A Stochastic Differential Equation for Neutron Count with Detector Dead Time and Applications to the Feynman-Α Formula

Detector dead time losses, caused by both physical components in the detection system and electronic data acquisition, are perhaps the most prominent effect in non-ideal detector behavior. As the detection rate grows, dead time has a dramatic effect on the regulation system and in-pile experiments. In particular, when conducting the Feynman-α experiments, sincethe dead time has a stronger effect on correlated neutrons, the dead time biases the variance to mean ratio by reducing it [1].

The influence of the dead time on the detection count distribution and on the Feynman-Y (or variance to mean) formula is a well studied topic, with treatment varying from full first principle modeling [2] to experimental numeric corrections [3]. Yet, the problem in its entirety is not solved: An applicable formula fully accounting for the dead time losses is not known.

Modeling reactor noise and the stochastic fluctuations of the neutron population and detection (often referred to as stochastic transport) is traditionally performed using the Probability Generating Function (PGF) formalism and the master equation. In the last decade, originating from the work of Hayes and Allen [4], a new modeling approach for reactor noise has been studied, via Stochastic Differential Equations (SDE). The approach is based on diffusion scale approximations, justified by the very large number of reactions and high reaction rate in a nuclear core, to model the stochasticity of the reactions as if they are driven by Brownian motion.

Whereas the original model introduced in [4] only refers to the neutron population size, in a recent study by the authors [5], the neutron population size was coupled with the detection count, resulting in a system of SDE that accounts for the pair: population and detection. It has been shown in [5] that the model is precise up to the second moment, in the sense that the first and second moments are in complete agreement with the classical results obtained using the PGF formalism.

In the talk, we introduce a generalization of the SDE model introduced in [5], accounting for a non-paralyzing dead time.

The quadratic approximation of the generalization can be fully analyzed- up to the second moment of the distribution- resulting with the correction term to the classic Feynman-Y function. When compared with experimental results, the corrected formula shows high correspondence for dead time losses up to 6%.

Bibliography

1. Hashimoto, K. Ohya, Y. Yamane: Experimental investigation of dead-time effect on the Feynman-α method. Annals of nuclear energy, Vol. 23 (1996), pp. 1099—1104
2. Kitamura and M. Fukushima. Correction of count-loss effect in neutron correlation methods that employ single neutron counting system for subcriticality measurement. Journal of Nuclear Science and Technology, Vol. 51, 766-782
3. Gilad, Y. Neumeier, C. Dubi. Dead time corrections on the Feynman-Y curve using the backward extrapolation method. Journal of Nuclear Science and Technology, 2017.
4. Hayes, E. Allen: Stochastic point-kinetics equations in nuclear reactor dynamics. Annals of Nuclear Energy, 32, 2005
5. Dubi and R. Atar. Modeling neutron count distribution in a subcritical core using stochastic differential equations. Annals of Nuclear Energy, 111 (2018).