Synthesis of an Elementary Function for the Spatially Dependent Neutron Slowing Down Distribution in Hydrogen

Abstract

An elementary function for the collision density of neutrons slowing down from a plane source in hydrogen is synthesized through a set of schemes incorporating several known explicit features of the exact solution. First, the Marshak distribution is assigned a distorted lethargy variable that makes its zeroth and second spatial moments exact at all lethargies while automatically preserving its detailed accuracy at large lethargies. The same exact moments are also required of a specific functional form able to assume the correct spatial dependence at small lethargies. Then, a linear combination of these functions is constructed with coefficients making the two moments and the first spatial derivative at the source plane exact at all lethargies. The resulting distribution automatically becomes correct at both lethargy extremes. In addition, a remaining lethargy-dependent parameter makes the fourth spatial moment exact at all lethargies except within a finite interval of intermediate values, where its error must reach a maximum of 2.7%. Extraneous roots from multiple bifurcations of the parameter are identified by their unphysical implications. For computational simplicity, both this parameter and the incorporated function for the exact spatial derivative at the source plane are replaced by fitted elementary functions. The resulting expression for the collision density agrees very closely with McInerney's Monte Carlo calculations. Some extensions are described in a separate Note.