On the stochastic dependence between photomultipliers in the TDCR method

The TDCR method (Triple to Double Coincidence Ratio) is widely implemented in National Metrology Institutes for activity primary measurements based on liquid scintillation counting. The detection efficiency and thereby the activity are determined using a statistical and physical model. In this article, we propose to revisit the application of the classical TDCR model and its validity by introducing a prerequisite of stochastic independence between photomultiplier counting. In order to support the need for this condition, the demonstration is carried out by considering the simple case of a monoenergetic deposition in the scintillation cocktail. Simulations of triple and double coincidence counting are presented in order to point out the existence of stochastic dependence between photomultipliers that can be significant in the case of low-energy deposition in the scintillator. It is demonstrated that a problem of time dependence arises when the coincidence resolving time is shorter than the time distribution of scintillation photons; in addition, it is shown that this effect is at the origin of a bias in the detection efficiency calculation encountered for the standardization of (3)H. This investigation is extended to the study of geometric dependence between photomultipliers related to the position of light emission inside the scintillation vial (the volume of the vial is not considered in the classical TDCR model). In that case, triple and double coincidences are calculated using a stochastic TDCR model based on the Monte-Carlo simulation code Geant4. This stochastic approach is also applied to the standardization of (51)Cr by liquid scintillation; the difference observed in detection efficiencies calculated using the standard and stochastic models can be explained by such an effect of geometric dependence between photomultiplier channels.