Kinetic simulation of ion thruster plume neutralization in a vacuum chamber

In this section, we briefly discuss the computational framework implemented in CHAOS to couple the PIC and DSMC approaches in order to calculate the self-consistent electric field, taking into account the reaction between ions and neutral particles. Many aspects of the CHAOS solver used in this study have already been verified in our previous work e.g. the fully kinetic PIC module [26], PIC-DSMC coupling [23], in-space computational BCs [29], and grounded wall BCs [24]. In the EP plume, collisions and electric fields operate on significantly different time and length scales, differing by at least two orders of magnitude. To mitigate the impact of these differences, CHAOS employs several computational techniques as previously described in our earlier papers [22–24, 26, 29].

The DSMC module models three types of collisions: MEX collisions between Xe-Xe and Xe-Xe⁺, and CEX collisions between Xe-Xe⁺. The collision cross sections for MEX between neutral particles and MEX and CEX collisions between neutral particles and ions are obtained from [37, 38]. The no-time-counter collision scheme proposed by Serikov et al [39] is used in this study since it accounts for the disparate timesteps and weighting factors of ions and neutral particles. The neutral particles move only when the DSMC module is executed, every 100 PIC timesteps, while the ions and electrons move every iteration.

In the PIC module, the electric potential is calculated using an explicit PIC technique. In the fully kinetic approach, the electric field, E, is self-consistently solved by:

$$\begin{equation} \rho = e\left(n_\mathrm{i}-n_\mathrm{e}\right), \end{equation} \tag{ 1 }$$

$$\begin{equation} \nabla^2\phi = - \frac{\rho}{\epsilon_0}, \end{equation} \tag{ 2 }$$

$$\begin{equation} E = - \nabla\phi, \end{equation} \tag{ 3 }$$

where ρ is the charge density, e is the elementary charge, $n_\mathrm{i}$ and $n_\mathrm{e}$ are the number density of ions and electrons, φ is the electric potential, and ε₀ is the permittivity of free space. A finite volume approach based on an unstructured octree grid is used to solve equation (2).

CHAOS has a number of PIC and DSMC coupling algorithms that save computational effort. First, using a Morton Z-curve, CHAOS constructs two separate grids with a linearized forest of octrees (FOT) in the PIC and DSMC modules, respectively, because the mean free path, λ, and Debye length, $\lambda_\textrm D$, differ by at least three orders of magnitude. The FOT for DSMC (C-FOT) is constructed to resolve the local mean free path, while the FOT for PIC (E-FOT) is constructed to resolve the local Debye length, where the refined cell size $x \lt\lambda_\textrm D$. We apply an adaptive mesh refinement method since the number density can vary widely in the computational domain. Both C- E-FOTs are reconstructed every 20 000 iterations before sampling starts. Second, weighting factors, W, are utilized to increase the number of charged computational particles compared to the neutral particles due to the disparate length scales of the C- and E-FOTs and disparate number densities of the neutral particles and CEX ions. In this study, the ratio of neutral and ion ($W_\mathrm{n}/W_\mathrm{i}$) is set at 200 000. Third, time-slicing of the DSMC and PIC modules and species-dependent timesteps are implemented due to the different timescales for collision and plasma frequencies. The positions of the neutral particles, ions, and electrons are updated with timesteps of $\Delta t_\mathrm{n} \gg \Delta t_\mathrm{i} = \Delta t_\mathrm{e}$ to reconcile these disparate timescales. In this study, we use a timestep of $\Delta t_\mathrm{n} = 1.12 \times 10^{-4}$ s for neutral particles and $\Delta t_\mathrm{i,e} = 2.8 \times 10^{-10}$ s for both ions and electrons.

