Approximate, analytic solutions of the Bethe equation for charged particle range

TL;DR

This paper derives approximate analytical solutions for the Bethe equation to calculate charged particle ranges in matter, providing a useful tool for validation and complementing numerical methods.

Contribution

It introduces polynomial and Taylor expansion methods to analytically integrate the Bethe equation, enabling quick and accurate range calculations.

Findings

Ranges match reference data within expected Bethe model accuracy

Analytic expressions for non-relativistic energy deposition rate

Methods can validate and complement numerical solutions

Abstract

By either performing a Taylor expansion or making a polynomial approximation, the Bethe equation for charged particle stopping power in matter can be integrated analytically to obtain the range of charged particles in the continuous deceleration approximation. Ranges match reference data to the expected accuracy of the Bethe model. In the non-relativistic limit, the energy deposition rate was also found analytically. The analytic relations can be used to complement and validate numerical solutions including more detailed physics.