Computation of some transport theory expressions for a line source in an infinite half-space

Abstract

In a previous paper, Williams [Williams, M.M.R., 2007. The transport and diffusion theory of a line source in an infinite half-space with internal reflection. Ann. Nucl. Energy 34, 910–921] has obtained transport theory closed form expressions for the particle density for a problem in which a filamentary source is placed within a half-space, perpendicular to the surface (this “line source” problem is closely related to the classical “searchlight” problem of radiative transfer). Some numerical results for limiting cases (surface, z → 0), where the expressions reduce to a single integral were also reported. In this paper, we report additional numerical results for arbitrary internal positions in the medium by evaluating the closed form expressions which have required efficient and accurate evaluations of the generalized H function and subsequent two fold integrations. We have verified the accuracy of the techniques and the results through comparisons with both the limiting case ( z → 0) reported in Williams [Williams, M.M.R., 2007. The transport and diffusion theory of a line source in an infinite half-space with internal reflection. Ann. Nucl. Energy 34, 910–921] and also independent results obtained by numerical solutions of the related integral equation via use of the singularity subtraction technique and quadratures. The present work should further aid in progress on related problems with Fresnel reflections and other complications, as well as to the validation of standard commercial reactor physics computer codes. For completeness, we should mention the recent book by Ganapol [Ganapol, B.D., 2008. Analytical benchmarks for nuclear engineering applications: case studies in transport theory, NEA, OECD] in which a number of three-dimensional transport problems are solved with special reference to acceleration techniques.