The Reflection Eigenvalues in the Criticality problem for Linear-Quadratic Anisotropic Scattering

Abstract

The recently studied Linear-Quadratic anisotropic scattering function in one-speed neutron transport theory is used to solve the reflection eigenvalues for the certain critical slab thicknesses. A slab reactor including the nuclear material is taken into account for the linear-quadratic anisotropic scattering with the reflection boundary condition. The aim is to find the reflection coefficients according to the secondary neutron numbers and the reactor thickness. The numerical results are obtained by using the Modified FN method.

Keywords

• Linear-Quadratic anisotropic scattering,Case’s eigenfunctions,Modified FN method,Criticality equation,Critical thickness,Reflection eigenvalues

Lineer-Kuadratik Anizotropik Saçılma için Kritiklik Probleminde Yansıtıcı Özdeğerleri

Abstract

Tek-hızlı nötron transport teoride son zamanlarda çalışılan lineer-kuadratik anizotropik saçılma fonksiyonu belirli kritik levha kalınlıkları için yansıtıcı öz değerlerini çözmek amacıyla kullanılır. Lineer-kuadratik anizotropik saçılma özelliği gösteren nükleer materyale sahip bir düzlem reaktör göz önüne alınmıştır ve problem yansıtıcı sınır koşullarını içermektedir. Amaç belirli ikincil nötron sayılarına ve belirli reaktör kalınlıklarına göre yansıtıcı katsayılarını bulmaktır. Sayısal sonuçlar Modified F_N yöntemi kullanılarak bulunmuştur. 

Keywords

• Lineer-kuadratik anizotropik saçılma,Case özfonksiyonları,Modified FN method,kritiklik denklemi,Kritik kalınlık,Yansıtıcı özdeğerleri

References

1. B. Davison, Neutron Transport Theory. London: Oxford University Press, 1958, ch. 1-8.
2. K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Physics, vol. 9, no. 1, pp. 1-23, 1960.
3. K. M. Case, and P. F. Zweifel, Linear Transport Theory. Reading Mass: Addition-Wesley, 1967, ch. 4-6.
4. M. M. R. Williams, Mathematical Methods in Particle Transport Theory. New York: Wiley-Interscience, 1971, ch. 3-8.
5. G. I. Bell, and S. Glasstone, Nuclear Reactor Theory. New York: Von Nostrand Reinhold, 1972, ch. 1-3.
6. W. M. Stacey, Neutron Transport Theory. 2nd ed., Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA, 2007. ch. 9.
7. J. Mika, “Neutron transport with anisotropic scattering,” Nucl. Sci. Eng., vol. 11, no. 4, pp. 415-427, 1961.
8. N. J. McCormick, and I. Kušcer, “Half-space neutron transport with linearly anisotropic scattering,” Journal of Mathematical Physics, vol. 6, no. 2, pp. 1939-1945, 1965.

