Heavy Moderator Approximations in Neutron Transport Theory

The flow of neutrons slowing down into a given angular and lethargy (or energy) interval equals an integral over the contributions from neutrons at other angles and lethargies within the medium. The two angular dependences of the integrand are commonly expanded in Legendre polynomials, and for heavy moderators, the lethargy dependence being integrated over is expanded in a power series about the lethargy of the exit neutrons. For each of these three expansions there is an index whose effect on the rate at which the resulting coefficients TLL'n vanish with increasing moderator mass is found. These estimates indicate where to terminate the lethargy expansions of the components of the current so as to maintain a consistent order of accuracy for each, and the sensitivity of the approximations to individual Legendre coefficients of the scattering cross sections follows. Exact recursion formulas derived for the TLL'n contain infinite sums, which can be terminated such that the uncertainty in the terms retained in the transport theory approximations are no larger than the estimated values of terms neglected. The familiar ``consistent P1 approximation'' is generalized as a particular example, and it is shown to be sensitive to only the first three Legendre coefficients of the cross section. The Greuling-Goertzel approximation is also generalized and its merits noted. The associated difficulties in treating recoil due to inelastic scattering are discussed.