High-Fidelity Approximations for Extinction Probability Calculations

A quantity that is frequently of interest in stochastic neutronics calculations is the probability of extinction (POE), or its complement, the survival probability. Even within the simplest stochastic point kinetics formulations, the POE is typically extracted from numerical calculations or approximated. An example of the latter strategy involves the truncation of the fission multiplicity distribution at two, resulting in the “quadratic approximation.” While this methodology yields closed-form results for the POE, it is valid only for supercritical multiplication near unity. In this technical note, we attempt to obviate fission multiplicity truncation in the construction of transient and infinite time limit closed-form POE solutions. In the infinite time limit, we arrive at the necessity of solving a quintic algebraic equation; we provide a brief discussion of the mature formalism available for solving quintic equations and generate a variety of simple representations using hypergeometric series. We evaluate and discuss both the new and existing approximations in the context of an example ²³⁵U system and compare their validity over a range of supercritical multiplication factors.