The spatial dependence of the energy spectrum of slowing down particles

An extensive investigation has been carried out of the energy spectrum of particles in a half-space irradiated with a monoenergetic beam, ie the classical albedo problem but extended to include energy dependence. Slowing down is described by elastic scattering and various models for dealing with the anisotropic nature of the scattering, especially important for mass ratios A<1, are proposed. To obtain analytical solutions the case of an energy dependent cross-section is considered when the Laplace transformed equation can be reduced to an effective one-speed problem with isotropic scattering. The equation is solved by the Wiener-Hopf technique and explicit expressions are obtained for the angular distribution of emitted particles and the spatial variation of the spectrum within the medium. Surface phenomena are considered and the emergent current and the energy reflection coefficient, γ, for mass ratios in the range are compiled. The leakage current for large A is peaked near the source energy but becomes progressively closer to the infinite medium, E-1, form as A decreases. A detailed calculation of γ shows that its value for hard-sphere scattering follows closely the results obtained by Sigmund who used a modified infinite medium approach. An extension of the method to include the effects of anisotropic scattering in the C.M. system is discussed and illustrated. Finally, the energy deposited by the infinite medium approximation of Sigmund is calculated and its accuracy can be assessed by comparison with the exact half-space results. It is found that for hard-sphere scattering, the infinite medium values of γ are always less than the exact values. However, for A<1, the error never exceeds 3.5%. It becomes progressively worse as A increases. (author)