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The Thermal-Neutron Milne Problem with a Two-Term Degenerate Kernel

M. M. R. Williams

Nuclear Science and Engineering

Volume 27 / Number 3 / March 1986 / Pages 511-519


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The nonabsorbing thermal-neutron Milne problem is solved for isotropic scattering in the laboratory system. The scattering kernel has been approximated by a two-term degenerate sum and the resulting equations are solved by analytic continuation, together with Wiener-Hopf factorization. The solution so obtained is not explicit in the sense of quadratures, but is in the form of a nonsingular Fredholm equation, which is ideally suited to solution by iteration once certain generalized energy-dependent H-functions have been tabulated. The energy transfer properties of the approximate kernel are discussed, and their effect on the structure of the total flux evaluated. In general, the complete solution consists of an asymptotic part, together with a rethermalization term, which is connected intimately with the energy exchange process, and the integral transient which depends markedly on the variation of the total cross section with energy. It is shown that, when the cross section is constant, the rethermalization term becomes zero and the solution reverts to the one-velocity one, multiplied by a Maxwellian. Certain properties of the energy-dependent H-functions are discussed in the Appendix.

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