Energy loss spectrum of swift charged particles penetrating a layer of material

TLDR: In this paper, a steepest-descent evaluation of the Bothe-Landau integral yields general expressions for the spectrum, the most probable energy loss, and the halfwidth, without reference to a particular collision cross section.

Abstract: The energy loss spectrum of a beam of charged particles penetrating a layer of random material has been analyzed theoretically. A steepest-descent evaluation of the Bothe-Landau integral yields general expressions for the spectrum, the most probable energy loss, and the halfwidth, without reference to a particular collision cross section. The procedure has been tested against rigorous results for model cross sections as well as the Landau-Vavilov theory for free-Coulomb scattering. The most probable energy loss is found to be systematically smaller than the mean energy loss by an amount which is independent of foil thickness for not-too-thin foils. This peak shift turns out to be insensitive to the angle of emergence from the foil but sensitive to nonuniformities in foil thickness and to detector resolution. The influence of shell corrections on the spectrum has been discussed, and the peak shift has been evaluated explicitly for the Landau-Vavilov case, for moderately relativistic ions, for slow ions stopped in a free electron gas, and for heavy ions stopped by elastic collisions. A procedure has been outlined for determining stopping power and straggling from measured peak energy losses and halfwidths. Systematic errors may be present in current stopping power data, caused by the use of most probable instead of mean energy loss values. Estimated peak shifts compare well with recent experimental results.