The thermal-neutron Milne problem

A variational principle devised originally for the isotropic Milne problem without capture has been developed for anisotropic scattering. The theory is described for linear anisotropy but the extension to higher-order anisotropy is, in principle, straightforward. The flux in the energy-dependent problem is expressed in terms of the corresponding one-velocity solution which is already well known; as a result some new one-velocity half-space Green functions are introduced. Then a variational principle for the extrapolation length may be written down and the angular distribution of emergent neutrons determined by iterating the trial function. Results are presented for both isotropic and anisotropic scattering models. Using linearly anisotropic scattering, there is excellent agreement with experiment, except at grazing incidence, for the spectrum emerging from a water surface.