SP ∞-A ∞ elementary solutions in general geometry

Abstract

For a restricted, but not trivial, class of neutron diffusing systems, namely those in which the total cross section is everywhere constant (“constant sigma”, or Cσ systems), the one-velocity, isotropic scattering transport equation in general 3D geometry can be transformed without any approximations into a second order integrodifferential equation that is formally identical to an energy-dependent diffusion equation. The latter equation, which can also be considered as the limit of the SP 2N-1 (or A N) system of equations as N→∞, allows for a further transformation into a boundary integral form. If the Cσ system is made of homogeneous regions (each one with its own value of the scattering cross section), the Green function, or fundamental solution, to be used in each region can be worked out explicitly and is shown to involve a bilinear expansion in terms of the Case eigenfunctions. Such a structure is mirrored by the structure of the solution of the transport problem, which can also be given the form of a (regionwise) superposition of these eigenfunctions.