Calculation of the diffusion lengths for one-speed neutrons in a slab with forward and backward scattering

Abstract

The diffusion lengths for one-speed neutrons in a slab are calculated using the first kind of Chebyshev polynomials approximation (T_N) method. The scattering models are constituted in place of the scattering function with an argument of the cosine of the neutron scattering angle. Therefore, the forward-backward-isotropic (FBI) scattering model is used as the scattering function in transport equation which describes the interaction and the conservation of the neutrons throughout a system. In the solution algorithm, first the neutron angular flux is expanded in terms of the Chebyshev polynomials of first kind. After inserting this expansion in the transport equation, the coupled differential equations are derived using the properties of the Chebyshev polynomials of first kind. These equations are solved together and then the diffusion equation is obtained by applying the first order approximation (N = 1) which is known as the diffusion approximation. Finally, the diffusion lengths for one-speed neutrons are calculated for selected values of the collision, backward and forward scattering parameters. The calculated diffusion lengths are given in the tables together with the ones already obtained in literature in order to indicate the applicability of the present method. The convenience and rapid convergence of the present method with its easily executable equations can be observed from the derived equations and the results in tables.

Keywords

• Forward and backward scattering,Diffusion length,Slab geometry

References

1. 1. Lamarsh, J.R., Baratta, A.J., Introduction to nuclear engineering. 2001, USA: Prentice Hall, Inc.
2. 2. Davison, B., Neutron transport theory. 1958, London: Oxford University Press.
3. 3. Case, K.M, Zweifel, P.F., Linear transport theory. 1967, USA: Addison-Wesley Publishing Company.
4. 4. Aspelund, O., On a new method for solving the (Boltzmann) equation in neutron transport theory. PICG, 1958. 16: p.530-534.
5. 5. Conkie, W.R., Polynomial approximations in neutron transport theory. Nucl. Sci. Eng., 1959. 6: p.260-266.
6. 6. Yabushita, S., Tschebyscheff polynomials approximation method of the neutron transport equation. J. Math. Phys., 1961. 2: p.543-549.
7. 7. Anlı, F., Yaşa, F., Güngör, S., Öztürk, H., TN approximation to neutron transport equation and application to critical slab problem. J. of Quant. Spectros. Radiat. Trans., 2006. 101: p.129-134.
8. 8. Öztürk, H., Anlı, F., Güngör, S., TN method for the critical thickness of one-speed neutrons in a slab with forward and backward scattering. J. of Quant. Spectros. Radiat. Trans., 2007. 105: p.211-216.

9. 9. Tıraş, M., Öztürk, H., Bülbül, A., T1 and U1 approximations to neutron transport equation in one-dimensional spherical geometry. Kerntechnik, 2014. 79: p.163-166.
10. 10. Yıldırım, S., Öztürk, H., Diffusion length calculations for one-speed neutrons in a slab with backward, forward and linear anisotropic scattering. Kerntechnik, 2014. 79: p.243-246.
11. 11. Arfken, G., Mathematical methods for physicists. 1985, London: Academic Press, Inc.
12. 12. Bell, G.I., Glasstone, S., Nuclear reactor theory. 1972, New York: VNR Company.