An exact solution to the problem of the time-energy distribution of fast particles slowing down in a multi-species medium

Abstract

Calculational methods are developed for obtaining analytical solutions of the transport equations which describe the slowing down of fast atoms in a host medium. The situation is quite general in that the injected particles may be present in the host and thus the history of the recoil atoms are recorded. Using a modified form of the momentum approximation for the inter-particle scattering cross-section, it is found possible to employ Mellin transforms to solve the equations. Further use of some methods developed in neutron transport theory enable an exact inversion of the Mellin transform to be obtained. The general time-energy distribution of a particle slowing down, the energy distribution arising from a steady source and the mean slowing down time of particles from a pulsed source have all been obtained. The case when the cross-sections are inversely proportional to the speed can be dealt with in a particularly simple form.