The effect of keeper geometry on the transport of electrons is investigated experimentally using electrostatic probes in the plume of a hollow cathode. Three keeper configurations—one single orifice and two multiple orifices—were studied. The multiple orifice cases were chosen to examine the influence of the hole-pattern radius while the total exit area and the number of holes remained constant. Two-dimensional maps of the plasma parameters and wave properties were inferred from the probe measurements and were used to evaluate a generalized Ohm’s law for the electron flow field. The contributions of pressure, fields, and drag on the transport of electrons were analyzed. The results indicate that increasing the hole-pattern spread reduces the electric field in the plume and increases the pressure contribution to the transport. A further analysis of turbulent wave energy conservation indicates that the multiple orifice keepers increase ion-neutral collisional damping, similar to auxillary flow injection. The implications of these findings on cathode plume modeling and keeper design are discussed.

## I. INTRODUCTION

Thermionic hollow cathodes^{1} are low-temperature plasma sources that are widely deployed in plasma science and technology, finding uses in basic plasma research^{2,3} and also in more applied fields like materials processing^{4} and space propulsion.^{5} The latter application motivates this work.

For spacecraft electric propulsion (EP) systems, such as ion and Hall thrusters,^{6} the hollow cathode supplies the discharge current. In these systems, it is desirable to provide a stable current while minimizing the coupling voltage and gas flow rate to optimize the power and propellant efficiencies of the overall thruster system. These efficiencies are critically linked to processes within the cathode and in its external plume region. Internally, power is primarily deposited into ionization and heating of the emitter—necessary processes to produce the plasma and the discharge current. In the plume, the onset of turbulent electrostatic waves affects the transport of electrons that ultimately causes a characteristic rise in plasma potential between the keeper exit and anode.^{7–9} This potential gradient requires additional electrical power to extract electrons from the cathode, an inefficiency in the system. Moreover, the turbulence also affects the stability of the discharge current and the onset of plume mode,^{10–13} a large-amplitude current instability correlated with energetic ion production. In sum, the onset of turbulence is a deleterious phenomenon that is integral to the efficiency and stability of the cathode system.

Previous work has shown that these turbulent electrostatic plasma waves can be damped out with additional propellant flow, either directly through the cathode or injected into the downstream plume region.^{14,15} However, this technique costs extra propellant—a premium for space propulsion applications. More recently, experimental investigations indicate that changing the keeper geometry can, in some configurations, produce similar effects to gas injection.^{16} This difference was noted when modifying the keeper orifice geometry from a single centrally located hole to a set of smaller holes, while preserving the total cross-sectional exit area (for example, see Fig. 1). These novel designs can in some cases produce a more stable and electrically efficient discharge, potentially improving upon cathode designs that require additional gas injection.

Earlier studies suggest improved global performance and stability of a hollow cathode with a multiple orifice (MO) keeper over the single orifice (SO) configuration but this is not the case for all designs. In general, the mechanism(s) by which a MO keeper may reduce the discharge voltage or oscillations is unknown. This makes it unclear how to approach a design, given the virtually unlimited number of potential configurations.^{16} This work undertakes a targeted fundamental investigation of the physics driving changes in discharge voltage and current stability in the MO keeper. We specifically focus on a small subset of keeper geometries, enforce a constant total orifice exit area and number of holes, and vary the hole-pattern radius. Due to the effects of the keeper design on oscillation and voltage levels (both connected to turbulence in the plume region), this study focuses on the cathode plume. With our electrostatic diagnostics, we measure key plasma and wave properties to examine electron transport across a 2D experimental domain to determine how the keeper pattern radius affects these processes and might lead to observed changes in performance.

In Secs. II–VI, we first discuss the electron transport model used for hollow cathodes with a SO keeper, providing an overview of the classical and non-classical (wave-driven) contributions. Then, we describe the experimental apparatus used to investigate that transport. Following are the results of our study, primarily presented as 2D surface plots of the plasma, wave, and transport properties. Last, we discuss the sources of systematic uncertainty in our measurements, some implications for cathode modeling, a physical understanding of how and why the keeper pattern radius is affecting plume properties, and some indications for how the keeper design might affect internal plasma properties.