To satisfy the objectives of this study, we use both in-space and in-chamber BCs for the outer edge domain boundary. For the in-space simulation, the charge-conserving energy-based BC (CCE BC) developed by Jambunathan and Levin [29] is used for the downstream boundary (z = 0.8 m), and the buffer BC is used for the other boundaries to simulate the infinite expansion of the thruster plume, similar to our previous calculations [22, 29]. The buffer BC simulates the inflow of electrons from outside by placing a buffer region outside the computational boundary and copying the particles inside the boundary out to a distance of $\lambda_\mathrm{D0}$ beyond it. The CCE BC specularly reflects some electrons arriving at the edge of the computational domain and eliminates others using the following approach. The baseline total charge, Q₀, and the average electron kinetic energy in the computational domain, $\langle E_\mathrm{e} \rangle$, are obtained at the time step just before the beam ions reach the downstream domain boundary. In subsequent timesteps, when the total charge in the domain is less than Q₀, electrons with energies less than $\langle E_\mathrm{e} \rangle$ are specularly reflected. In [29], it was verified that these BCs satisfy the requirements needed for the plume modeling by changing the size of the domain (see table II in [29]). In terms of electrical BCs, the inhomogeneous Neumann BC for the electric potential is implemented on all domain boundaries in the in-space simulations. For each boundary, the normal potential gradient $({\partial \phi}/{\partial n})_\mathrm{bc}$ is computed based on the current density through the boundary as follows;

$$\begin{equation} \left(\frac{\partial \phi}{\partial n}\right)_\mathrm{bc} = \frac{e \left(N_\mathrm{i,bc}-N_\mathrm{e,bc}\right)}{A_\mathrm{bc}\epsilon_0}, \end{equation} \tag{ 4 }$$

where $N_\mathrm{i,bc}$ and $N_\mathrm{e,bc}$ are the number of ions and electrons that cross the boundary, and $A_\mathrm{bc}$ is the area of the boundary.

For the in-chamber BCs, we use a fully diffuse reflection condition with a 300 K accommodation for ions and neutral particles on the chamber walls at the edge of the computational domain. The walls absorb all electrons by removing them from the domain and neutralize all incident ions by returning them into the domain as neutral particles with a temperature of 300 K. The 0 V Dirichlet BC is implemented on every boundary surface in the PIC module. The plasma screen, which is the housing of the thruster assembly and is normally electrically grounded, has the same BC as the vacuum chamber walls. We also assign a charge-absorbing BC for particles impinging on the thruster and neutralizer exits and a Dirichlet potential BC for the electric potential with a baseline value of 0 V. The detailed settings about the potential boundaries are described in section 3.2.

Figures 1 and 2 show the three- and two-dimensional schematics of the in-chamber cases investigated in this study. Note that only the geometry of the neutralizer differs in figures 1 and 2. The GIT is placed in a cubic vacuum chamber with a length of 0.8 m per side. In this study, only a half of the domain is simulated due to symmetry to save computational effort, i.e. a specular reflection BC and a Neumann BC ($\partial \phi/ \partial n = 0$) are implemented on the x = 0 m plane. Numerical pumps, shown as green volumes, are installed at all corners of the downstream face to remove heavy neutral particles from the vacuum chamber. Since the focus of this paper is not on detailed modeling of how a vacuum chamber pump works, the pump is simplified to adjust the background pressure. Computational particles entering the numerical pump volume are deleted from the calculation, utilizing the same method employed in our previous studies [24, 40]. The cross-sectional area of the numerical pump is assumed to be $1.25 \times 1.25$ cm², which produces a typical vacuum chamber pressure of 10⁻⁶ Torr. The pump is located at the downstream corner so as not to disturb the plasma sheath on the walls.

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Since the purpose of this study is to improve our understanding of how the thruster plume changes from the space to the ground-test environment, the dimension of the thruster, which emits Xe neutrals and Xe⁺ ions, is the same as that used in our previous calculations of an ion thruster system [22–24, 29]. The center position of the cylindrical thruster exit with a radius, R₀, of 0.0625 m is located at ($x_0, y_0, z_0$) = (0.0, 0.4, 0.1) m. The domain contains the plasma screen offset from the inlet plane at z = 0.1 m. This study examines two neutralizer sizes and number densities. The first configuration is designated as 'A,' which has the same characteristics as in our previous work [23], as illustrated in figure 1; (a): three-dimensional view, (b): two-dimensional view seen in the x–y plane, and (c): in the z–y plane. This neutralizer has the same radius as the thruster and is placed side by side on the same plane. In addition to the type A configuration, this study analyzes a more realistic configuration designated as 'B.' This neutralizer has a higher electron density and a smaller outlet downstream from the thruster, as shown in figure 2; (a): three-dimensional view, (b): two-dimensional view seen in the x–y plane, and (c): in the z–y plane. The neutralizer exit radius, $R_\mathrm{e0}$, is set to 1 cm based on the size of a typical neutralizer [41] and the exit electron number density is 32 times greater than type A to obtain the same current. Unlike the type A cases, the neutralizer exit shifts far from the thruster exit by a value of d₀ in the z-direction to investigate the effect of the electron exit position on the plume.

