A variational treatment of the asymptotic flux behavior in a halfspace

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Abstract
We obtain a variational estimate of the flux at infinity for a subcritical homogeneous halfspace driven by both an incident flux and an internal source. The angular dependences of the scattering phase function, the incident flux, and the internal source are allowed to be arbitrary. The basic variational functional is derived incorporating the transport equation and boundary conditions in the defining problem as constraints, using a Lagrange multiplier technique. Explicit results are obtained using asymptotic (Case discrete mode) trial functions. Improvements in the basic result are obtained by treating the uncollided and once collided fluxes exactly, and by iterating a lower order variational result. A by-product of this analysis is two approximations to Chandrasekhar's H-function, and we compare these simple analytic approximations with exact results in the special case of isotropic scattering.