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Spectral analysis of the anisotropic neutron transport kernel in slab geometry with applications

Leonard, A. and Mullikin, T. W. (1964) Spectral analysis of the anisotropic neutron transport kernel in slab geometry with applications. Journal of Mathematical Physics, 5 (3). pp. 399-411. ISSN 0022-2488. doi:10.1063/1.1704132.

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A spectral analysis of the transport kernel for anisotropic scattering in finite slabs is achieved by first solving a type of generalized scattering problem for a subcritical slab. Initially, the scattering problem is stated as an inhomogeneous integral transport equation with a complex-valued source function. This is readily transformed to singular integral equations and linear constraints in which the space and angle variables enter as parameters. Dual singular equations appear in applications of Case's method to transport problems, but we cannot yet completely explain this duality. The singular equations are transformed to Fredholm equations by an extension of Muskhelishvili's standard method and by analytic continuation. It is shown that, for a wide class of scattering functions, this particular Fredholm reduction yields equations which converge rapidly under iteration for all neutron productions and slab thicknesses. The ultimate solution of the singular equations contains arbitrary constants which, when evaluated by the aforementioned linear constraints, display explicitly the Fredholm determinant and the eigenfunctions of the transport kernel. An immediate consequence of this result is the criticality condition and the associated neutron distribution. Specific applications to linear anisotropic and isotropic scattering in slab geometry are discussed. In addition, it is seen that the case of isotropic scattering in spheres can be treated with this method, and, in fact, the spectral analysis of the kernel for the slab problem immediately applies to the sphere kernel.

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Additional Information:© 1964 American Institute of Physics. Received 15 October 1963. This research is sponsored by the United States Air Force under Project RAND-Contract No. AF 49(638)-700 monitored by the Directorate of Development Planning, Deputy Chief of Staff, Research and Development Headquarters, United States Air Force.
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United States Air ForceAF 49(638)-700
Issue or Number:3
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11848
Deposited By: Tony Diaz
Deposited On:09 Oct 2008 16:50
Last Modified:08 Nov 2021 22:22

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