Table 2 summarizes the conditions of each species at the thruster and neutralizer exits. Similar to previous mesothermal studies [17, 26, 29], we chose a ratio of the initial ion temperature, $T_\mathrm{i0}$, to the initial electron temperature, $T_\mathrm{e0}$, of 0.01. All species are initialized at their sources with a stationary half-Maxwellian distribution in the streamwise (z-direction) and full-Maxwellian in the cross-stream directions (x and y-direction). The velocity distribution based on temperature for each species temperature shown in table 2 is isotropic in all directions, but in the streamwise direction, it has the bulk velocities shown in table 2 as well. The reference plasma number density is considered to be the same as the ion number density at the thruster exit of $n_\mathrm{0} = 1.0 \times 10^{13}$ m⁻³. These selected values for $T_\mathrm{e0}$ and $n_\mathrm{0}$ result in an initial Debye length, $\lambda_\mathrm{D0} = 3.32 \times 10^{-3}$ m, initial electron plasma frequency, $\omega_\mathrm{pe} = 1.78 \times 10^8$ rad s⁻¹, and initial electron thermal velocity, $v_\mathrm{te0} = 592,892$ m s⁻¹. It should be noted that the Debye length of $3.32 \times 10^{-3}$ m obtained for the conditions modeled in this work is approximately 10 times larger than that of an actual GIT thruster [8, 9]. In the kinetic simulations, the ion and electron timesteps should follow the requirements of $\Delta t \lt 0.1 \omega_\mathrm{pe}$. In this work, we use a timestep of $\Delta t = 0.05 \omega^{-1}_\mathrm{peo} = 2.8 \times 10^{-10}$ s for ions and electrons. The distance traveled by an electron with $v_\mathrm{te0}$ at this time step is $1.7 \times 10^{-4}$ m, which is sufficiently smaller than the minimum mesh size of approximately $2 \times 10^{-3}$ m, and therefore, the CFL condition is also satisfied. The superparticle parameter, $F_\mathrm{num}$, is set at 2500 for all simulations such that there are at least 15 particles per species per cell for both the C-FOT and E-FOT. To obtain a time-averaged steady solution, 15 particles/cell is adequate since we take time-averaged values over 200 000 steps. We also verified that with 15 particles/cell, the electric field calculation converged over a few time steps. As a result, the total number of computational ion and electron particles is about 20 million at steady state. Table 2 also gives the Xe neutral particle parameters for the cases with a background pressure; otherwise, the plume is assumed to be collisionless. The exit number density of neutrals is $n_\mathrm{n0} = 1.0\times 10^{17}$ m⁻³, which is the typical order for actual GITs giving a total number of computational neutral particles of about five million at steady state.

Table 2. Parameters of the species at the thruster and neutralizer exits.

Thruster exit conditions	Xe	Xe+	e− (Type-A)	e− (Type-B)
Source center a (m)	(0.0, 0.4, 0.1)	(0.0, 0.4, 0.1)	(0.0 ,0.525, 0.1)	(0.0, 0.4725, 0.1+d0)
Source radius (cm)	6.25( = R0)	6.25( = R0)	6.25( = $R_\mathrm{e0}$)	1.0( = $R_\mathrm{e0}$)
Number density (m−3)	$1.0\times 10^{17}$	$1.0\times 10^{13}$	$1.0\times 10^{13}$	$3.2\times 10^{14}$
Bulk velocity (m s−1)	200	30 000	0	0
Temperature (K)	300	232	23 210	23 210
Total current (mA)	—	0.59( = $I_\mathrm{i0}$)	4.65( = $I_\mathrm{e0}$)	4.65( = $I_\mathrm{e0}$)

^a The source centers are (x₀, y₀, z₀) for Xe and Xe⁺ and ($x_\mathrm{e0}$, $y_\mathrm{e0}$, $z_\mathrm{e0}$) for e⁻.

