ON THE RATE OF CONVERGENCE OF THE SPHERICAL-HARMONICS METHOD FOR A SANDWICH REACTOR OF SMALL LATTICE PITCH

Abstract

There are some cases where the spherical-harmonics-method calculations can be carried out purely algebraically, without specifying numerically the order of approximation used. Such is the problem of determining the spatial distribution of the thermal neutron flux in an infinite sandwich reactor of a small lattice pitch. The spherical-harmonics-method solution of this problem, in an arbitrary order of approximation, is compared with the exact solution. It is shown that if both are expanded in the Fourier series in terms of the optical depth, the nth term of the Fourier series for the spherical-harmonics-method solution differs from the corresponding term for the exact solution by the factor[Formula: see text]where N is the order of approximation used in the spherical-harmonics method and ε is one-half of the lattice pitch on the optical scale.This result should provide some guidance in assessing the rate of convergence of the spherical-harmonics method also for the more complex (and realistic) cases.