Analytical method for calculating the efficiency and solid angle of an NaI(Tl) detector
Introduction
Monte Carlo codes are widely used to calculate efficiency for a wide range of detectors. The optimization of simulation methods using these codes is still under development (Salgado et al., 2012; Mouhti et al., 2018; Chuonga et al., 2019; Tam et al., 2017; Haj-Heidari et al., 2016; Olivares et al., 2017). However, work on developing and improving mathematical analytical methods for determining this efficiency and other parameters, such as solid angle and self-absorption, are still being developed (Abbas et al., 2020; Noureldine et al., 2016; Barrera et al., 2017; Hamzawy 2015; Gouda et al., 2016 El-Kourghly et al., 2021). Mathematical methods are typically based on the calculation of the probability of interaction of gamma-rays with the material of the detector (Selim et al., 1994, 1996, 1998, 2000; Jehouani et al., 2000; Yalcin et al., 2007). Another method is based on calculating the average path length of photons in a detector (El-Khatib et al., 2013). In this paper, we present an analytical method distinguished from previous ones by being based on calculating the balance of the inflow and outflow of gamma-rays through a detector. Moreover, the method enables the calculation of the solid angle of the detector for point or disk-shaped sources, unlike the method of Thabet et al. (2020) which dealt only with a point source. The method is applied to a NaI(Tl) detector because it is widely used, but it can be applied in principle to other types of gamma-ray detectors.
Section snippets
Mathematical model
Let us consider a cylindrical detector of radius R and length L, irradiated by a point source placed on the principal axis of the cylinder at a distance d from the first face of this detector as shown in Fig. 1. Throughout this work, we assume that the source emits one photon per second over the entire solid angle of 4π.
The detector efficiency can be formulated as:where represents the probability that a photon will undergo an interaction within the detector material and the
Results and discussion
The detector efficiency calculations were compared to those obtained by Jehouani et al. (2000) and Yalcin et al. (2007) using the Monte Carlo method. The results plotted in Fig. 3 show good agreement, within the statistical fluctuations of the Monte Carlo results.
For a disk source, the effect of the dimensions of the source on the absolute efficiency is shown decreases as the source moves away from the detector, as the results of Fig. 4 demonstrate. The Figure also shows the convergence of
Conclusions
An analytical formula was introduced to estimate the efficiency of a detector. The method was shown to produce the same results as other methods. The advantage of our analytical formula is that it provides the value of the solid angle of the detector. The method is easy to generalize to much more complex geometric shapes. However, when the source is in the form of a disk with a radius close to the length of the detector radius, our analytical formula requires a more complex numerical
Declaration of competing interest
None.
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