The transport equation with anisotropic scattering in finite slab geometry

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Abstract
We consider one‐speed neutron transport or monochromatic radiative transfer in a slab of finite width, with particles incident on both faces. The scattering kernel is represented as a finite sum of Legendre polynomials. An exact expression is obtained for the expansion coefficients of the flux in a singular eigenfunction representation. It involves the X and Y functions of Chandrasekhar and the ql and sl polynomials of Sobolev. Case’s full‐ and half‐range expansions are obtained as limits when the slab thickness becomes zero or infinite, respectively. Orthogonality relations for the singular eigenfunctions and their adjoints are given. Also, we derive a number of orthogonality relations of the Inönü‐type involving moments of the eigenfunctions. Expressions for surface densities, currents, and higher order Legendre moments of the surface fluxes are obtained. Singular integral transform relations between the Chandrasekhar φ and ψ functions and the polynomials gl(ν), are given. Also, we give the solution of slab problems with one face having diffuse or specular reflecting boundary conditions.

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