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Exact Solutions of the Neutron Slowing Down Equation

Trine-Yie Dawn, Chio-Min Yang

Nuclear Science and Engineering

Volume 61 / Number 2 / October 1976 / Pages 142-158


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The problem of finding the exact analytic closed-form solution for the neutron slowing down equation in an infinite homogeneous medium is studied in some detail. First we consider the existence and unique properties of the solution of this equation for both the time-dependent and the time-independent cases. A direct method is used to determine the solution of the stationary problem. The final result is given in terms of a sum of indefinite multiple integrals by which solutions of some special cases and the Placzek-type oscillation are examined. The same method can be applied to the time-dependent problem with the aid of the Laplace transformation technique, but the inverse transform is, in general, laborious. However, the solutions of two special cases—(a) where the scattering and absorption cross sections both vary as 1/υ and (b) where the scattering cross section is assumed to depend on lethargy, u, in the form Σs(u)υ(u) = (Σsυ)0 exp(-κu) (κ > 0) and a 1/υ absorption cross section—are obtained explicitly. We also compare our results with previously reported works in a variety of cases. The time moments for the positive integral n are evaluated, and the conditions for the existence of the negative moments are discussed.

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