## 4 Results and Discussion
### Limit on the Time Resolution
Another time resolution limiting factor could be in the electron detector where the fast electrons interact inelastically at a stochastic depth which in combination with the finite drift velocity leads to a timing uncertainty of the electron arrival event. For a sensor thickness of 300 \(\upmu\)m and a drift velocity in the order of 10\({}^{6}\) cm/s [40] this would result in a broadening of approx 30 ns [41; 42]. To verify our assumptions on the dominant role of the drift velocity in the SDD we tilt the sample holder in such a way that one detector receives a smaller count rate than the other one. This happens since the holder shadows a part of the detector hence a smaller area is illuminated. In Figure 8b, both time coincidence histograms are shown from which the X-ray count rate on the detector is indicated. The width of the coincidence window for the non-shadowed detector (blue) is around twice as high as compared to the shadowed one (red) confirming that the illuminated area plays the most important role on the time resolution for the current coincidence setup. One way to overcome this limitation could be to design new types of X-ray detectors which are similar to the pixelated electron detector with more channels, each with an integrated preamplifier stage, with a smaller active area to improve the time resolution while maintaining a high solid angle. Highly integrated electronics could potentially make this a viable solution in the near future [43]. An alternative method could be to shadow some part of the detector where a smart choice on the masked area could improve the SNR. Since the detector is circular with radius \(R\), we can choose a circular mask which  
Figure 8: **(a)** Zoom in on the shape of the time coincidence histogram where an asymmetric profile is seen. The dotted black line shows a Gaussian blurred sawtooth function which would be the theoretical shape of the histogram. **(b)** The time coincidence histogram of two detectors. The sample holder is titled such that the count rate of X-rays on detector \(1\) is lower which is due to the shadowing of the detector. The width of the time coincidence peak is significantly lower for the shadowed detector (detector 1). is centered on the detector with a radius \(r\) between 0 and R. The efficiency (\(e_{x}^{\prime}\)) and time resolution (\(\tau^{\prime}\)) of the new masked detector is:  
\[e_{x}^{\prime}(r)=\frac{R^{2}-r^{2}}{R^{2}}e_{x} \tag{11}\]  
\[\tau^{\prime}(r)=\frac{R-r}{R}\tau+\Delta\tau \tag{12}\]  
where \(\Delta\tau\) is the broadening which arises from other time broadening influences in the experimental setup. These equations were derived for a geometry in which the detector is not tilted with respect to the sample. For tilted detectors, the situation would be different. By inserting the two values into Equation (5) the influence on the radius of the mask on the SNR can be investigated. To do this, values on the other parameters should be known. Here the values are approximated from the data shown in Figure 2. The parameter values used for the SNR simulation in Figure 9 are shown in Table 10. Please note that some parameters values given relative to C\(\sigma\), for instance \(B_{e}=\) C\(\sigma\) which indicates that the number of background electrons is equal to the number of electrons in the core-loss edge. In Figure 9, the SNR gain, defined as the ratio of the SNR with the SNR when the mask radius is zero, as a function of mask radius for different incoming electron currents is shown. The gain in SNR is maximal when the current is higher and has an increase of 7%. Further with higher current, the optimal mask radius increases to 0.37 for the highest current used. From these calculations it is seen that the detector masking using an annular detector can increase the SNR at the expense of detection efficiency. However, the gain provided by this trick is minor and depends on the incoming current hence making this option not very attractive.