T N approximation on the critical size of time-dependent, one-speed and one-dimensional neutron transport problem with anisotropic scattering Hakan Öztürk a, * , Süleyman Güngör b a Osmaniye Korkut Ata University, Technical and Vocational School of Higher Education, Osmaniye 80000, Turkey b Çukurova University, Faculty of Science and Letters, Department of Physics, Adana 01330, Turkey article info Article history: Received 29 July 2008 Received in revised form 2 January 2009 Accepted 9 January 2009 Available online 23 February 2009 abstract The criticality problem is studied based on one-speed time-dependent neutron transport theory, for a uniform and finite slab, using the Marshak boundary condition. The time-dependent neutron transport equation is reduced to a stationary equation. The variation of the critical thickness of the time-dependent system is investigated by using the linear anisotropic scattering kernel together with the combination of forward and backward scattering. Numerical calculations for various combinations of the scattering parameters and selected values of the time decay constant and the reflection coefficient are performed by using the Chebyshev polynomials approximation method. The results are compared with those previ- ously obtained by other methods which are available in the literature. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Criticality type eigenvalues are needed for various applications in reactor physics such as pulsed neutron experiments. In this phenomena, the number of neutrons decreases with time after a short neutron pulse. Then, by considering the behavior of the angular flux of neutron population at later times, fundamental and higher order time eigenvalues (decay constants) are com- puted to determine the criticality conditions of the system. Fun- damental time eigenvalue is defined as the smallest number of k’s for the time-dependent system which have a neutron distribu- tion with a term exp(kt)(Sahni and Sjöstrand, 1990). In homoge- neous systems, determining the criticality conditions and computing the time decay constant can be considered to be iden- tical. If a relation between the decay constant and criticality parameters is established then the results obtained for a critical system could be used for a time-dependent system. Therefore, when the neutron flux density in a pulsed neutron experiment decays exponentially it is possible to remove the time depen- dence of the neutron flux from the transport equation. Then, the time-dependent transport problem can be reduced to a stationary one. The time eigenvalues of many systems can be determined by calculating the critical size of those systems in stationary condi- tion (Carlvik, 1968; Dahl et al., 1983; Sahni and Sjöstrand, 1990; Sahni et al., 1992; Yildiz, 1999). There are many cases in which an extended knowledge about the scattering of neutrons through the media should be taken into consideration in order to investigate the solution of the transport equation for a more accurate analysis. Among the polynomial expansion based techniques, the spherical harmonics or P N method is the most widely used in transport theory applications. However, in some circumstances, the P N expansion of angular flux near mate- rial boundaries may be a poor representation. Hence, Aspelund (1958), Conkie (1959) and Yabushita (1961) presented some disad- vantages of the P N method, especially in the case of anisotropic scattering and computation of the extrapolated end points, and thus they used the Chebyshev polynomials approximation (T N method) in their studies. Recently, Anli et al. (2006a,b) applied a modified version of the T N method to the critical and reflected crit- ical slab problems (Öztürk et al., 2007). In this work, the reflected critical slab problem is studied by using Chebyshev polynomials of first kind, i.e., the T N method. Therefore, the work previously carried out by Öztürk et al. (2007) is extended to a time-dependent system with linearly anisotropic scattering together with both forward and backward scattering. In the method, first the time-dependent transport equation is re- duced to a stationary one, and then the neutron angular flux is ex- panded in terms of the Chebyshev polynomials of first kind. Finally, the critical slab thicknesses are calculated for various values of the anisotropy parameters a, b, and b 1 , selected values of time decay constant K, and the reflection coefficient R. The numerical results obtained for the reflected critical slab thickness of the time-dependent system will be tabulated together with the corresponding results obtained by various other methods and the ones previously reported in the literature. 0306-4549/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2009.01.012 * Corresponding author. Tel.: +90 328 825 1818; fax: +90 328 825 0097. E-mail addresses: hozturk@cu.edu.tr, hakanozturk3@hotmail.com (H. Öztürk). Annals of Nuclear Energy 36 (2009) 575–582 Contents lists available at ScienceDirect Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene