Radiation transport between two concentric spheres

The eigenfunctions for the transport of radiation between concentric spheres are proved to be the spherical harmonics and Legendre polynomials. The corresponding eigenvalues are then derived for the cases of both an outer emitting sphere and an inner absorbing sphere, and an inner emitting sphere and an outer absorbing sphere. Simple approximations to these eigenvalues are found and their various properties are derived. These results are then applied to the problem of ICF capsules inside hohlraums.