A non-Gaussian treatment of radiation pencil beams

The problem of describing steady-state transport of a perpendicularly incident particle beam through a thin slab of material is considered For a scattering kernel sufficiently peaked in momentum transfer to allow a Fokker-Planck description of the scattering process in both energy and angle, an approximate closed form solution to this problem was obtained almost 50 yr ago and is referred to as the Fermi-Eyges formula. It is shown that a Fermi-Eyges-like formula can be derived for a broader class of scattering kernels. This class consists of scattering described by the continuous slowing-down approximation (the Fokker-Planck description in energy), but not sufficiently forward peaked in angle to allow an angular Fokker-Planck representation. This generalized formula reduces to the classic Fermi-Eyges result for scattering operators,vith a valid Fokker-Planck limit and also describes problems that, while involving a forward-peaked scattering kernel, do not possess a Fokker-Planck description. A classic example of such a kernel is the Henyey-Greenstein kernel, and the Fermi-Eyges-like solution in this case exhibits more beam spreading than that predicted by the classic Fermi-Eyges formula. In particular, the scalar flux is non-Gaussian in the radial coordinate, as contrasted with the Gaussian Fermi-Eyges result.