On the spectrum of discrete–ordinates neutron transport problems

Over the last six decades, the discrete spectrum of the neutron transport operator has been widely studied. Significant theoretical results can be found in the literature regarding the one–speed linear transport equation with anisotropic scattering. In this study, the discrete–ordinates (SN) transport problem with anisotropic scattering has been considered and the discrete spectrum results in multiplying media have been corroborated. The numerical results obtained for the dominant SN eigenvalues agreed with the ones for the analytic problem reported in the literature up to a triplet scattering order. A compact methodology to perform the spectral analysis to multigroup SN problems with high anisotropy order in the scattering and fission reactions is also presented in this paper.


INTRODUCTION
presented a general procedure with the intent to analytically solve the Boltzmann Transport Equation (BTE) by the expansion of its solution into a complete set of eigenfunctions [1]. This is, undoubtedly, one of the most remarkable studies in the field of neutral particle transport theory [2]. Even though eigenfunctions might present a lack of utility in the solution to practical nuclear engineering problems, this method has been applied among different fields in physics seeking a comprehensive mathematical understanding [2,3]. Case found an analytic solution to the steadystate, homogeneous BTE in slab geometry considering one-speed particles and isotropic scattering.
Over the past 60 years, the method has been applied to more complex problems that may consider energy dependence, multiplying media, anisotropic scattering, heterogeneity, and/or multiple dimensions [2,4].
Case proved that the solutions are given by two discrete modes corresponding to a ± pair of eigenvalues that lie outside interval (−1; +1), in addition to a complimentary continuous eigenvalue spectrum over the interval [−1; +1] [1]. This pair of discrete dominant eigenvalues are conventionally referred to as -eigenvalues [5] since they depend on the material cross section by = + .
In regard to obtaining a discrete eigenvalue spectrum, besides the monoenergetic problem with isotropic scattering [1], the main results have been applied to problems considering linearly anisotropic scattering [6], and more recently, arbitrary-order anisotropic scattering [4,7,8]. In Section 2 of this paper, the methodology described by Sahni and Tureci [4] is summarized and the main results for all the mentioned cases are presented.
To the best of this author's knowledge, no published work has extended these results to include multigroup these results to multigroup transport problems in the discrete-ordinates ( ) formulation considering arbitrary order of anisotropy on the scattering and fission reactions. The procedures described here present a general solution to the problem cited before, including the possibility of obtaining eigenvalues over the complex plane. The spectral analysis for the BTE that supports the discrete eigenvalues of the Case's spectrum has also been performed.

MATERIALS AND METHODS
According to the notation used by Sahni and Tureci [4], the steady-state BTE for one-speed neutrons in a slab-geometry homogeneous media can be written as In Equation (1), the conventional terms apply: is the total macroscopic cross-section, c is the mean To solve Equation (1), the method of separation of variables is applied by the substitution that yields At this point, the recursion relations for Legendre polynomials are applied along with some algebraic manipulations to obtain the transcendental equation whose roots are the discrete values of ξ and appear in ± pairs. Moreover, if is a complex number, then its complex conjugate ̅ is also a root.

Spectral analysis of the transport equations
The time-independent multigroup BTE with −'th order of anisotropy on both the scattering and fission terms within a region of a multiplying slab [9,10] is considered with appropriate boundary conditions ∈ , = 1: , = 1: .
The angular quadrature of order is defined by the discrete directions ( ) and their associated with the definitions: To solve the homogeneous equation, Equation (6), it is considered the function It is noteworthy that Equation (8) is analogous to the ansatz in Equation (2). Now, this expression is substituted into Equation (6) to obtain, after some operations, an eigenvalue problem of order . (9) By solving this eigenvalue problem, a set of linearly independent eigenfunctions defined in Equation (8) for ∈ is obtained. The eigenvalue problem from the BTE, Equation (9), was solved for all the examples reported by Sahni and Tureci [4]. It was started at low orders of the Gauss-Legendre quadrature that were increased up to obtain results in agreement with the discrete Case's spectrum within a range of less than 100 . The RMatrixEVD subroutine from the ALGLIB library [11] was used, in order to find the eigenvalues (real and imaginary parts) and eigenvectors of a general matrix. Tables 2 to 4 show the results obtained from solving the eigenvalue problem published by Sahni and Tureci [4]. In all cases, the moduli of the dominant eigenvalues and the relative deviations in with respect to the reference values are presented. One can observe that for a quadrature order = 64, the relative deviation is less than 80 in all cases.  a Only the magnitude of real or purely imaginary eigenvalue pair is tabulated. b relative deviation ( ) with respect to the discrete analytic eigenvalue [4].

Solution to the BTE
As a result of the previous analysis, in this subsection, a methodology to obtain the analytic solution to the slab-geometry multigroup BTE in multiplying media [12][13][14] is proposed. It is remarked that the presented procedures can also be used to derive the homogeneous component of the general solution in fixed-source problems [15][16][17][18][19][20]. Equation (5) can be represented in matrix form as where is the -order square matrix with entries and is a column matrix whose entries are ( ). The solution to the homogeneous system of ordinary differential equations in Equation (10) can be written as where are the eigenvectors associated to the eigenvalues ξ of matrix . Depending on the material parameters, the eigenvalues ξ can appear in ± real pairs, imaginary or complex conjugate. Since the input of matrix are real numbers, the real eigenvalues will be associated to real eigenvectors, and the complex conjugate eigenvalues will be associated to complex conjugate eigenvectors.
As it is the case, when = + and = − are a pair of eigenvalues, the eigenvectors associated to and are = + and = − , respectively [21]. After some operations, two real-valued solutions are obtained 1 ( ) = ( cos − sin ) and = ( cos + sin ) .
Therefore, if real eigenvalues and complex conjugate pairs are found, i.e., + 2 = , a set of linearly independent eigenfunctions is obtained, and the solution to Equation (5) where , and ′ are arbitrary constants to be determined. Equation (14) can be represented in matrix form as with the definition of matrices The terminology presented here for the spectral analysis can be used in the development and implementation of spectral nodal methods to obtain accurate and efficient numerical solutions to the BTE in slab-geometry. It is noteworthy that the notation used here is general and compact, nevertheless, it can be modified, and one should perform the construction of the matrices and the order of the operations aiming at computational efficiency.

CONCLUSION
In this paper, the spectral analysis of the BTE with anisotropic scattering has been performed and the results with the discrete Case's eigenvalues from the analytic transport problem have been compared. As one could anticipate, for a quadrature order high enough, the dominant eigenvalues agree with the discrete spectrum. A simplification for the procedures to obtain the analytic solution of the BTE considering high-order anisotropic events in the scattering and fission sources was also presented. This simplification has been presented in a compact form and includes the possibility of obtaining complex eigenvalues from the spectral analysis. The present methodology shall be applied not only to slab-geometry problems but also to multidimensional spectral nodal methods that use transverse integration procedures.