandt (ARB) wrote it in terms of a non-relativistic polarization function, and Jackson and McCarthy (JM) obtained the relativistic counterpart. Both expressions involve intricate triple integrals. The present manuscript describes how the polarization functions of JM and ARB can be reduced to compact analytical formulas. The corresponding asymptotic behaviours are also found. The accuracy of two currently-used empirical fits to the ARB and JM polarization functions is assessed.
ER -