In this study, the following assumptions are made to simplify the model. Since the apertures on the thruster optics are not modeled, ions are uniformly emitted from the thruster wall surface. In reality, the apertures cause variations in ion density, and electrons present to some extent inside the aperture [14, 42]. Unlike a hollow cathode, electrons are emitted from the neutralizer wall surface similar to a filament neutralizer [10, 11], a direct emission neutralizer [43], or a diode mode neutralizer [44]. Collisions between Xe and e⁻, Coulomb collisions between Xe⁺ and e⁻, the presence of multiply-charged ions (i.e. Xe$^{++}$ or more), electron emission due to secondary electron emission (SEE) and ion-induced electron emission (IIEE) , and reflection of ions and electrons at the wall surface with their charges preserved are neglected.

We test eight conditions to investigate the effects of in-space versus in-chamber geometries, neutralizer locations, and the presence of background neutral particles, as defined in table 3. The in-space geometry simulation, designated as '1,' is compared to the in-chamber geometry simulation, designated as '2,' for the neutralizer types A and B. The difference in the BC settings was described in section 2.2. Then we discuss the ion beam and neutralizer coupling, changing the neutralizer exit positions at four locations: 2B-0, 2B-1/4, and 2B-1/2 cases indicating $\frac{d_0}{R_0} = 0$, $\frac{d_0}{R_0} = \frac{1}{4}$, and $\frac{d_0}{R_0} = \frac{1}{2}$, respectively, where d₀ is the distance between the neutralizer exit and thruster exit, and R₀ is the thruster radius. The above-mentioned cases do not model neutrals to differentiate the electrical effect from the high-background pressure effect. Therefore, we also simulate a case with a finite background pressure designated as 2B-BP with $\frac{d_0}{R_0} = 1$.

Table 3. Test conditions of each case ID for facility effect and neutralizer position study.

Case ID a	Outer boundary	Distance, $d_0/R_0$	Background neutral
1A	In-space	0	Not present
1B	In-space	1	Not present
2A	In-chamber	0	Not present
2B	In-chamber	1	Not present
2B-0	In-chamber	0	Not present
2B-1/4	In-chamber	1/4	Not present
2B-1/2	In-chamber	1/2	Not present
2B-BP	In-chamber	1	Present

^a The neutralizer types, A and B, are described in table 2.

An electrical schematic diagram for GIT ground operation modeled in this study is shown in figure 3. Ions with a potential of $V_\mathrm{dc}$ generated in the discharge chamber inside the thruster are accelerated by ion optics and emitted from the external grid with a voltage of $V_\mathrm{th}$, where the beam ion kinetic energy in the axial direction corresponds to $V_\mathrm{dc}-V_\mathrm{th}$. Since this study uses 30 000 m s⁻¹ for the beam ion velocity when $V_\mathrm{th} = 0$ V, $V_\mathrm{dc}$ is set at 612 V, assuming that there is only ion motion outside the thruster, for simplicity. This study further investigates how the thruster plume is affected by using similar $V_\mathrm{dc}$ and $V_\mathrm{th}$ potential conditions to that in actual experiments. Table 4 shows these three additional conditions with respect to the 2B-BP case. The 2B-ACC case corresponds to the situation where there is no outermost decel grid so that the accel grid with $V_\mathrm{th} = -200$ V is exposed to the plasma plume. The exit density and velocity for the 2B-ACC case are corrected to $8.68\times 10^{12}$ m⁻³ and 34 553 m s⁻¹, respectively, because the incoming ions are considered to have a 200 V higher axial energy compared to the baseline case 2B-BP. The 2B-NM and 2B-NP cases simulate the case where the neutralizer is negatively and positively biased relative to the chamber and plasma screen by $V_\mathrm{ne}$, which corresponds to the neutralizer bias voltage applied in GIT experiments.

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Table 4. Test conditions of each case ID for electric potential boundary study.

Case ID	Electric potential
	Thruster exit, $V_\mathrm{th}$	Neutralizer exit, $V_\mathrm{ne}$
2B-BP	0 V	0 V
2B-ACC a	−200 V	0 V
2B-NM	0 V	−5 V
2B-NP	0 V	5 V

^a The ion bulk velocity is corrected to 34 553 m s⁻¹, and the ion exit density to $8.68\times 10^{12}$ m⁻³. All other conditions are the same as the 2B-BP case.

