An Approximate Solution to the Asymptotic Slowing Down Equations of Fast Neutrons

The relation is proposed as an approximate solution to the asymptotic slowing down equations in an infinite, homogenous, and weakly absorbing mixture of elements, in the energy range of fast-reactor neutrons. q(E) is the slowing down density; a is the absorption ratio Σ_a/Σ_t/; s_λ(E) and Q_λ are, respectively, the scattering ratio Σ_s,λ/Σ_t and the excitation energy of the λ'th level; ξ is equal to the average logarithmic energy loss per elastic scattering with the element containing the λ'th level; the sum extends over all elastic (Q_λ = 0) and inelastic (Q_λ > 0) levels. The above relation is constructed to reduce to the approximate solutions both in the limit of purely elastic scattering and in the limit of inelastic scattering by infinitely heavy scatterers. The relation is shown to be an approximate solution also in intermediate cases, where both target recoil and level excitation are important, provided that the mixture contains a substantial amount of medium-mass or light scatterers. Higher order terms may be included in the relation to better account for the effects of absorption.