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The SPN-PL Method for Neutron Transport in Homogeneous Slabs with Anisotropic Scattering

H. Kschwendt

Nuclear Science and Engineering / Volume 44 / Number 3 / Pages 423-434

June 1971


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A synthesis and generalization of several recently developed methods for the numerical solution of the neutron transport equation in a homogeneous slab, assuming anisotropic scattering and energy dependence is presented. The generalization lies in the explicit inclusion of anisotropic scattering. After a Fourier transformation, a system of linear integral equations is obtained, the kernal of which is expanded in spherical Bessel functions. To process the final result in the direction of numerical evaluation, an approximation is proposed that results in the SPN - PL method where the flux is given by a double sum over spatial and angular Legendre polynomials. The expansion coefficients are determined from a system of linear integral equations. Treating the energy dependence by means of the multigroup concept, this system is reduced to a linear system of algebraic equations. Corresponding matrix elements depend on the optical thickness of the slab and can be computed from expansions available for arbitrary slab thicknesses. The SPN - PL method is of great practical importance since it is possible to obtain the solution of the transport equation with low computational effort. For example, assuming monoenergetic neutrons and isotropic scattering, the first and second eigenvalues of the transport equation can both be obtained with five exact digits from 3 × 3 or 4 × 4 matrices. The influence of the mean value of the linear anisotropy on the first and second eigenvalue and the decay constant is studied in detail. The validity of our approach is confirmed by comparing it with the SN and other methods. For certain mean values and optical thicknesses the second eigenvalue is found to be a complex number. Critical flux distribution is determined with great accuracy and shows perfect agreement with other published values. The flux due to a δ source, and a combination of a δ source with a flat one, is analyzed; it is confirmed that the SPN - PL method is not only applicable to small systems, but also (in most cases) to very large assemblies.

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