After the simulations reach a steady state, the current is calculated for ions and electrons by directly sampling the computational particles, as shown in figure 3. From the conservation laws, ion and electron currents are given by:

$$\begin{equation} I_\mathrm{i0} = \left\{I_\mathrm{ne}+I_\mathrm{th}+I_\mathrm{ps}+I_\mathrm{vc}\right\}_\mathrm{ion}, \end{equation} \tag{ 5 }$$

$$\begin{equation} \kern14.2pt I_\mathrm{e0} = \left\{I_\mathrm{ne}+I_\mathrm{th}+I_\mathrm{ps}+I_\mathrm{vc}\right\}_\mathrm{electron}. \end{equation} \tag{ 6 }$$

CHAOS has multiple GPUs with MPI-Cuda parallelization strategies [26]. This study uses 16 NVIDIA A100 GPUs on the Delta machine at the National Center for Supercomputing Applications for all cases. In the cases without neutral particles, we simulate 500 000 steps before sampling and then sample 200 000 steps to obtain the field macro-parameters and currents. In contrast, in the case with neutral particles, we simulate 5000 000 steps prior to sampling due to the slow CEX particle motion and then sample 500 000 steps. The total simulation runtimes are; about 30 h for the 1A and 1B cases, about 12 h for the 2A, 2B, 2B-0, 2B-1/4, and 2B-1/2 cases, about 90 h for the 2B-BP, 2B-ACC, 2B-NM, and 2B-NP cases.

This section examines how the neutralizer position affects the coupling between ion and electron sources in ground-based testing. Specifically, we consider only the type B neutralizer and perform calculations for the in-chamber case, where the neutralizer position is the only variable, i.e. the cases 2B-0, 2B-1/4, and 2B-1/2. The charge density in the x = 0 m plane is shown in figure 9 for different distances between the neutralizer exit and the plasma screen wall, d₀. In all cases, electrons enter slightly above the ion source, form a negative charge density region, and then move toward the lower-right direction, as shown by the green arrows. Electrons begin to slow down and accumulate when they pass through (y, z) = (0.4, 0.2) m, where the potential is at its maximum. A negative charge density region around (y, z) = (0.3, 0.35) m is formed on the opposite side of the electron source from the point of maximum potential, which we refer to as an 'electron pool' in this study. This electron pool has also been observed in previous studies [25, 28] and is a unique phenomenon in ion thruster plume neutralization with neutralizers adjacent to the thruster.

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When $d_0/R_0 = 0$, the electron pool is formed near the plume center, but as d₀ increases, the negative region shifts towards the lower right. However, these characteristics of electron motion are mainly observed on the thruster center plane, including the neutralizer exit (the x = 0 m plane). Figure 10 shows the charge density distribution in the x–y plane at the z = 0.45 m. Although there is significant variation in charge density around $(x,y) = (0,0.25)$ m or $(0,0.6)$ m in figure 10, the plume is uniformly distributed in most radial directions.

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Next, figure 11(a) shows the normalized potential along the thruster axis for each neutralizer-type position. The plume potential increases as d₀ decreases, and the peak potential is particularly high for $d_0/R_0 = 0$. This result indicates that the best neutralizer-ion beam coupling is obtained where $1/2 \lt d_0/R_0 \lt 1$ because the position of the maximum potential is seen to occur approximately $(z-z_0) = R_0$, and there is no significant difference between the $d_0/R_0 = 1/2$ and 1 cases. Figure 11(b) shows the normalized potential on the neutralizer axis near the neutralizer exit. In all cases, the potential decreases just after the neutralizer exit and increases after the potential reaches a minimum value at approximately $(z-z_\mathrm{e0})/R_\mathrm{e} = 0.2$, forming a virtual cathode. The electron space charge limits the low-energy electron transport in the virtual cathode region, as explained in a previous experimental study [35]. As d₀ increases, the minimum potential in figure 11(b) increases. This is due to the relaxation of the space-charge limitation as the exit of the neutralizer approaches the high potential space. In addition, when the neutralizer exit is on the wall ($d_0/R_0 = 0$), there is no path for electrons to travel upstream to the neutralizer exit and into the plume, thereby reducing the neutralizer-ion beam coupling and increasing the electric potential in the plume. The results shown in this section conclude that moving the neutralizer downstream contributes more to neutralizing the ion beam. This trend is consistent with an experiment in which the plume potential decreased as the neutralizer moved toward the downstream [35].

