## 2 Methodology
### Numerical Approach
#### 2.2.2 Plasma Modeling  
The plasma simulation includes five species and reactions: an overview of each individual species and reaction is given in Figure 7. From the preceding DSMC simulation, the neutral gas background density is used in order to avoid _direct_ collision modeling of charged particles with neutral particles since \(n_{\text{Xe}}\gg\left[n_{\text{e}^{-}},\,n_{\text{Xe}^{+}}\right]\). However, collisions with the background gas are still considered using a cross-section database, where for each time-step, artificial neutral particles are created from the background gas and paired with other ions, while electron-neutral collisions are modeled with an elastic-cross-section-based collision probability approach (see [26] for details). The relation between the volumetric and SEE processes of electron production can be estimated based on the data in Figure 7. The values of the cross-sections for collision processes in which electrons are produced (all except of charge exchange (CEX)) are smaller than \(1\times 10^{-20}\,\text{m}^{2}\). Together with the density of neutral atoms of \(n_{\text{Xe}}\approx 1\times 10^{17}\,\text{m}^{-3}\), this yields the minimal mean free path length of the processes \(\lambda\geq 1000\,\text{m}\). The probability of such a collision to happen over the path \(\Delta x\) is \(P=\Delta x/\lambda\). The probability to produce an electron inside the vacuum facility in any type of collision listed in Figure 7 is smaller than 0.001. On the other hand, the probability of a 1 keV Xe ion producing an electron in the secondary electron emission (SEE) lies at approximately 0.03 [27]. This estimation shows that the SEE is at least one order of magnitude more efficient than the volumetric collision processes and is the main source of electrons inside the vacuum chamber. For solving the (static) electric field, a third-order HODG electrostatic Poisson solver is applied, which neglects the temporal evolution of the magnetic field **B**:  
\[\
abla\cdot\textbf{E}=\
abla^{2}\phi=-\frac{\rho}{\epsilon}\,. \tag{5}\]  
In Equation (5), \(\textbf{E}\), \(\phi\), \(\rho\) and \(\epsilon\) denote the electric field, electric potential, charge density and permittivity \(\epsilon=e_{0}e_{r}\), respectively. The deposition of charges is performed by a linear, \begin{table}
\begin{tabular}{l l} \hline \hline
**BC Name** & **Boundary Condition** \\ \hline Inlet & \(\textbf{u}=(4;\,0;\,0)\text{m}\,\text{s}^{-1}\), \(T=320\,\text{K}\), \(n=1\times 10^{20}\,\text{m}^{-3}\) \\ Thruster walls & Diffusive reflecting wall with \(T=300\,\text{K}\) and extended Maxwellian \\  & model according to [25] \\ Vacuum chamber & Open boundary condition (every crossing particle is removed) \\ Probes & Specularly reflecting wall \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical boundary conditions for neutral xenon DSMC simulations. Figure 7: Overview of species and collision-cross-section-based reactions used in the PIC-DSMC simulations, detailing mass and charge for neutral xenon (Xe), electrons (e\({}^{-}\)), xenon ions (Xe\({}^{+}\) and Xe\({}^{++}\)), and charge exchange ions (CEX). This table underpins the simulation’s ability to model plasma interactions, including ionization and SEE, which are essential for modeling and understanding ion beam self-neutralization. locally volume-weighted deposition technique. The local mesh size and (fixed) time-step are chosen based on the so-called PIC-CFL condition [26] given by  
\[\frac{\Delta t}{0.4\cdot\Delta x}\sqrt{\frac{k}{m_{e}}}<1\qquad k=k_{B}T_{e}\,, \tag{6}\]  
where \(\Delta t\) and \(\Delta x\) denote the time-step and spatial discretization, respectively. The numerical boundary conditions for the plasma simulation are shown in Table 2. Note that in the simulation without the probes, the _floating probes_ (i.e., metal wires) and _dielectrics_ (i.e., probe mounts) are omitted.