The Moments Method for the one-group transport and diffusion equations

Abstract

We analyze the Moments Method for the one-group transport equation in an infinite, isotropic and homogeneous medium and generalize some of the results found in the literature. We derive general recursion relations for global moments of the form of a product of a polynomial in ( x, y, z) of order n times a spherical harmonic Y kl ( Ω) and of the form r n times a Legendre polynomial of order k of Ω· r/ r. We determine which moments can be obtained by a finite recursion and obtain the maximum order of the eigenvalue of the scattering operator that enters the recursion relation, defining thus an equivalence class between scattering operators. We have also investigated and derived general recursion relations between one-dimensional transport like equations for transverse moments of the form ρ n ( e φ · ρ/ρ ) k , where ρ is the radius vector in the xy plane and e φ is the unit vector in the direction of the projection of Ω onto the xy plane. In particular, we investigate the expression for the mean distance to absorption < r 2>. The results are extended to the case of the one-group diffusion equation and an expression for < r 2> is also derived for the particular case of a medium in thermal equilibrium.