Neutron Penetration and Slowing Down at Intermediate Distances through Medium and Heavy Nuclei

This paper reports new theoretical results and numerical calculations of neutron slowing down and penetration to fairly large distances from the source assuming constant mean free path and no absorption. The Placzek one-velocity problem which results from a Laplace transformation of the Boltzmann equation is solved approximately by keeping only a finite number of harmonic terms in the Laplace transform of the scattering function but not restricting the expansion of the neutron density. The Laplace inversion is carried out by the method of steepest descent. Numerical results are given for nuclei of mass infinity and 9, carried out to three orders of approximation, namely zeroth, first, and second. They show the expected transition behavior between age theory and the asymptotic exponential behavior. The accuracy of the method is high, being greater than that of the numerical work over the useful range of the calculations. Rough agreement with Wick's asymptotic results for $M=12\mathrm{}$ is found.