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To understand the effect of background pressure in the vacuum chamber, we use the 2B case as the baseline. Figure 12 shows the background pressure, p, in the x = 0 m plane, where p, is calculated from the ideal gas equation of $p = n_\mathrm{n}kT_\mathrm{n}$, and $n_\mathrm{n}$ and $T_\mathrm{n} = 300$ K are the neutral number density and neutral particle temperature, respectively. As a result of the neutral number density at the thruster exit and the size of the numerical pump, the highest pressure is about 4 µTorr at the thruster exit, with a minimum pressure of about 1.6 µTorr around the center of the vacuum chamber. This is similar to the typical background pressures found during ground test experiments.

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CEX and MEX collisions with the background neutrals change the ion velocity, which, in turn, alters the plume. CEX ions, originally neutral particles, have much smaller velocities than the beam ions and are the main cause of facility effects due to the high background pressure. Figure 13 shows the number density distribution of CEX ions in the x = 0 m plane. CEX ions are produced in all regions where beam ions exist, but many CEX ions are particularly produced immediately downstream of the thruster, where the neutral number density is a maximum. However, high-density regions also appear outside the plume core, indicating an asymmetric structure. This occurs because the CEX ions produced with a very small velocity remain in the electron pool region (y, z) = (0.3, 0.35) m, as indicated in figure 9. Similarly, CEX ions are trapped in the virtual cathode near the neutralizer exit.

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Figure 14(a) shows the potential on the thruster axis, with the maximum potential decreasing in the presence of background neutrals. The decrease is due to two reasons. First, more electrons are present in the plume core. When collisions with neutral particles are modeled, the charge density on the thruster axis for ions and electrons in figure 14(b) indicates that the presence of CEX ion increases the ion density of the plume by up to 14%, while the electron density increased by nearly 35%. The second reason for the decrease in electric potential is the decrease in electron temperature. The electron temperature at ($x, y, z$) = (0, 0.4, 0.1625) m in the 2B-BP case obtained by a Maxwellian fitting (equation (7)) of 15 eV is nearly half that obtained in the 2B case due to the increase in the minimum potential of the virtual cathode. The simulations show that the accumulation of CEX ions in the virtual cathode increases its minimum potential of the virtual cathode from $\phi = -3.2$ to −2.7 V. The smaller decrease in voltage at the virtual cathode means that fewer electrons return to the neutralizer, and the kinetic energy of the electrons that can pass through the virtual cathode is lower, resulting in greater neutralization of the plume.

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Another interesting difference is the ion sheath formed on the side walls. Figure 15 shows the normalized volume charge density, $\rho/\rho_0$, of the 2B and 2B-BP cases. Only when background neutral particles are present, the $\rho/\rho_0 = 0$ line appears between the chamber side wall and the plume core region. This is due to slow CEX ions that create an ion sheath in front of the chamber wall, where we define the ion sheath as the volume near the wall where $\rho/\rho_0\gt0$. The thickness of the ionic sheath is approximately 0.1 m. A 1D analytical expression for the Child-Langmuir sheath thickness, s, can be calculated as [45]

$$\begin{equation} s = \lambda_\mathrm{D,s}\frac{0.79}{\sqrt{\alpha_i}}\left(\frac{e\phi_\mathrm{s}}{T_\mathrm{e,s}}\right)^{3/4} \end{equation} \tag{ 8 }$$

where $\alpha_i = 0.61$ is the number density factor [45]. $\lambda_\mathrm{D,s}$, $\phi_\mathrm{s}$, and $T_\mathrm{e,s}$ are the local Debye length, the electric potential, and the electron temperature at the sheath edge ($\rho/\rho_0 = 0$ line), respectively. Using ($x, y, z$) = (0.3, 0.4, 0.3) m as the reference point, we obtain from our simulations: $n_i = n_e \sim 1.0\times10^{11}$ m⁻³, $T_\mathrm{e,s}\sim 2$ eV, and $\phi_\mathrm{s} \sim 7$ V. Thus, the analytical sheath thickness is calculated as $s \sim 0.086$ m, which is close to our simulated sheath thickness.

