## 4 Results and Discussion  
### Collection Efficiency EELS and EDX  
The coincidence setup gives the ability to obtain the collection efficiencies for the EELS and EDX setup. This is possible if the background signals \(B_{e}\) and \(B_{x}\) are known. In EELS, knowledge of \(B_{e}\) is obtained via a power-law fit such that only the electrons arising from the core-loss remain (\(N_{EEL}^{\prime}\)). For EDX, \(B_{x}\) is in general relatively small and consists out of overlapping characteristic X-rays and a continuum. The continuum can be removed by fitting a linear function to two windows on each side of the X-ray line and the background arising from the overlapping [21] of characteristic X-ray lines can be estimated by fitting the X-ray spectrum to a sum of Gaussians. When both background values can be estimated, the collection efficiency of an EELS setup can be determined by taking the ratio of Equations (2) and (3) which gives:  
\[\epsilon_{e}=\frac{N_{C}}{N_{EDX}^{\prime}}\frac{\sigma_{T}}{\sigma} \tag{9}\]The ratio \(\sigma_{T}/\sigma\) is determined by the considered energy window and collection angle of the EELS acquisition. This value can be determined theoretically by computing the total scattering cross section of the particular edge (\(\sigma_{T}\)) and the cross section where the limited energy window and collection angle are incorporated. In this work, the Hartree-Slater model from the Gatan Digital Micrograph 3 was used to determine this ratio.  
The collection efficiency of the EDX setup is determined by taking the ratio of Equations (1) and (3):  
\[\epsilon_{x}=\frac{N_{C}}{\omega N_{EL}^{\prime}} \tag{10}\]  
from which the fluorescence yield should be known to determine \(\epsilon_{x}\).  
This method was used to determine the collection efficiencies of the experimental setup (\(\epsilon_{e}\) and \(\epsilon_{x}\)) used in this work using an Al-Mg-Si-Cu alloy as an example [22]. In Figure 1a, the conventional EEL spectrum is shown where \(N_{ELEL}^{\prime}\) is standardly determined by subtracting the background events from the signal where the background is determined by fitting a power law before the edge onset. \(N_{EDX}^{\prime}\) (see Figure 1b) is determined by fitting the EDX spectrum with the sum of three Gaussian and a linear slope where one Gaussian contains the characteristic aluminum peak. The other characteristic peaks come from low-concentration copper and silicon. Finally, \(N_{C}\) is calculated by first determining the time coincidence interval which is indicated in Figure 1c which shows the histogram of time difference between the X-ray and electron events. This histogram is similar to the second order correlation g2 which is used to measure the photon bunching [23] and anti-bunching [24; 25]. The next step is to determine \(N_{RUP}\) which is done by taking the average counts outside the coincidence window and multiplying this with the time coincidence window. \(N_{C}\) is then calculated by subtracting the total number of events inside the time coincidence window by \(N_{RUP}\). Using the three values shown in Figure 1, the collection efficiencies can be determined. The two collection efficiencies are \(\epsilon_{e}=26\%\) and \(\epsilon_{x}=4.8\%\). For \(\epsilon_{e}\), a value between 20-50% is expected [19] and for \(\epsilon_{x}\) a value of 5.2% is predicted [26]. Compared to the standard methods which are used to determine the collection efficiencies for EELS and EDX, this method has the benefit that no reference sample is required, and knowledge of the beam current is not needed [21] but the ratio (\(\sigma_{T}/\sigma\)) should be known which can be computed theoretically.