A Dead Time Correction Formula for Non-Constant Paralyzing Dead Time in Sub-Critical Systems

Detector dead time refers to a time period following a detection where the detector is non-operational, typically due to the physical components in the detection system. Detector dead times are perhaps the most prominent effect in non-ideal detector behavior and are known to have a significant effect in reactor monitoring. Most mathematical models consider the dead time to be constant, and in response the correction formulas depend on two parameters: the average count rate and the dead time period. In practice, the dead time itself may be a random variable. In these cases, the common approach is to neglect any statistical variations in the dead time, and to use the average value of the dead time in the correction formula. If the variance of the dead time is sufficiently small, this approach naturally gives favorable results. However, as the statistical variation of the dead time increases, this approach may create noticeable deviations. In the present study, we will introduce a novel correction formula for a non-constant paralyzing dead time, introduced in the setting of a sub-critical system. The formula has two theoretical features: first, the formula can be derived (under the point model assumptions) as a second order approximation from a stochastic differential model. Second, it serves as a generalization of a well-known formula for a constant dead time: the resulting formula generalizes the classical exponential dead-time correction to arbitrary dead-time distributions through their Laplace transform. The theoretical result is then compared with numerical simulations.