ON THE RATE OF CONVERGENCE OF THE SPHERICAL HARMONICS METHOD: (FOR THE PLANE CASE, ISOTROPIC SCATTERING)

Abstract

The rate of convergence of the spherical harmonics method in neutron transport theory, for the plane case with isotropic scattering, is investigated, for an arbitrary number of layers of different materials, and both in the one-group and in the multi-group theory. The attention is centered upon the cases when there is no external supply of neutrons, so that one deals with an eigenvalue problem. The error in the eigenvalue due to the truncation of the spherical harmonics expansion is found to vary essentially as 0(1/N2), where N is the order of the highest harmonics retained. The proof of this result consists of two parts. Firstly it is shown that the use of the spherical harmonics method, under the conditions stated, is strictly equivalent to making a certain approximation in the kernel of the integral equation for the flux; and then the error due to this approximation is assessed.