## 2 Instrument  
The operation principle of the PES is that atoms in a low-density matter target are ionized whereupon photoelectrons are emitted with a kinetic energy (\(E_{K}\)) that is proportional to the photon energy (\(\hbar\omega\)) according to the photoelectric effect, \(E_{K}=\hbar\omega-E_{B}\) where \(E_{B}\), is the binding energy of the electron before its emission. Flight tubes with an applied retardation voltage act as dispersive elements where the time is measured from the instant of ionization until electrons reach a microchannel plate-based detector. Data acquisition is performed with digitizer boards (Teledyne SP Devices ADQ 412-4G) in a \(\mu\)TCA rate. The boards can be operated either in four-channel mode with 2 GSs\({}^{-1}\) (0.5 ns binning), or optionally, two channels can be bridged in order to increase the sample rate to 4 GSs\({}^{-1}\) (0.25 ns binning). The dynamic range is \(\pm\)400 mV. A fluxgate-type magnetometer is mounted near the source region. A three-axis Helmholtz coil structure surrounds the device to compensate for the background magnetic field, which would otherwise affect the electron trajectories. The design was imported into the commercial software _Simion 8.2_ to optimize the collection efficiency and resolving power for a range of electron kinetic energies spanning several keV [25]. Figure 1 shows simulated electron trajectories for (a) sxPES and (b) hxPES. In the hxPES, generally, the flight tube segments can be modeled as field free regions when not considering field penetration. An exception is the second segment, which is a retardation region designed to dramatically reduce the electron kinetic energy. In a non-relativistic treatment, the spectral distribution of each pulse is encoded in the time of flight (TOF) according to the analytical expression  
\[T_{n}=\frac{L_{n}}{v_{n}} \tag{1}\]  
for \(n\in 1,3,4,5\), where \(L_{n}\) and \(v_{n}\) are the flight-tube length and velocity in region \(n\). \(v_{n}\) is given by \(E_{K}\) and the applied voltage \(V_{n}\),  
\[v_{n}=\sqrt{\frac{2\cdot e^{-}\cdot(E_{K}-c_{n}\cdot V_{n})}{m_{e}}}. \tag{2}\]  
\(c_{n}\) are correction factors--typically very close to 1--that take into account field penetration between segments. They are found for each segment by comparing with electron trajectory simulations and, if the fixed voltage ratio is kept, are valid for a large voltage range. In addition to field free regions, the second stage (\(n=2\)) is a retardation region with a constant electric field. The acceleration of an electron is given by the Lorentz force, that in the absence of magnetic fields is \(\vec{a}=\frac{e^{-}}{m_{e}}\cdot\vec{\
abla}V\), which leads to retardation, velocity and distance traveled as a function of time  
\[a_{2}=\frac{e^{-}}{m_{e}}\cdot\frac{c_{2}\cdot V_{2}-c_{1}\cdot V_{1}}{L_{2}} \tag{3}\]  
\[v_{2}(t)=a_{2}\cdot t+v_{1}, \tag{4}\]  
\[x_{2}(t)=\frac{a_{2}}{2}\cdot t^{2}+v_{1}\cdot t \tag{5}\]  
At the end of the retardation region, we have the relation \(x_{2}(T_{2})\equiv L_{2}\); thus, the time an electron spends in the retardation region is  
\[T_{2}=\frac{-v_{1}+\sqrt{v_{1}^{2}+2\cdot L_{2}\cdot a_{2}}}{a_{2}}. \tag{6}\]  
Figure 1: Cross-section of flight tubes with simulated electron trajectories (red lines) for sxPES (**a**) and hxPES (**b**). Potential gradients are indicated with blue contours. The total TOF is  
\[T=\sum_{n=1}^{5}T_{n} \tag{7}\]  
which applies also to sxPES, although there, the retardation region must be omitted (\(L_{2}=0\)). Inverting the relation to derive \(E_{K}(T)\) analytically, which is necessary for calibration, is complicated. Therefore, we employ numerical methods that continuously update the inversion in real time while the retardation voltages are being tuned, ensuring that reliable spectral distribution is provided without relying on libraries of computer-simulated calibrations. Auger-Meitner transitions occur at fixed electron kinetic energies, providing features suitable for testing the validity of the calibration.