## 4 Results and Discussion
### Low Concentration Detection  
In this section, when the coincidence method can have advantages compared to conventional EELS and EDX is investigated. This is done by comparing the SNR between coincidence and conventional EELS/EDX which is calculated using Equations (A1) and (A2). The two main parameters which define the SNR are the incoming current \(I\) and signal-to-background ratio (\(SBR\)) which for EELS is defined as \(C\sigma/B_{e}\). Similarly for EDX, the \(SBR\) is defined as \(C\sigma_{T}\omega/B_{x}\). The other parameters chosen for the calculations are shown in Table A1 where the value for C\(\sigma\) is approximated using Equation (1) on experimental data from which \(\epsilon_{e}\) and B\({}_{e}\) are the values obtained from Section 4.1, \(I\) is measured from the read out of fluorescence screen and time \(t\) is acquisition time. All these values are plugged into Equations (6) and (A1) while varying \(I\) and \(SBR\). The formula for calculating the SNR  
Figure 4: **(a)** The EDX spectrum of GaAs where the Ga peak is fitted with a Gaussian to indicate the extent of the X-rays originating from the de-excitation of the Ga atom. The red and blue line indicate the extent of the X-ray lines in the energy spectrum. **(b)** The background-subtracted EEL spectra using the X-rays in the regions are indicated by the legend. When only using X-rays from _region 1_, it is seen that only the Ga L\({}_{23}\) edge is selected without the As L\({}_{23}\) edge (brown). When _regions 1+2_ are selected, electrons from both edges are seen (green). **(c)** The same as (a) where the X-ray regions, out which the coincidence events are chosen, have been modified. **(d)** The background-free EEL spectrum where, by selecting the proper X-ray window (_region 2_), only the As L\({}_{23}\) edge is selected (brown). When _regions 1+2_ are selected, electrons from both edges are seen (green). Figure 5: **(a)** The conventional EEL spectrum from GaAs where the two edges are indicated. **(b)** The background-subtracted EDX spectra using only electrons from the two regions indicated in **(a)**. The red spectrum only shows X-rays originating from Ga whereas the blue region shows both Ga and As since the core-loss edges are delocalized in energy. **(c)** The As L\({}_{a}\) line can be disentangled from the Ga L\({}_{a}\) line by normalizing the blue and red spectrum from **(b)** in the region where only Ga L\({}_{a}\) X-rays are present (brown region). The difference between the normalized blue and red spectrum results in an X-ray spectrum containing only the As L\({}_{a}\) X-rays. for EELS and EDX are given in Appendix A. In Figure 6a,b both \(SNR_{C}\) and \(SNR_{EELS}\) are shown where the acquisition time has been kept constant for every calculation. An increase in SNR is seen when the incoming current and \(SBR\) increases, as could be expected. To verify under which circumstances coincidence detection outperforms conventional EELS, the ratio \(SNR_{C}/SNR_{EELS}\) is shown in Figure 6c. This shows that there is a well-defined parameter space for which coincidence detection outperforms conventional EELS, more specifically in the low current and low \(SBR\) regime. The same analysis can be applied for EDX since the main difference for the SNR is the fluorescence yield \(\omega\) (see Equation (A2)). Hence similar conclusions can be obtained albeit that typically for EDX the \(SBR\) is significantly higher than for EELS explaining why it is commonly accepted that EDX is preferable when it comes to trace element detection even though the detection efficiency is considerably lower than for EELS [33]. Please note that this derivation takes only Poisson noise statistics into account and assumes an ideal detector and perfect background subtraction with no extrapolation and fitting error [34; 35; 36; 37]. From Figure 6c one can conclude that also with conventional EELS we can detect low-concentration species in a surrounding matrix (low SBR situation) by choosing the current high enough.