Spectral PN(λ) approximation(s) to the transport problems in spherical media

In this study, the applicability of ultraspherical-polynomials (PN(λ)) approximation as a spectral approach to spherical media transport/transfer problems is investigated for strongly anisotropic scattering media in which the scattering kernel is an admixture of linearly anisotropic and strongly backward/forward anisotropic parts. It is shown that the PN method (corresponding to the sub-case λ→1/2 in the PN(λ) approximation), among all ultraspherical-polynomials methods, is the unique method converting the one-speed neutron transport equation into an algebraic eigenvalue problem in spectral functions Gn(ν). It is also demonstrated that the TN method (corresponding to the sub-case λ→0 in the PN(λ) approximation) is another algebraic spectral approach suitable for spherical transport/transfer problems with restriction of strongly peaking anisotropy in scattering. The PN(λ) approximation as λ→−1/2 is found to be a prospective candidate, which can be converted into an algebraic spectral case by forcing some spectral functions into pre-selected proper forms. Even though the PN(λ) approximations applicable as a spectral approach to the spherical media transfer problems are shown to be limited to the sub-cases aforementioned, the unifying PN(λ) formalism makes it possible to study spherical media transfer problems in general. As an example, the PN(λ) method with any variable λ is demonstrated to be applicable to the bare homogeneous sphere criticality problems by using pseudo-slab approach. Critical radii calculated are in very close agreement with literature values and equiconvergent.