Spherical harmonics convergence studies for critical spheres

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The use of spherical harmonics approximations of the monoenergetic Boltzman neutron transport equation to determine the critical radius of a spherical reactor is considered with emphasis placed upon the even-order approximations. Numerical computations indicate that in terms of the prediction of a critical radius the P 2 approximation is usually superior to the P 1 approximation. For higher order approximations the odd-order approximations are invariably superior to their corresponding even-order approximations. When Mark's vacuum-interface boundary conditions are used for a bare spherical reactor or the appropriate interfacial boundary condition applied to a reactor with a thick reflector, the even-order approximations provide a lower bound on the exact critical radius. Under the same circumstances, the odd-order approximations provide an upper bound. A particular weighted average of these counter convergent even and odd-order approximations is found to converge more rapidly than the use of either independently.