Anisotropic scattering in half-space transport problems

Abstract

Case's singular normal mode analysis of half-space transport problems is extended to include anisotrophic scattering. The cases of absorbing and non-absorbing media are treated separately. By way of illustration, the solution to the Milne problem is obtained in each instance. It is shown that the results may always be written in terms of an appropriate X-function, and a rapidly convergent iterative scheme for evaluating the X-functions is exhibited. However, differences from the isotropic scattering problem arise: For the nonabsorbing medium the zeros of the characteristic function coalesce at infinity; for the absorbing medium, the normal mode analysis contains undetermined constants which are seen to be related to coefficients in the Laurent series of the X-function. In each case the solution is carried out with the appropriate modifications, and Case's three X-function identities are rederived.