## 3 Theoretical Description
The first one is described by a core-loss electron and core-loss X-ray arising from another electron but inside the time coincidence window (\(C\sigma\cdot C\sigma_{T}\omega\)). The second type of coincidence is when a background electron and core-loss X-ray is detected inside the window (\(B_{e}\cdot C\sigma_{T}\omega\)). The opposite can also happen where a core-loss electron and background X-ray are detected inside the time coincidence window (\(C\sigma\cdot B_{x}\)). Finally, the background electron and X-ray can be detected inside the window (\(B_{e}\cdot B_{x}\)). From Equation (4) it is clear that the RUP goes as \(I^{2}\) whereas \(N_{C}\) scales linearly with \(I\) (see Equation (3)). Also note the linear dependence on the time window which is directly related to the time resolution of the event detection system (taking it smaller misses events, taking it larger adds only RUP). From Equations (3) and (4), a SNR can be determined which describes if the coincidence events are statistically significant. Since this is fundamentally a Poisson process (Multiple detected events can be triggered for a single arriving electron or X-ray, but this is dealt with in a discrimination stage in the detector, leading to a stream of pure Poisson single detection events), the standard deviation of this signal is given by its square root. The SNR of the coincidence setup (\(SNR_{C}\)) when acquiring for exposure time \(t\) is given by following equation:  
\[SNR_{C}=\frac{N_{C}\cdot\sqrt{t}}{\sqrt{N_{RUP}+N_{C}}} \tag{5}\]  
\[SNR_{C}=\frac{C\omega\sigma\cdot\sqrt{I\cdot\epsilon_{e}\epsilon_{x}\cdot t}}{ \sqrt{(C\sigma+B_{e})\cdot(C\sigma_{T}\omega+B_{x})\cdot\tau\cdot I+C\omega \sigma}} \tag{6}\]  
where the signal of interest is the total number of detected coincidence events. Using Equation (6) we can determine how the incoming current influences the SNR and if there is an optimal current to perform the coincidence experiments. Therefore, Equation (6) is reduced to following:  
\[SNR_{C}(I)=a\sqrt{\frac{I}{bI+c}} \tag{7}\]  
where \(a\), \(b\) and \(c\) are the multiplicative constants which makes the notation easier. Equation (7) is a monotonically increasing function. Hence when performing experiments, maximizing the current maximizes the SNR which also applies for conventional EELS or EDX. However, in the limit of infinite current, the SNR converges to \(a/\sqrt{b}\) which indicates that at some point, which depends on the parameters, no gain in SNR is expected by increasing the current. However, increasing the current is not always viable due to the occurrence of electron beam damage for some materials. So instead of maximizing the SNR in a certain time, the total amount of incoming electrons (\(N=I\cdot t\)) is fixed where the current is left as a free parameter. For conventional EELS and EDX, the SNR only depends on \(N\) so taking a very long acquisition with a low current or a short acquisition with a high current results in the same SNR, if the rest of the setup is ideal. On the other hand, for the coincidence experiment this current does play a role in the final SNR. This can be seen by plugging \(N\) into Equation (6) and giving the following:  
\[SNR_{C,N}(I)=\frac{a^{\prime}}{\sqrt{bI+c}} \tag{8}\]  
where it is observed that this function is always monotonically decreasing, hence when maximizing the SNR for a given amount of dose, the current should be as low as possible. This can be understood from the observation that at low current, the chance that two uncorrelated events will be close to each other diminishes rapidly while the correlated events will remain at the same short time interval, making it easier to discriminate between real coincidence events versus unwanted RUP signal. This however has the obvious downside of very long acquisition times which would make the experiment more prone to sample drift.