Numerical Evaluation of the Three-Dimensional Searchlight Problem in a Half-Space

Abstract

The linear Boltzmann equation for the transport of neutral particles is investigated with the objective of generating a benchmark-quality calculation for the three-dimensional searchlight problem in a semi-infinite medium. The derivation assumes stationarity, one energy group, and isotropic scattering. The scalar flux (both surface and interior) and the current at the surface are the quantities of interest. The source considered is a pencil-beam incident at a point on the surface of a semi-infinite medium. The scalar flux will have two-dimensional variation only if the beam is normal; otherwise, it is three-dimensional. The solutions are obtained by using Fourier and Laplace transform methods. The transformed transport equation is formulated so that it can be related to a one-dimensional pseudo problem, thus providing some analytical leverage for the inversions. The numerical inversions use standard numerical techniques such as Gauss-Legendre quadrature, summation of infinite series, H-function iteration and evaluation, and Euler-Knopp acceleration. The numerical evaluations of the scalar flux and current at the surface are relatively simple, and the interior scalar flux is relatively difficult to calculate because of the embedded two-dimensional Fourier transform inversion, Laplace transform inversion, and H-function evaluation. Comparisons of these numerical solutions to results from the MCNP probabilistic code and the THREEDANT discrete ordinates code are provided and help confirm proper operation of the analytical code.