On Energy-Dependent Neutron Diffusion

TL;DR

The paper addresses the ambiguity in energy-dependent neutron diffusion by presenting a formal second-order asymptotic derivation of the multigroup diffusion equation and, crucially, an exact inversion of the $P_1$ flux–current relation to obtain a matrix diffusion coefficient $D_{g,g'}$ that can be computed from cross-section data. The diffusion-coefficient expression is obtained via a Neumann-series inversion of $(Id - T^{-1} S)$, yielding a physically interpretable diffusion process as a sequence of energy-changing scatterings with an average migration length across groups, $D_{g,g'} = ilde{\Lambda}_{g,g'}/3$. A hydrogen-infinite-medium benchmark demonstrates reasonable agreement with established methods like CMM and OS, illustrating practical applicability and potential benefits for Monte Carlo cross-section generation and heterogeneous-model simulations. The work relaxes restrictive assumptions of prior approaches and provides a straightforward, exact route to compute accurate multigroup diffusion coefficients, with future work focusing on broader benchmarking and angular-dependence effects.

Abstract

While the energy-dependent neutron diffusion equation is widely employed in nuclear engineering, its status as an approximation to the transport equation is not yet completely understood, and several different approximations are in use to determine the diffusion coefficients. Past work on the theory underlying the diffusion approximation has often made use of asymptotic arguments; in the energy-dependent case, however, papers have appeared that differ substantially in their findings. Here we present a formal asymptotic derivation of the multigroup diffusion equation which addresses these differences, along with the varying and sometimes physically stringent assumptions employed in these works. Further, we show a way to exactly invert the relationship between flux and current in the P1 approximation, giving a matricial expression for the multigroup diffusion coefficient which is formally exact, has clear physical meaning, and which can be easily computed to arbitrary precision on the basis of cross-section data already produced by lattice calculations. The resulting 2-group diffusion coefficient for an infinite medium of hydrogen is calculated with Monte Carlo, and compared to the those deriving from the Cumulative Migration Method and from the out-scatter approximation.