The energy dependent Pál–Bell equation and the influence of the neutron energy on the survival probability in a supercritical medium

Abstract

The survival probability of a neutron injected into a supercritical fissile medium is studied with respect to the energy dependence of the incoming neutron. We assume a point model but allow the energy dependence to be included through a general energy exchange model. We have studied the effect of slowing down on the survival probability by means of the Pál–Bell equation and the Goertzel-Greuling kernel, the separable kernel, the Hansen–Roach dataset and a WIMS generated dataset for a homogenised reactor to approximate the slowing down process. The Goertzel-Greuling model is known to be exact for A=1 and reverts to age theory for large mass ratios. It is also accurate for all intermediate mass numbers, except possibly when strong resonances are present. The separable kernel is a simple model of energy exchange, corresponding to the thermalisation of neutrons in a single collision, which ignores the slowing down process but provides a simple result allowing both ends of the reactor spectrum to be included. The Hansen–Roach data set is obtained from a realistic slowing down model in a fast system and the thermal system is modelled by homogenised reactor data, generated using WIMS, and is typical of the material found in a PWR. Using a scattering cross section which is an arbitrary function of energy, and capture and fission cross sections which are proportional to 1/v, we find that the survival probability is energy independent insofar as it depends only on the values of the ratio of the fission and capture cross sections. For non-1/v cross sections there is an energy dependence which we discuss below. The formalism developed is robust enough for studies to be made of the influence of resonance cross sections and inelastic scattering on survival probability.