A Rayleigh quotient method for criticality eigenvalue problems in neutron transport

•A Rayleigh quotient method reduces number of transport sweeps.•The discretized eigenvalue transport equations can be shown to be primitive.•The Rayleigh quotient method can solve subcritical alpha-eigenvalue problems.•The method can easily be implemented into existing discrete ordinates transport codes. The alpha- and k-effective eigenproblems describe the criticality and fundamental neutron flux mode of a nuclear system. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. We present Rayleigh quotient methods that are applied to demonstrably primitive discretizations of the one-dimensional slab, multigroup in energy, neutron transport equation. These methods are capable of solving subcritical and supercritical alpha- and k-effective eigenvalue problems. The derived eigenvalue updates are optimal in the least squares sense and positive eigenvector updates are guaranteed from the Perron-Frobenius Theorem for primitive matrices.