## 3 Theoretical Description  
To understand the count rate of coincidence events, we start with the description of the detected count rate for a conventional EELS and EDX setup. We assume that the inelastic electrons which fall onto the detector can be described by one core-loss edge and a background signal (\(B_{e}\)) where the origin of the background signal is of no interest apart from the assumption that it scales with the collection efficiency \(e_{e}\) and current \(I\) (electrons/s). The number of detected electrons per second on the detector (\(N_{EEL}\)) covering a certain energy range \(\Delta E\) is then given by the following equation [19]:  
\[N_{EEL}=(C\cdot\sigma+B_{e})\cdot e_{e}\cdot I \tag{1}\]  
where \(C\) is the areal density, equal to the product of the concentration and the specimen thickness. \(\sigma\) is the inelastic scattering cross section depending on the beam energy, energy integration range \(\Delta E\) and collection angle of the EELS setup, \(I\) is the incoming electron current and \(e_{e}\) is the electron collection efficiency. In Equation (1), the effect of multiple scattering which occurs for thicker samples has not been taken into account. For X-ray detection the same reasoning can be applied to formulate the count rate in a given energy interval (\(N_{EDX}\)) [20]:  
\[N_{EDX}=(C\cdot\sigma_{T}\cdot\omega+B_{x})\cdot\epsilon_{x}\cdot I \tag{2}\]  
where \(\omega\) is the fluorescence yield which incorporates the relative line intensity depending on the specific excitation that was chosen, \(\sigma_{T}\) the total inelastic scattering cross section for excitation of a particular edge and \(B_{x}\) is the number of background X-ray events. The X-ray collection efficiency \(\epsilon_{x}\) consists out of the geometric solid angle as well as the energy dependence of the X-ray detection efficiency. The difference between \(\sigma\) and \(\sigma_{T}\) is that for X-rays the total cross section is used. However, for the electron detection not all inelastic electrons arising from a particular edge are detected due to the limited energy field of view and limited collection angle. Please note that Equation (2) neglects the effect of absorption where due to the finite thickness of the sample, X-rays can be absorbed. A coincidence event is measured when both an electron and X-ray is detected at the'same' time. Hence the count rate of coincidence events is given by the following equation:  
\[N_{C}=C\cdot\epsilon_{x}\epsilon_{e}\cdot\omega\cdot\sigma\cdot I \tag{3}\]  
Please note that the background signals \(B_{e}\) and \(B_{x}\) do not take part in the coincidence detection as we assume the background (\(B_{e}\) and \(B_{x}\)) to be caused by unrelated scattering processes which have no time correlation with the electrons and X-rays of interest. Moreover, \(\sigma\) is used instead of \(\sigma_{T}\) since not every inelastic electron, which has a correlated X-ray, is detected due to the finite collection angle and energy window of the EELS setup. Besides the correlated events, there are uncorrelated events which occur when an electron and X-ray are detected within a certain time interval even though they possess no underlying correlation. This happens since the electron emission is a Poisson process hence two electrons can be emitted very close in time. The number of uncorrelated events is the product between \(N_{EEL}\) and \(N_{EDX}\) which is time invariant. These random uncorrelated process events (RUP) have a probability to occur inside the time coincidence window (\(\tau\)). This window is set by the time resolution of the experimental setup. Hence the number of uncorrelated events occurring inside the time coincidence window is given by  
\[N_{RUP}=N_{EEL}\cdot N_{EDX}\cdot\tau=\left((C\cdot\sigma+B_{e})\cdot(C\cdot \sigma_{T}\cdot\omega+B_{x})\right)\cdot I^{2}\cdot\tau\cdot\epsilon_{e} \epsilon_{x} \tag{4}\]  
It is clear from Equation (4) that there exist four types of coincidences which can create a RUP event.