Analytical calculations of neutron slowing down and transport in the constant-cross-section problem

Some aspects of the problem of neutron slowing down and transport in an infinite medium consisting of a single nuclide that scatters elastically and isotropically and has energy-independent cross sections were investigated. The method of singular eigenfunctions was applied to the Boltzmann equation governing the Laplace transform (with respect to the lethargy variable) of the neutron flux. A new sufficient condition for the convergence of the coefficients of the expansion of the scattering kernel in Legendre polynomials was rigorously derived for this energy-dependent problem. Formulas were obtained for the lethargy-dependent spatial moments of the scalar flux that are valid for medium to large lethargies. In deriving these formulas, use was made of the well-known connection between the spatial moments of the Laplace-transformed scalar flux and the moments of the flux in the ''eigenvalue space.'' The calculations were greatly aided by the construction of a closed general expression for these ''eigenvalue space'' moments. Extensive use was also made of the methods of combinatorial analysis and of computer evaluation, via FORMAC, of complicated sequences of manipulations. For the case of no absorption it was possible to obtain for materials of any atomic weight explicit corrections to the age-theory formulas for the spatial moments M/sub 2n/(u) of the scalar flux that are valid through terms of the order of u-5. The evaluation of the coefficients of the powers of n, as explicit functions of the nuclear mass, is one of the end products of this investigation. In addition, an exact expression for the second spatial moment, M2(u), valid for arbitrary (constant) absorption, was derived. It is now possible to calculate analytically and rigorously the ''age'' for the constant-cross-section problem for arbitrary (constant) absorption and nuclear mass. 5 figures, 1 table