# Studies and Rejection of Intercrystal Crosstalk on FPGA in a High-Energy Photon-Counting System
## 3 Studies of Intercrystal Scatter
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#### 3.2.3 Time  
We first take the instant at which 0 V is exceeded as the beginning time of each event. Under this condition without a trigger, the timing difference has a narrow dispersion with a full width at half maximum (FWHM) of just 0.66 ns. However, the start time cannot be acquired directly. The leading-edge timing method is usually employed in the electronic readout to discriminate times of response signals. The response signals are created using a PSpice electronics model of the readout circuits [20]. The FWHM of the distribution expands to 4.41 ns after using a leading-edge timing circuit with a 5 mV trigger voltage, as shown in Figure 9. As the trigger voltage rises, the dispersion widens even further.  
There are always negative distributions in the three curves of Figure 9, indicating the possibility of incident events being detected following crosstalk events. To explain this phenomenon, we should first clarify factors that influence the timing of events. There are three main factors that can have an impact: (1) the response times of detectors, (2) the time discrimination method employed, and (3) the flight times of the detected photons. SiPMs utilized in the system have the same specifications, guaranteeing a consistent performance, including the time response characteristics. In addition, the leading-edge timing method with the same trigger voltage is applied to each signal. Therefore, the leading factor impacting the timing of events is the flight time.  
The flight trajectories of the photons that make up an intercrystal interaction pair are shown in Figure 10 to better illustrate the time difference between them. L1 and L3 are the minimum flight distances for the recoil electron and scattered photon to reach the SiPM device, respectively. L2 is the flight distance of the scattered photon before its deposition. Therefore, their time difference can be expressed as \(\Delta\)T = (L2 + L3 \(-\) L1)/c, where c is the speed of light. As can be seen in Figure 10, the difference in the flight time between the incident event and its accompanying crosstalk event is mainly related to the scatter angle. L2 + L3 tends to be shorter than L1 with a smaller scatter angle \(\theta\), which results in a negative distribution of \(\Delta\)T. Meanwhile, the flight distance of the scattered photon will very easily exceed that of the recoil electron with larger scatter angles, particularly in back-Compton scattering.  
Figure 10: Flight trajectories of two photons that make up an intercrystal interaction pair.  
Figure 9: Time difference between events that make up intercrystal interaction pairs, in which the times of incident events are baselines for comparison.  
The correlation between the time difference and the scatter angle of over 1500 pairs of intercrystal interaction events is shown in Figure 11a, aligning with the previous discussion regarding negative time differences, which tend to appear in interactions with small scatter angles. Conversely, there is only a minimal likelihood of a negative distribution when the scatter angle is large. Further illustrating this trend, the distribution in Figure 11a is transformed into Figure 11b.