# Analytic saddlepoint approximation for ionization energy loss distributions

## TL;DR

This paper introduces a saddlepoint approximation for ionization energy loss distributions applicable across a range of relativistic velocities, providing a computationally efficient method that closely matches detailed models.

## Contribution

A novel saddlepoint approximation method for ionization energy loss distributions that is accurate and computationally efficient across various relativistic velocities.

## Key findings

- Accurately reproduces Vavilov's most probable energy loss within 1%.
- Matches the full width at half maximum within 10%.
- Performs better as Vavilov's kappa approaches 1.

## Abstract

We present a saddlepoint approximation for ionization energy loss distributions, valid at arbitrary relativistic velocities of the incident particle 0<v/c<1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal's distribution. This distribution is intended for use in simulations with relatively low computational overhead. The approximation generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov's kappa approaches 1.