9. N. J. McCormick, and I. Kušcer, “Singular eigenfunction expansions in neutron transport theory,” Advances in Nuclear Science and Technology, vol. 7, pp. 181-282, 1973.
10. V. Protopopescu, and N. G. Sjöstrand, “On the solution of the dispersion equation for monoenergetic neutron transport with linearly anisotropic scattering,” Prog. Nucl. Energ., vol. 7, no. 1, pp. 47-58, 1981.
11. C. E. Siewert, and P. Benoist, “The FN method in neutron-transport theory, Part I: Theory and applications,” Nucl. Sci. Eng., vol. 69, no. 2, pp. 156-160, 1979.
12. P. Grandjean, and C. E. Siewert, “The FN method in neutron-transport theory, Part II: Applications and numerical results,” Nucl. Sci. Eng., vol. 69, no. 2, pp. 161-168, 1979.
13. A. Kavenoky, “The CN method of solving the transport equation: Application to plane geometry,” Nucl. Sci. Eng., vol. 65, no. 2, pp. 209-225, 1978.
14. C. Tezcan, A. Kaşkaş, M. Ç. Güleçyüz, “The H-N method for solving linear transport equation: theory and applications,” J. Quant. Spectrosc. Ra., vol. 78, no. 2, pp. 243-254, 2003.
15. N. G. Sjöstrand, “Complex eigenvalues for neutron transport equation with quadratically anisotropic scattering,” J. Nucl. Sci. Technol., vol. 18, no. 1, pp. 1-5, 1981.
16. R. G. Türeci, and D. Türeci, “Time dependent albedo problem for quadratic anisotropic scattering,” Kerntechnik, vol. 72, no. 1-2, pp. 59-65, 2007.
17. R. G. Türeci, and M.Ç. Güleçyüz, “The slab albedo and criticality problem for the quadratic scattering kernel with the H-N method,” Kerntechnik, vol. 73, no. 4, pp. 171-175, 2008.
18. R. G. Türeci, “Solving the criticality problem with the reflected boundary condition for the triplet anisotropic scattering with the Modified FN method,” Kerntechnik, vol. 80, no. 6, pp. 583-591, 2015.
19. R. G. Türeci, and D. Türeci, “Half-space albedo problem with Modified FN method for linear and quadratic anisotropic scattering,” Kerntechnik, vol. 82, no. 2, pp. 239-245, 2017.
20. R. G. Türeci, and D. Türeci, “Slab albedo for linearly and quadratically anisotropic scattering kernel with Modified FN method” Kerntechnik, vol. 82, no.5, pp. 605-611, 2017.
21. R. G. Türeci, “Half-space albedo problem for the (İnönü, Linear and Quadratic) anisotropic scattering,” Kerntechnik, submitted for publication.
22. G. J. Mitsis, “Transport solutions to the one-dimensional critical problem,” Nucl. Sci. Eng., vol. 17, no. 1, pp. 55-64, 1963.
23. I. Carlvik, “Monoenergetic critical parameters and decay constants for small homogeneous spheres and thin homogeneous slabs,” Nucl. Sci. Eng., vol. 31, no. 2, pp. 295-303, 1968.
24. D. C. Sahni, and N. G. Sjöstrand, “Criticality and time eigenvalues for one-speed neutron transport,” Prog. Nucl. Energ., vol. 23, no. 3, pp.241-289, 1990.
25. D. C. Sahni, N. G. Sjöstrand, and N. S. Garis, “Criticality and time eigenvalues for one-speed neutron in slab with forward and backward scattering,” J. Phys. D. Appl. Phys., vol. 25 no. 10, pp. 1381-1389, 1992.
26. D. C. Sahni, “Some new results pertaining to criticality and time eigenvalues of one-speed neutron transport equation,” Prog. Nucl. Energ., vol. 30, no. 3, pp.305-320, 1996.
27. C. E. Siewert, and M. M. R. Williams, “The effect of anisotropic scattering on the critical slab problem in neutron transport theory using a synthetic kernel,” J. Phys. D. Appl. Phys., vol. 10, no. 15, pp. 2031-2040, 1977.
28. E. B. Dahl, V. Protopopescu, and N. G. Sjöstrand, “On the relation between decay constants and critical parameters in monoenergetic neutron transport,” Nucl. Sci. Eng., vol. 83, no. 3, pp. 374-379, 1983.
29. C. Tezcan, and R. Sever, “The critical slab with the backward-forward-isotropic scattering model,” Ann. Nucl. Energy., vol. 12, no.10, pp. 573-576, 1985.
30. C. Tezcan, and C. Yıldız, “The criticality problems with the FN method for the FBIS model,” Ann. Nucl. Energy., vol. 13, no. 6, pp. 345-348, 1986.
31. D. C. Sahni, and N. G. Sjöstrand, “Criticality and time eigenvalues in one-speed neutron transport,” Prog. Nucl. Energ., vol. 23, no. 3, pp. 241-289, 1990.
32. N. S. Garis, “One-speed neutron transport eigenvalues for reflected slabs and spheres,” Nucl. Sci. Eng., vol. 107, no. 4, pp. 343-358, 1991.
33. C. Tezcan, and C. Yıldız, “The transformation of PL solutions for extremely anisotropic scattering to solutions with isotropic scattering and application to the critical slab problem,” Journal of Quantative Spectroscopy and Radiative Transfer, vol. 49, no. 4, pp. 411-416, 1993.
34. M. A. Atalay, “The critical slab problem for reflecting boundary conditions in one-speed neutron transport theory,” Ann. Nucl. Energy., vol. 23, no. 3, pp. 183-193, 1996.
35. M. A. Atalay, “The reflected slab and sphere criticality problem with anisotropic scattering in one spedd neutron transport theory,” Prog. Nucl. Energ., vol. 31, no. 3, pp. 229-252, 1997.
36. R. G. Türeci, M. Ç. Güleçyüz, A. Kaşkaş, and C. Tezcan, “Application of the H-N method to the critical slab problem for reflecting boundary conditions,” Journal of Quantative Spectroscopy and Radiative Transfer, vol. 88, no. 4, pp. 499-517, 2004.
37. R. G. Türeci, M. Ç. Güleçyüz, A. Kaşkaş, C. Tezcan, “The singular eigenfunction method: the critical slab problem for linearly anisotropic scattering,” Kerntechnik, vol. 70, no. 5-6, pp. 322-326, 2005.
38. M. Ç. Güleçyüz, R.G. Türeci, and C. Tezcan, “The critical slab problem for linearly anisotropic scattering and reflecting boundary conditions with the H-N method,” Kerntechnik, vol. 71, no. 1-2, pp.149-154, 2006.
39. R. G. Türeci, “The criticality problem with linear-quadratic anisotropic scattering for reflected boundary condition,” Kerntechnik, submitted for publication.