Calculating Alpha Eigenvalues and Eigenfunctions with a Markov Transition Rate Matrix Monte Carlo Method

Abstract

For nuclear systems with or without fissile material, the alpha eigenvalues and eigenfunctions describe the exponential time-dependent behaviour of the neutron flux within the problem resulting from any arbitrary initial source. The transition rate matrix method (TRMM) uses a steady-state, general purpose Monte Carlo code to tally rates and probabilities to estimate a transition rate matrix (TRM) that describes neutrons transitioning among the position-energy-direction phase space. The eigenvalues of the TRM are estimates of the alpha eigenvalues of the underlying system. With this information, we use classical eigenfunction expansion methods and bi-orthogonality to compute expansion coefficients that characterize the initial conditions and time-dependent source, yielding a method for calculating the transient behaviour within the problem.

In applications of the TRMM to infinite media, where the TRM describes neutrons transitioning among the energy phase space, calculated alpha eigenvalues agree with analytical spectra for multi-group problems, and eigenfunction expansions match time-dependent Monte Carlo (TDMC) solutions for continuous-energy problems. In applications of the TRMM to one-speed, one-dimensional media, calculated alpha eigenvalues match those obtained by other semi-analytical methods, and eigenfunction expansions match TDMC solutions. The TRMM shows similar success for continuous-energy, one-dimensional media. These verifications of the TRMM use smaller research Monte Carlo codes.

Implementation of the TRMM into the open-source Monte Carlo code OpenMC demonstrates the abilities of the TRMM in obtaining the alpha eigenvalues and eigenfunctions of large, three-dimensional problems. For these problems, the large multi-dimensional phase space increases the size of the TRM, and sparse storage techniques become important in avoiding memory limitations. Also, the TRMM must now efficiently determine the eigenvalues of very large matrices. We present results for applications to a small, fast burst reactor and a large, graphite-moderated reactor, where the TRMM is able to calculate the appropriate fundamental shape eigenfunctions and a few higher shape eigenfunctions.