The synthetic kernels of neutron slowing-down theory and a related expansion

Abstract

Various approximate solutions of the integral equation for neutron moderation are obtained by truncating an asymptotic expansion for the collision density, a process which is shown to be equivalent to replacing the scattering kernel by an alternative synthetic kernel which permits "exact" solution of the integral equation. Truncation after one or two terms, and approximation of a constant in one case, yield the classical kernels. It is shown that the appropriate kernel to be used depends on the particular quantity under investigation. In particular, an illustration is given showing why the Wigner kernel is superior to that of GoertzeLand Greuling for calculating resonance absorption. Difficulties may arise when using the entire non—truncated series. It is shown that, for the sources of practical interest in reactor theory, these difficulties may be removed by splitting the source into two "components", one rapidly decaying and the other not. When the source is not known sufficiently accurately for this splitting to be done, the series must, in general, be truncated to avoid divergence. The method of obtaining a kernel by truncation is consistent with the matching of moments method.