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Comparing the ion and electron current result with and without background neutral particles in Cases 2B vs. 2B-BP, we find that the current flowing to the plasma screen increased for both ions and electrons due to the backflow of CEX ions. Additionally, we demonstrated that the presence of CEX ions increases the potential of the virtual cathode, which results in a decrease in the current flowing back to the neutralizer. The current flow to the downstream and the side walls ($I_\mathrm{vc,end}$ and $I_\mathrm{vc,side}$, respectively) exhibits an interesting behavior due to the presence of the background neutral particles. With respect to ions, 24% of the ion currents change their direction toward the side wall (see the difference in $I_\mathrm{i,vc,side}$) due to collisions with neutral particles, whereas only 3% of the electron currents (see the difference in $I_\mathrm{e,vc,side}$) change their direction. Consequently, ion-neutral particle collisions have little effect on the destination of electrons. However, note that this study ignores electron-neutral particle collisions, which may affect the electron current.

Since the 2B-BP case, which includes both the grounded chamber wall and background pressure, is the closest condition to actual ground testing among the cases that we consider, it is useful to compare the predicted maximum plume potential and electron temperature with experimental studies [8–12]. Our maximum plume potential of 110 V is higher than that of Foster et al [8], Foster et al [9, 12], Polansky et al [10], and Conde et al [11] who measured 4-10, 3.5, 20-70, and 27, respectively. Our maximum electron temperature of 15 eV can similarly be compared with measured values of 2-2.7, 2-20, 2-7, and 2, respectively. Note that although the density and geometry conditions are different, our potentials and electron temperatures are only a few times larger than those obtained by experiments. This may be due to factors already mentioned in section 3.1. First, electron emission may be an important factor, especially on the chamber downstream wall at z = 0.8 m, where electrons incident with an energy of 6.37 eV on average are observed in this study. In this energy region, according to [46], the total emission yield, the sum of probabilities of SEE and inelastic reflection on the wall, from carbon is between 0.1 and 0.5, which means more than 10% of electrons are recovered when they hit the wall. In addition, the IIEE yield is also approximately 0.1 when 600 eV xenon ions hit the carbon [47]. The small energy secondary electrons emitted from the grounded walls would be trapped inside the high potential plume and could reduce the plume potential. Second, due to the apertures on an actual GIT exterior grid, neutralization is initiated closer to or inside the exterior grid by electrons inside the grid apertures, as shown in [14, 42]. The third possibility is the axial location of the neutralizer exit. According to [35], the angle between the normal beam axis and the neutralizer axis can affect the plume potential. Finally, another past study [48] suggested that the neutralizer-ion beam coupling was enhanced by ions generated in electron-neutral collisions, which is also not modeled in this study. In addition, if electron collision with other particles were simulated, electrons could lose their kinetic energy inside the plume, preventing their escape from the high-potential regions, which would also reduce the electron temperature and electrical potential. The influence of these processes, however, would need to be modeled in the fully kinetic/DSMC context.

First, we present the results of changing the potential at the thruster exit ($V_\mathrm{th}$), simulating the case when a negatively biased accel grid is exposed to the plasma without a decel grid Case 2B-ACC. As shown in figure 16, low potentials are observed near the negatively biased outlet, while a higher potential region is seen downstream. In figure 17, the potential plotted on the thruster axis shows that in the 2B-ACC case, it is initially −200 V but reaches a larger maximum potential than that in the 2B-BP case at $z\sim0.2$ m or 0.05 m downstream from the peak of the 2B-BP case.

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In the 2B-ACC case, the ion inflow velocity increases by an amount proportional to the square root of the accel grid potential of 200 V because there is no deceleration by a decel grid (2B-ACC). The axial ion velocity, $w_\mathrm{i}$, in the y = 0.4 m plane is shown in figure 18(a), where $w_\mathrm{i}$ is normalized by the reference beam ion velocity of $w_\mathrm{i0} = 300~000$ m s⁻¹ (612 eV). The maximum potential difference seen in figure 17 is due to the large axial velocity of the ions in 2B-ACC despite the same radial velocity distribution, which reduces the divergence of the beam and increases the ion density in the center of the plume. As ions move downstream, their axial velocity decreases, as shown in figure 18(a), resulting in radial divergence as in the 2B-BP, and the difference in potentials decreases. While the 2B-ACC case shows larger velocities near the thruster exit (z < 0.2 m), there is almost no difference between the two cases downstream. The line plots in figure 18(b) also show that the velocities are almost identical, but the beam width is narrower in the 2B-ACC case. This is also due to the convergence of the plume near the thruster caused by the external electric field induced by the potentials between the plasma screen of 0 V and the thruster exit of -200 V. This was also observed in an experiment [49], where the grounded thruster cover around the accel grid reduced the beam divergence angle.

