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Orthogonal Polynomial Angular Expansion of the Boltzmann Equation

G. C. Pomraning, M. Clark, Jr.

Nuclear Science and Engineering

Volume 17 / Number 1 / September 1963 / Pages 8-17


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The angular dependence of the solution of the monoenergetic Boltzmann equation in slab geometry with isotropic scattering is expanded classically in the set of Jacobi polynomials which are orthogonal in the interval −1 to +1 with respect to the weight function w(μ) = (1 − μ)α (1 + μ)β. The low order solution obtained by retaining only the first two terms in the expansion is investigated in detail. In this low order it is shown that a proper choice of α and β leads to the exact asymptotic transport eigenvalue. With this choice of α and β a significant improvement in the linear extrapolation distance and the critical size of a bare slab over the usual (P − 1) diffusion theory is obtained. However, it is shown that, in general, the truncated set of classical Jacobi equations does not conserve neutrons. A modification in the truncation procedure is made in order to obtain neutron conservation while retaining the advantages of the Jacobi expansion. The choices α = β = -½ and α = β = −1 are discussed in some detail and shown to have advantages over the corresponding Legendre (α = β = 0) expansion.

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