The critical slab problem for linearly anisotropic scattering with the HN method
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Date
2006
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Publisher
Azerbaijan National Academy of Sciences Institute of Radiation Problems ve Turkish Atomic Energy Authority
Abstract
Hn method, is used to calculate the critical slab problem for linearly anisotropic scattering. HN method is a combination of CN, FN and the method of elementary solutions. In CN method; the integral equations of the Fredholm type, that is; the third form of the transport equation has been used. The integral equations mentioned above have been related with the angular flux at the boundary of the given medium. They have been expressed in terms of the infinite medium Green function. Infinite medium Green function has been obtained by the Fourier transform technique. In FN method; the integro-differential form of the linear transport equation has been used, the values of eigenvalues have been selected. This selection has been based on an equal spacing scheme for choosing the first two eigenvalues as (v0,l) and the remaining eigenvalues have been spaced equally in interval (0,1).
In Hn method, the integro-differential form of the transport equation has been used as in FN method, the eigenvalues have been calculated using the normalization condition of the eigenfunctions and a different numerical procedure has been considered. In this procedure the expansion of the angular flux ju) in terms of the eigenfunctions of the method of elementary solutions and their orthogonality relations have been used.
In our work we have used HN method. We consider the angular flux Ψ(a, µ), -1 ≤ µ ≤ 1 at one of the boundaries of the slab together with the entering flux Ψ(a, -µ), 0 ≤ µ ≤ 1. Then, we multiply the angular flux Ψ(a, µ) with the singular eigenfunctions and integrate over µ Ɛ(-1, +1), to calculate the expansion coefficients in Ψ (x, µ) Here the orthogonality relation of the singular eigenfunctions of the method of elementary solutions are used. After obtaining these coefficients we consider the entering angular flux Ψ(a, -µ), 0 ≤ µ ≤ 1. We multiply this equation with µ m+1 and integrate over 0 ≤ µ ≤. Thus we get a system of linear algebraic equations. Therefore replacing the appropriate coefficients into the linear algebraic equations the final critically condition is obtained.
It is shown that the method leads to concise equations and accurate numerical results even in the lowest order of approximations.
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Keywords
HN method, HN yöntemiyle, Critical slab problem, Kritik slab sorunu, Linearly anisotropic scattering, Doğrusal anizotropik saçılma
Citation
Tezcan, C. (2006). The critical slab problem for linearly anisotropic scattering with the HN method. The Fourth Eurasian Conference on Nuclear Science and Its Application : Presentations, (s. 204-209). 31 October-3 November 2006. Baku, Azerbaijan.