## II. GOVERNING RELATIONS FOR ELECTRON TRANSPORT

In this section, we adopt a three-fluid description (electrons, ions, and neutrals) of the plasma to discuss the transport of electrons in the cathode plume, following the approach of the highest fidelity SO cathode models to-date.^{17,18} An inherent assumption in our analysis is that this existing picture for standard cathodes applies to the MO configurations in our study, most notably that the discharge is axisymmetric, which appears to be borne out by our measurements of plasma density shown later in Sec. IV A. With this in mind, we first overview the classical expression for electron transport and the simplifications that we employ to analyze the plume. Then, we discuss how the presence of electrostatic turbulence in these devices leads to anomalous wave-driven transport and deviations from the standard governing equations.

### A. Classical transport

Figure 2 shows our experimental configuration, a canonical setup for stand-alone testing of hollow cathodes for EP devices. The cathode is positioned on the left of the image and the discharge anode is placed on the right. The flow of electrons from the cathode to the anode is dictated by the electron momentum equation. In this work, there is no applied magnetic field. Hollow cathode codes often neglect electron inertia due to their small mass in describing conservation of electron momentum.^{7,17,19} We adopt the same convention in our analysis, giving

In Eq. (1), $ne$ is the electron density, $q$ is the unit charge, $E$ is the electric field vector, $Te$ is the electron temperature in eV, $me$ is the electron mass, $ue,i$ are the electron and ion velocity vectors, $\nu ei$ is the electron–ion collision frequency, and $\nu en$ is the electron-neutral collision frequency. The first term in Eq. (1) describes how changes in pressure affect the flow of electrons. In our cathode system, the plasma is expanding from a very dense internal plasma to the relatively sparse plume region. Thus, the pressure force pushes electrons out of the cathode and toward the anode. The second and third terms are drag forces resulting from Coulomb and particle collisions that prevent the flow of electrons to the anode. The last term in Eq. (1) is the electric force. Physically, an electric field is required to pull electrons across the plume and conserve current if the drag forces from collisions exceed the pressure gradient. This field then causes the potential to rise from cathode to anode and dictates the voltage (power) required to extract electrons from the internal plasma.

### B. Expressions for collision frequency in the cathode plume

Numerical simulations indicate a high degree of ionization in the plume region of the cathode.^{18} This implies electron-neutral collisions are relatively infrequent compared to their Coulombic counterpart, and we therefore neglect this term in Eq. (1). Classically, the Coulomb collision frequency can be calculated using^{20}

where $ln\u2061(\Lambda )$ denotes the Coulomb logarithm. Extensive numerical modeling and experimental measurements have shown this classical contribution is typically 1–3 orders of magnitude too small to explain the observed changes in plasma potential.^{7,17} Subsequently, it has been shown through theory,^{21} simulation,^{17} and experiment^{8,9} that wave-driven processes are responsible for enhancing the electron–ion collision frequency in the plumes of hollow cathodes. Through this previous work, the growth of broadband ion acoustic turbulence (IAT) has been identified as the primary source of enhanced collision frequency. Physically, the onset of these waves results from inverse electron Landau damping, a kinetic process where electrons resonant with the IAT exchange momentum with the waves leading to growth. The oscillations are damped through nonlinear ion Landau damping and ion-neutral collisions. These processes lead to the following linear growth rate for the IAT:^{21}

In this expression, $k$ is the wave number, $cs\u2261qTe/mi$ is the ion acoustic speed, $Ti$ is the ion temperature, and $\nu in$ is the ion-neutral collision frequency. The first term on the right-hand side is the inverse electron Landau damping term that is responsible for the growth of the waves. The second and third terms represent nonlinear ion Landau damping and ion-neutral collisional damping of the waves.

