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

The aim of this paper is to show 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 complementary 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 anisotropy are added to this law.