The three-dimensional transport equation with applications to energy deposition and reflection

A detailed investigation of the energy deposition in, and surface reflection from, an infinite half-space has been made. Two types of source are considered: the first is a line source embedded in the medium perpendicular to the surface and the second is an incident pencil, of arbitrary direction, incident at a point. The resulting problem involves three dimensions in space and therefore requires description by a transport equation in the appropriate coordinates. The physical problem considered is that of a beam of incident ions or a line ion source in the medium. Only the fate of the foreign incident ions is considered and no attempt is made to follow the recoil atoms generated. Progress is made in the analytical solution of the problem by assuming an energy-independent mean free path and the transport approximation for the scattering kernel. The Wiener-Hopf method is used together with Fourier transforms for transverse directions. Considerable success has been achieved in obtaining exact solutions for some special limiting cases, and the numerical results which emerge are tabulated.