Through quasilinear theory, it has been shown that the growth of IAT results in an effective force on the electrons acting on longer time-scales.^{22} In a fluid description, this process can be interpreted as an effective collision frequency between the electrons and ions such that $\nu e=\nu Coulomb+\nu eIAT$, where $\nu eIAT$ is the wave-driven contribution. By applying the dispersion relation, the growth rate in Eq. (3), and the definition of wave momentum the effective collision frequency for IAT can be expressed as^{22}

where the sum over $k$ represents a sum over the electrostatic modes in the broadband IAT spectrum with wavenumber $k$, $\omega k$ is the angular frequency of oscillation, and $Wk$ is the wave energy density of the mode. By applying the dispersion relation of ion acoustic waves^{22} to the definition of wave energy density,^{22} the quantity $WkneTe=(\varphi ~k/Te)2$, where $\varphi ~k$ is the amplitude of a plasma potential oscillation with wavenumber $k$. Executing a sum over modes, we define the total wave energy density as $W=\u2211kWk$ and the spectral-average frequency as $\omega 0=\u2211k(\omega kWk)/W$. We can further define and average growth rate for the IAT as $\gamma \u2261\u2211k(\gamma kWk)/W$. This quantity, $\gamma $, determines the gains and losses of the total wave energy density as the IAT propagates through the plasma. Applying the conservation of wave energy leads to following the relation^{22}

where $v\u2192g=csu^i+ui$ is the IAT group velocity. For slowly drifting ions, this relation allows us to estimate the time-average growth rate of the IAT as $\gamma \u223ccs\u2207(W)/2W.$

### C. Current flux

Applying our assumptions and using experimental measurements of density, temperature, potential, and oscillation amplitude, we can invert Eq. (1) to solve for the net current density in the plasma,

where $\nu e=\nu ei+\nu eIAT$ is the total electron collision frequency and $q2nme\nu e\u2261\sigma e$ can be identified as the electrical conductivity of the plasma. Physically, this expression indicates that the pressure gradient and electric field control the direction and flow of net current density in the plasma while the collision frequency (primarily dictated by the anomalous contribution from IAT) scales the flux. Under the assumption that the electrons are the primary carriers of electrical current in this system (justified since $me/mi\u226a1\u21d2ue/ui\u226b1$), we have approximately solved Eq. (6) for the electron current density.

## III. EXPERIMENTAL SETUP

Following the theory outlined above, only a relatively small set of experimental measurements is required ($ne$, $Te$, $\Phi $, $\nu an$) to visualize the flow of current in the plasma and how it is affected by modifications to the keeper. We describe in this section the experimental methods used to investigate the effects of keeper configuration on current (electron) flow in the plasma. First, we discuss the apparatus we employed and then detail the analysis techniques we used to infer plasma properties from measured quantities.

### A. Cathode and probes

The test article employed in this study was a LaB$6$ hollow cathode with a nominal design point of 20 A and was operated at 15 A discharge current. The keeper was made of stainless steel and was modified with the orifice geometries shown in Fig. 1. The single orifice configuration had radius $ro$. The multiple orifice keepers preserved the total cross-sectional exit area by evenly dividing it across six smaller holes of radius $rm=ro/6$. For the MO 1 case, the hole-pattern radius $rp=r1=ro$, and for the MO 2 case $rp=r2=1.77ro$. The hollow cathode was operated on high-purity argon at 20 sccm. The discharge voltage was found to change as a function of keeper geometry, as indicated in Fig. 1. The vacuum facility was 1 m in diameter and 1 m in length and was pumped using turbomolecular and cryogenic pumps, achieving a total pumping speed of 4.5 kL/s on Ar. The discharge anode was placed 40 mm downstream of the keeper exit and was 19 cm in diameter and 30.5 cm long.

The experimental configuration is illustrated in Fig. 2. A collection of three electrostatic probes was placed in the plasma spaced radially 70 mm apart. This configuration is meant to reduce the perturbative effects while acquiring data from each probe. As a result, the probe measurements are acquired at different times. The probes were translated spatially in the radial ($r^$) and axial ($z^$) directions. The grid spacing varied between 1–10 mm in the radial direction and 1–5 mm in the axial direction, with finer spacing near the keeper orifice. The first probe was a swept planar Langmuir probe,^{23,24} used to measure the electron temperature. The second was a floating point emissive probe^{25} to infer the plasma potential. This measurement was acquired with an oscilloscope and then averaged in time. The final probe was a high-speed ion saturation probe,^{26,27} where the plasma oscillations were acquired using an oscilloscope at 20 MHz. We used the time-average of this signal as our estimate of the ion saturation current. Together, these experimental measurements allow us to examine Eq. (1) to understand how varying the keeper geometry affects the transport of current through the plasma.

