Calculation of Effective Delayed Neutron Fraction Using a Modified k-Ratio Method

A modified k-ratio method, which is applicable to continuous-energy Monte Carlo simulations, is proposed to estimate rigorous values of the effective delayed neutron fraction eff in the present paper. While reactivity in dk=k units is numerically calculated from static neutron transport equations, reactivity in dollar units is used in the reactor kinetics theory. In experiments, reactivity is measured in dollar units. Since eff is used as a reactivity unit converter between dollar and dk=k, accurate calculations for eff are important. Currently, a continuous-energy Monte Carlo simulation for neutron transport becomes much more practical by virtue of progress in the computer technologies. While it requires longer calculation time than a deterministic simulation, it has a great advantage, that is, it allows a rigorous treatment of a system with complex geometry and with complex energy and angular dependence of neutron behavior. Hence, the continuous-energy Monte Carlo simulation is widely used for studies, such as nuclear data assessments, in which calculation accuracy is more important than calculation time. In a eff calculation, an adjoint neutron flux is necessary. It is, however, quite cumbersome for a continuous-energy Monte Carlo code to calculate the adjoint neutron flux. There has never been a production Monte Carlo code for this. On the other hand, much work has been carried out on eff calculations based on a continuous-energy Monte Carlo method without calculating the adjoint neutron flux. Nauchi and Kameyama assume that eff is the ratio of the average number of fission neutrons generated by delayed neutrons to the average number of fission neutrons generated by all neutrons. On the other hand, Meulekamp et al. assume that eff is the ratio of the average number of fissions generated by delayed neutrons to the average number of fissions generated by all neutrons. In these methods, the number of fission neutrons or the number of fissions are counted at each neutron generation in continuous-energy Monte Carlo calculations. They consider only the fission neutrons or fissions until the subsequent fission and neglect the iterated fission event. While Meulekamp et al. indicate that the error due to this approximation is less than 1% in eff , 2) Irwanto et al. point out that the error is not negligible. There is another approach in Monte Carlo-based eff calculations. This approach can be categorized as the ‘‘k-ratio method’’ in which two eigenvalues at reference and fictitious states are used to obtain eff . Bretscher’s method 4) is used to consider a fictitious state in which delayed neutron contributions are ignored. On the other hand, Spriggs’ method, which is based on the work of Jones et al., considers another fictitious state in which the total fission spectrum is replaced by the delayed fission spectrum. Both methods introduce an approximation that makes the neutron flux at the fictitious state the same as the flux at the reference state. Bretscher’s method is expected to be more accurate than Spriggs’ method due to the similarity of the fictitious state to the reference state. It, however, requires more neutron histories to obtain highly significant solutions. Nagaya and Nakajima have applied a correlated sampling method to Bretscher’s method to reduce statistical uncertainties. While their attempt succeeded in doing so, they introduce additional approximations.