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  Discovering, Making, and Testing New Materials: SRNL’s Center For Hierarchical Waste Form Materials
  Savannah River National Laboratory researchers are building on the laboratory’s legacy of using cutting-edge science to effectively immobilize nuclear waste in innovative ways. As part of the Center for Hierarchical Waste Form Materials, SRNL is leveraging its depth of experience in radiological waste management to explore new frontiers in the industry.
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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 Σ_s(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.