Calculation of Averages for Primary Recoil Distributions

Abstract

The slowing down of an energetic particle in a random assembly of scattering centers is discussed. Certain averages for physically interesting quantities can be calculated in a very general form, including unequal masses of scattering and slowing down particles. The averages refer to quantities such as the total distance traveled (L) and the vector distance to the final positions (x_i) and powers thereof. Two assumptions simplify the evaluation of the general results considerably: (α) The mean free path λ is a simple power of the energy E: λ(E)∼E^μ; (β) the distribution g(E; ε) of energies ε after one collision with initial energy E does not change its shape with E: Eg (E; ε)=κ(ε/E). A special distribution of this kind is the hard core distribution. Under assumptions α and β all calculations are reduced to the evaluation of an integral like ∫κ(η)η^μdη. For equal masses and the hard core distribution, one can easily calculate averages of any power of L and obtain the distribution function of L, this being proportional to L^1/μ exp(−L/λ). Under the assumption β alone, with arbitrary λ(E), results are given for special distributions and equal masses (including the hard core distribution). The evaluation of the averages now requires simple integrals over λ for linear averages and double integrals for quadratic averages with known integral kernels. For unequal masses, approximations are discussed.