Real criticality eigenvalues of the one-speed linear transport operator

Abstract

The criticality eigenvalue problem has been studied for the one-speed neutron transport equation. Convex bodies of arbitrary shape and with vacuum boundary conditions have been considered. The cross sections may be space dependent, and the scattering is assumed to be anisotropic. Several new conditions have been derived which ensure that a point spectrum of eigenvalues exists and that all the eigenvalues are real. The most general such condition is that the even order coefficients in the development of the scattering function have a different sign than the odd order ones. As a consequence, for linearly anisotropic scattering the eigenvalues are all real if the average cosine of the scattering angle is negative. Numerical results computed for homogeneous bodies in the form of spheres and infinite slabs and cylinders confirm the theoretical considerations.