# Studies and Rejection of Intercrystal Crosstalk on FPGA in a High-Energy Photon-Counting System
## 4 Intercrystal Scattering Rejection Algorithm  
### Determination of Time Window  
A wide time window is helpful to retain the majority of intercrystal scatter events, but meanwhile, the events coming from various photons (different decays) are more likely to be misjudged. Therefore, determining an appropriate coincident time window width is necessary. According to Figure 9, we can derive a time window with a width of around 5 ns.  
Equation (1) is utilized to estimate the probability of \(n\) decays occurring in time \(t\). \(\lambda\) is the decay constant of the radioactive source, \(n_{0}\) is the number of nuclei in the radioactive source, and \(n_{0}\lambda\) is the number of decays per unit time. Here the probability of more than two decays (\(P(n\geq 2)\)) occurring within three adjacent detector units is calculated, so \(n_{0}\lambda\) is three times the no-load count rate per detector unit (\(N\)), which can be calculated by (2). In Equation (2), \(N_{0}\) is the initial intensity of \({}^{60}\)Co (unit: Ci, i.e., 3.7 \(\times\) 10\({}^{10}\)/s), typically 30\(\sim\)300 Ci, \(d^{2}\) is the entrance window area of the detector unit (1 \(\times\) 1 cm\({}^{2}\)), \(r\) stands for the distance between the radioactive source \({}^{60}\)Co and the detector array (6 m), and \(\eta\) is the detection efficiency (about 70%). \(N_{0}\) is multiplied by two, for the reason that \({}^{60}\)Co emits both 1.17 and 1.33 MeV \(\gamma\) photons [20].  
\[P(n)=\frac{(n_{0}\lambda t)^{n}}{n!}e^{-n_{0}\lambda t}, \tag{1}\]  
\[n_{0}\lambda=3\cdot N=3\times 2N_{0}\cdot\frac{d^{2}}{4\pi r^{2}}\cdot\eta \tag{2}\]  
The calculated results are listed in Table 2. For the 300 Ci source intensity, the probability of more than two decays happening within 5 ns in three adjacent detectors is only 0.13%. And this probability decreases further under conditions of a lower source intensity and non-empty load. If the time gap extends to 10 ns, the probability reaches 0.50%. Therefore, the probability of more than two decay events occurring within the 2\(\sim\)10 ns time window is quite low. Combined with the results of Figure 9, a time window of 5\(\sim\)10 ns can include more intercrystal scatter events.  
Figure 11: Correlation of scatter angle and the time difference between events that make up intercrystal interaction pairs. (**a**) Distribution of more than 1500 pairs. (**b**) Probability of negative time difference.  
As evidenced by the simulation results, when the criterion of \(\Delta\text{Det}=1\) is applied with a 5 ns time window, the RMSE under 30 Ci no-load conditions increases from 5.44 to 26.06. Meanwhile, when \(\Delta\text{Det}=1\) with a 10 ns time window, the RMSE rises to 26.10. When the no-load condition is adjusted from 30 Ci to 300 Ci, the RMSE increases from 5.44 to 26.89 and 26.49 under the 5 ns and 10 ns time windows, respectively. Therefore, the 5 ns and 10 ns time windows exhibit nearly equivalent levels of effectiveness. With the aim of retaining the majority of intercrystal scatter events, decreasing the potential for misjudging events from different photon decays, as well as improving the counting accuracy, the smaller 5 ns time window is selected as the optimal criterion.