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Next, we investigate the cases where the neutralizer exit potential ($V_\mathrm{ne}$) is 5 V lower (2B-NM) and 5 V higher (2B-NP) with respect to the ground voltage of 0 V of the 2B-BP case. As shown in figure 16, the potential of the plasma plume changes significantly even though the exit potential is only changed by ±5 V. Figure 19(a) shows the potential along the thruster axis where it can be seen that the maximum potential is increased by $e\phi / kT_\mathrm{e0} \sim 50$ for the 2B-NM case, while for 2B-NP it is decreased by $e\phi / kT_\mathrm{e0} \sim 50$. Additionally, as shown in figure 19(b), a comparison of the potentials near the neutralizer indicates that as the neutralizer potential is lowered, the downstream potential well becomes smaller, and it disappears completely in the 2B-NM case. The reason that this occurs is related to the current distributions, as we discuss next.

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Table 6 shows the currents to different parts of the GIT for three cases (2B-BP, 2B-NM, and 2B-NP). Regarding ion currents, the large plume potential for Case 2B-NM shown in figure 16 causes more divergence of the ion beam, and a larger number of CEX ions return to the chamber side walls and the thruster, compared to case 2B-BP. However, a more significant difference is observed in the electron currents than in ion currents. As shown in figure 19(b), when $V_\mathrm{ne}$ is small, all electrons are allowed to flow downstream, while when $V_\mathrm{ne}$ is large, more electrons return to the neutralizer, as a result of virtual cathode formation. Here, $I_\mathrm{e0}-I_\mathrm{e,ne}$ can be considered an effective emission current from the neutralizer, $I_\mathrm{e,eff}$, which increases as $V_\mathrm{ne}$ decreases. This trend is consistent with experimental results showing that lowering the neutralizer bias voltage releases more electron current [35]. Furthermore, there is a large difference in $I_\mathrm{vc,end}$, but a small difference in $I_\mathrm{vc,side}$ because electrons do not return to the neutralizer.

Despite the increase in $I_\mathrm{e,eff}$, the plume has a very high potential in the 2B-NM case. The electric potential where electrons are emitted with respect to the vacuum chamber, $V_\mathrm{ne}$, changes the electron motion in the vacuum chamber. Electrons produced at lower potentials than the vacuum chamber ($V_\mathrm{ne} = -5$ V, 2B-NM) do not return to the plume because of the sheath in front of the vacuum chamber walls and are all absorbed by the wall, resulting in a shortage of electrons for neutralization. On the other hand, the sheath reflects almost all electrons produced at higher potentials ($V_\mathrm{ne} = 5$ V) except for those with high kinetic energy. As a result, a sufficient number of electrons remain in the chamber when $V_\mathrm{ne}$ is large (2B-NP), as shown in figure 20. In table 6, $I_\mathrm{vc}$ decreases as $V_\mathrm{ne}$ increases, which confirms this electron confinement effect inside the vacuum chamber from the viewpoint of the currents.

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Figure 21 shows the EVDFs in the z-directions obtained by sampling computational electrons at R₀ downstream on the thruster axis for the 2B-BP, 2B-NM, and 2B-NP cases. Two distributions are clearly seen in the cases 2B-BP and 2B-NM, but almost all electrons are thermalized in the 2B-NP case. Fitting the electron temperature to the distribution for the population centered around zero velocity reveals that $T_\mathrm{e}$ decreases as $V_\mathrm{ne}$ increases. This is because the plume potential is reduced by the above-mentioned change in electron motion (see figures 16 and 19(a)), allowing the otherwise-trapped electrons to move with less energy.

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There are only a few papers where the effect of changing the neutralizer potential is discussed. In the ground testing, Criado et al [50] measured electron temperature and density in the plume of a GIT, which had a grounded current circuit, using a Langmuir probe. They showed that the electron temperature decreased when the neutralizer bias potential increased from −50 to 0 V (see figure 9 in [50]). This electron temperature trend is consistent with that obtained by our numerical simulation, suggesting that the electron motion observed in our result also occurs in the actual experiment.