Probability Distribution Functions for Collision Energy Loss

Abstract

In this chapter, the pdfs describing charged particle collision energy losses of three different conditions are derived. These results are of $f\left( {x,\Delta E} \right)$ where $f\left( {x,\Delta E} \right)\mathrm{ d}\left( {\Delta E} \right)$ represents the probability that a charged particle has lost kinetic energy between $\Delta E$ and $\Delta E + \mathrm{ d}\left( {\Delta E} \right)$ while penetrating into a depth x within a medium.

The differentiator between these pdfs will be the depth of penetration x or, equivalently, the thickness of the medium x. Large thicknesses result in a symmetric Gaussian pdf (under the requirement that the assumptions specified in the previous chapter are met). Reduced thicknesses result in the asymmetric pdfs of the Vavilov and Landau theories. All three conditions are derived in this chapter.

While the asymmetric results noted above provide analytical results that can be solved readily through numerical means, practical calculations, particularly for the rapid requirements of Monte Carlo simulations, are of interest. Such methods are reviewed.