Time eigenvalues of the one-speed neutron transport equation for spherically asymmetric modes in a homogeneous sphere

Abstract

The integral transform (IT) method has been generalized to obtain spherically symmetric as well as asymmetric eigenmodes of the one-speed, integral neutron transport equation in a homogeneous sphere with isotropic scattering. The IT method as developed originally is a applicable only to symmetric modes with real eigenvalues below the Corngold limit. The method presented in this paper can be used to investigate the discrete time eigenvalue spectrum in the full complex domain. The total neutron flux is expanded in spherical harmonics of the spatial unit vector r r and decoupled integral equations are obtained for the spherical harmonic moments. The kernel in each of these equations is then decomposed in a bilinear form. This decomposition reduces the integral equation to an infinite system of algebraic equations, which is then truncated for numerical solution. The matrix elements for this sytem of equations are shown to be identical to those encountered in the case of spherically symmetric modes. These are evaluated by generalizing Hembd's procedure to the complex domain. The time eigenvalues corresponding to the first few eigenmodes are presented as a function of system size.