A Conventional Phase Function with the Chebyshev Polynomials of Second Kind for the Criticality Problem in Transport Theory

Öz

The criticality calculations for one-speed neutrons in a finite homogenous slab are done using the conventional Henyey-Greenstein (HG) phase function in transport theory. After defining the phase function in transport equation, the neutron angular flux is expanded in terms of the Chebyshev polynomials of second kind (U_N method). Then, the critical half-thicknesses of the slab are calculated for various values of the scattering parameters. The numerical results obtained from the present method are given in the tables together with the ones obtained using an alternative phase function (Anlı-Güngör, AG) for comparison.

Anahtar Kelimeler

• Henyey-Greenstein Function,Critical Slab,UN Method,Transport Equation

Transport Teoride Kritiklik Problemi için İkinci Tip Chebyshev Polinomları ile Konvansiyonel Bir Faz Fonksiyonu

Öz

Transport teoride konvansiyonel Henyey-Greenstein (HG) faz fonksiyonu kullanılarak sonlu homojen bir dilimdeki nötronlar için kritiklik hesaplamaları yapılmıştır. Transport denkleminde faz fonksiyonu tanımlandıktan sonra, açısal nötron akısı ikinci tip Chebyshev polinomları cinsinden seriye açılmıştır (U_N metodu). Daha sonra, saçılma parametrelerinin farklı değerleri için dilimin yarı-kalınlıkları hesaplanmıştır. Mevcut metotla elde edilen nümerik sonuçlar, alternatif faz fonksiyonu (Anlı-Güngör, AG) kullanılarak elde edilen sonuçlar ile karşılaştırma yapmak için tablolarda verilmiştir.

Anahtar Kelimeler

• Henyey-Greenstein Fonksiyonu,Kritik Dilim,UN Metodu,Transport Denklemi

Kaynakça

1. Davison, B., Neutron Transport Theory, Oxford University Press, London, 1958.
2. Case, K. M., Zweifel, P. F., Linear Transport Theory, Addison-Wesley Publishing Company, 1967.
3. Bell, G. I., Glasstone, S., Nuclear Reactor Theory, VNR Company, New York, 1972.
4. Sahni, D. C., Sjöstrand, N. G., Garis, N. S., Criticality and time eigenvalues for one-speed neutrons in a slab with forward and backward scattering, J. Phys. D: Appl. Phys., 25, 1381-1389, 1992.
5. Henyey, L. G., Greenstein, J. L., Diffuse radiation in the galaxy, Astrophys. J., 93, 70-83, 1941.
6. Reynolds, L. O., McCormick, N. J., Approximate two parameter phase function for light scattering, J. Op. Soc. Am., 70, 1206-1212, 1980.
7. Williams, M. M. R., A synthetic scattering kernel for particle transport in absorbing media with anisotropic scattering, J. Phys. D: Appl. Phys., 11, 2455-2463, 1978.
8. Liu, P., A new phase function approximating to Mie scattering for radiative transport equations, Phys. Med. Biol., 39, 1025-1036, 1994.

9. Öztürk, H., The reflected critical slab problem for one-speed neutrons with strongly anisotropic scattering, Kerntechnik, 73, 66-74, 2008.
10. Öztürk, H., Anlı, F., Güngör, S., Application of the UN method to the reflected critical slab problem for one-speed neutrons with forward and backward scattering, Kerntechnik, 72, 74-76, 2007.
11. Öztürk, H., Anlı, F., Diffusion approximation for certain scattering parameters of the Anli-Güngör phase function, Kerntechnik, 77, 381-384, 2012.
12. Öztürk, H., Application of the Henyey-Greenstein and Anlı-Güngör phase functions for the solution of the neutron transport equation with Legendre polynomials: Reflected critical slab problem, Kerntechnik, 78, 447-453, 2013.
13. Anlı, F., Yaşa, F., Güngör, S., General eigenvalue spectrum in a one-dimensional slab geometry transport equation, Nucl. Sci. Eng., 150, 72-77, 2005.
14. Arfken, G., Mathematical Methods for Physicists, Academic Press, Inc., London, 1985.
15. Lee, C. E., Dias, M. P., Analytical solutions to the moment transport equations-I; one-group one-region slab and sphere criticality, Ann. Nucl. Energy, 11, 515-530, 1984.
16. Öztürk, H., Ege, Ö., An application of the alternative phase function with the second kind of Chebyshev polynomials: The criticality problem in transport theory (submitted to Kerntechnik).
17. Yıldız, C., Variation of the critical slab thickness with the degree of strongly anisotropic scattering in one-speed neutron transport theory, Ann. Nucl. Energy, 25, 529-540, 1998.
18. Sahni, D. C., Sjöstrand, N. G., Non-monotonic variation of the criticality factor with the degree of anisotropy in one-speed neutron transport, Transp. Theory Statist. Phys., 20, 339-349, 1991.