Solution to the monoenergetic neutron Boltzmann equation for a finite parallelepiped

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Abstract
A solution to the integro-differential Boltzmann equation, which governs the steady-state space-angle distribution of the flux of monoenergetic neutrons in a rectangular parallelepiped of an isotropically scattering material, is presented. Results for both the angular and the total fluxes are obtained by resorting to a three-dimensional Fourier transform technique. The non-singular linear integral equation for the transform of the total flux is studied by considering the integral transformation associated with its kernel. This transformation is recognized to be compact. In view of this compactness, the kernel of the given non-singular transform equation can then be approximated, as closely as desired, by a kernel of finite rank. The solution thus obtained is finally transformed back to yield the original total flux distribution, from which the angular flux distribution also follows.

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