Analytic and Monte Carlo random walk assessments of neutron fission chains and the probability of extinction

Abstract

Conducting safety studies of reactors and other nuclear systems using mainstream Monte Carlo (MC) codes for space-time kinetics (STK) and probability of extinction (POE) analyses is challenging. Left unchecked, lengthy finite and divergent prompt-neutron (PN) fission chains can cause lengthy execution times and code crashes. PN fission chains are examined here using simple models. First, static point POE discrete-step random walk stochastic models of unrestricted and restricted chains for constant reactivity are derived and solved analytically. Embedded in both models are polynomials, the degree of which corresponds to the maximum number of neutrons emitted by a fission event and the roots of which are POE candidates. The root structure of the quintic formulation for 235U fission multiplicity data is examined using algebra and calculus techniques to determine the POE and show that only a single POE exists for a system. The analytic unrestricted and restricted models are used in a simple expression which gives the chain length LMAX required to calculate a POE to a prescribed accuracy. The four-factor formula is used to express the analytic expressions in terms of k-infinity (kinf) to enable reactivity-dependent chain analysis. Second, analog MC models which have the analytical model physics are developed. The analytic and MC models are coded in the experimental Fortran program poemp.f, with the Bailey arbitrary-precision arithmetic package used to execute the challenging analytic restricted-chain calculations. Parametric simulations are executed using the 235U analytic and MC models. The analytic models accurately predict MC chain behavior. It is seen that LMAX is highly dependent upon the state of reactivity, exhibiting symmetry about kinf = 1, is large in the vicinity of kinf = 1, and decreases for far-subcritical and highly supercritical reactivity regimes. Third, an experimental version of MCNP6 is created whereby chain behavior can be monitored and controlled during source-mode execution. Parametric analog simulations are conducted using a bare highly enriched uranium (HEU) sphere model for a several subcritical and supercritical states. The analytic models reasonably predict MCNP6 chain and POE behavior while executing in a fraction of the time required by MCNP6. The analytic models give valuable insights into fission chain behavior, and should be of use in providing chain control guidance for mainstream MC code analog simulations of subcritical and supercritical systems while aiding in the understanding of chain behavior encountered in analog MC STK.