Complete Spherical Harmonics Solution of the Boltzmann Equation for Neutron Transport in Homogeneous Media with Cylindrical Geometry

Abstract

It is shown that the treatment of the transport equation in cylindrical geometry does not involve essentially more tedious calculations than the treatment in plane geometry. A complete solution is given for homogeneous media including the solutions. Every partial solution contains in its expansion of spherical harmonics some functions of a parameter with appropriate coefficients. It will be shown that these functions are Legendre polynomials and Legendre functions of the second kind as in the case of plane geometry for the main solution, and derivatives of these functions for the complementary solutions. They are solutions of the recursion relations for the expansion and yield a further recursion relation for the coefficients. Tables of these coefficients are given up to the eleventh spherical harmonic approximation and a general formula is derived for them. Two examples are worked out, a first based upon the supposition of a linearly anisotropic scattering law, and a second in which two higher terms of anistropy are added to this law. (auth)