Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry (Q1031804)

An equation for a neutron angular flux (a neutron transport equation) involves integration of the flux function \(\psi(z,x)\) multiplied by a neutron scattering kernel \(\sigma_S(\mu_0)\) over an angular variable. Here, \(\mu_0=\cos\theta_0=\Omega\cdot\Omega'\) is the cosine of the scattering angle. Thus someone's ability to solve the neutron transport equation analytically or numerically is limited due to the lack of information on the structure of \(\sigma_S(\mu_0)\). A conventional approach involves a Fourier expansion of it w.r.t.\ Legendre polynomials, \(\sigma_S(\mu_0)=\sum_{n=0}^{\infty}\sigma_nP_n(\mu_0)\), with the subsequent use of several initial terms of it. However, neglecting all Fourier coefficients \(\sigma_n\) except for \(n=0,1,2\), may lead to a significant loss of information on essential features of an anisotropic flux. To explore the properties of the highly anisotropic fluxes, the authors suggest using a one-parameter model scattering kernel with the known, simple and complete Legendre expansion. Namely, they propose using the generating function for the Legendre polynomials, \(\sigma_S(\mu_0)=\sigma_S/\sqrt{1-2\mu_0 t+t^2}=\sigma_S\sum_{n=0}^{\infty}t^nP_n(\mu_0)\). In spite of such a choice is not supported by theoretical or experimental data, the idea may have certain potential in applications.