The transport theory of beams

Abstract

In this paper we review analytic solution methods for the problem of a monoenergetic pencil beam of radiation impinging perpendicularly upon a thin source-free slab, infinite in the transverse directions. If the scattering is highly peaked in the forward direction, and if the slab is sufficiently thin, this beam will remain nearly collimated as it passes through the slab. Making the continuous slowing down and straightahead approximations, the energy variable can be analytically eliminated from the problem. If a simplified Fokker-Planck approximation in angle is introduced into the resulting monoenergetic transport equation, a very simple analytic solution is possible. This result for the scalar flux is known as the Fermi-Eyges solution, and is Gaussian in radius. For scattering phase functions which do not admit a Fokker-Planck representation in angle, a very recent formalism is discussed for addressing this monoenergetic problem. A classic example is the widely used Henyey-Greenstein scattering phase function, and in this case an analytic solution, algebraic in radius, is obtained for the scalar flux. For screened Rutherford scattering, this formalism leads to a quadrature result for the scalar flux. An asymptotic numerical evaluation of this result shows good agreement with benchmark Monte Carlo calculations.