Transfer-Matrix Method for Gamma-Ray and Neutron Penetration

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Abstract
The problem of radiation transport is formulated in terms of a transfer matrix H which is a 2 × 2 matrix of operators. H is simply related to the more intuitive transmission and reflection operators T and R. An explicit expression for H is derived in slab geometry for radiation distributions that depend on the angle with the slab normal and on energy. H for a multilayer slab is the matrix product of the transfer matrices for the individual layers. A formal expression for H for a homogeneous slab of finite thickness is found in terms of the T and R appropriate to an infinitestimally thin slab. These in turn are related to the single-scattering distribution and therefore can be computed from the microscopic cross section. For purposes of computation, finite matrix representations for the operators must be introduced corresponding to the finite vectors which approximate the distributions. Expansions of the distributions in the cosine of the angle and group representations in energy were chosen in the present work. Some numerical results are presented for gamma rays on aluminum. The extension to problems with internal sources and to nonplanar geometries is outlined.