Diffusion Length and Criticality Problems in Two- and Three-Dimensional, One-Speed Neutron Transport Theory. I. Rectangular Coordinates

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Abstract
Several problems in linear transport theory have been solved involving more than one dimension. We have considered the diffusion‐length problem in a slab system and have studied how the flux in the transverse direction varies as a function of distance from the source. The solution is found to consist of an asymptotic term, composed of a finite number of harmonics, plus a transient which is nonseparable in the x and z coordinates. When the slab is sufficiently thin only the transient term survives and the concept of a diffusion length loses its value. A method for overcoming this difficulty is presented. In another problem, the thickness of the slab is allowed to become semi‐infinite and the effect of decay in the z direction on the emergent angular distribution and surface flux are assessed. By approximating the flux in the y and z directions by a function of the form exp {iByy + iBzz} and solving the resulting one‐dimensional problem in the x direction exactly, it has been possible to obtain a statement of the critical conditions in a bare rectangular parallelepiped system. The application of this method to the diffusion length problem in a system with a rectangular cross section is discussed. Finally, by comparison with the exact solution for the slab, we estimate the accuracy of a reduced Boltzmann equation deduced by the author in a previous publication.

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