### B. Time-average probe analysis technique

We analyze the Langmuir probe measurements using the method described in detail by Ref. 23, which we outline here. The Langmuir probe I–V curve is broken down into three regions: ion saturation, transition (electron retarding), and electron saturation. We first apply a linear fit to the ion saturation region and subtract this ion saturation current ($Isat$) from the I–V curve such that only the electron contribution to the collected $Ie$ current remains. We then curve fit the transition region between the floating and plasma potentials to determine the temperature such that

where $Vb$ is the probe bias voltage. In the thin sheath limit, we use this temperature to calculate the ion density, $ni$, as

where $Ap$ is the probe area. We use the time-average value of the high-speed ion saturation probe as the estimate for $Isat$ as it tended to give more consistent and smoother results than the swept planar Langmuir probe. We then assume quasi-neutrality to infer the electron density $ni=ne=n$. Finally, we use the floating emissive probe voltage, $Vem$, and the electron temperature to estimate the plasma potential,

This temperature correction term comes from sheath effects near the surface of a hot emissive probe.^{25}

### C. High-speed probe analysis

The high-speed ion saturation probes are used to examine the spectral content of plasma waves and evaluate Eq. (4). We split the raw 50 ms acquisition into 1000 sections of 50 $\mu $s each and averaged their Fourier spectra. The resulting power spectrum has a frequency resolution of 20 kHz. To calculate an anomalous collision frequency from these measurements, we then assume that any temperature fluctuations occur on long time-scales compared to the IAT. This is justified for two reasons: first, since $\nu Coulomb<\omega k$, significant Ohmic heating cannot be driven classically;^{12,13} second, for the linear theory of IAT to apply we require that the IAT linear growth rate follow $\gamma k\u226a\omega k$ and that the wave amplitude follows $(\varphi ~k/Te)2\u226a1$. Together, a small IAT growth rate and amplitude imply that an individual mode cannot drive significant heating on its own time-scale (although the entire spectrum of waves can collectively drive heating). Under this assumption, the fluctuations in ion saturation current are proportional to changes in plasma density. If we further assume a collisionless isothermal sheath near the probe surface (justified under our earlier assumption that individual IAT modes do not affect the electron temperature on their own timescale), then we can approximate $(Wk/nTe)2$ in Eq. (4) as $(I~sat,k/Isat,0)2$, where $I~sat,k$ is the amplitude of the mode with wavenumber $k$ (or equivalently angular frequency, $\omega k$, from the assumed ion acoustic dispersion relation) in the Fourier spectrum of the ion saturation current and $Isat,0$ is the time-average value.

In the measured plasma oscillation spectra (see Fig. 3), there are some conditions where high-frequency resonances (typically 1–10 MHz) coexist with the underlying IAT in the same frequency band. The nature of these peaks is not known (true plasma instability, spurious noise in the signal picked up by the probe, probe perturbation); thus, we cannot apply the theory above to understand their potential effect on the plume. In cases where these resonances occur, we curve fit the underlying IAT spectrum to estimate the wave-driven collision frequency, $\nu eIAT$. An example of these curve fits is shown in Fig. 3. We execute a sum [see Eq. (4)] over the fitted domain (200 kHz–7 MHz), as has been done in previous works.^{9,28} The effect of this curve fitting method on our results is further discussed in Sec. V and Appendix A, where we compare to an upper bound on the calculated IAT-driven collision frequency. This uncertainty in our analysis should be borne in mind as we present and discuss the results below.

## IV. RESULTS

In this section, we present the results of the measurements and analysis described above for each keeper orifice configuration from Fig. 1, followed by the calculation of electron streamlines and a summary of key findings.

We first apply 2D loess fitting to the measurements to produce smooth gradients for the evaluation of Eq. (1).^{29} The plasma parameters are then mirrored across the $z$ axis to produce 2D contour plots that cover the majority of the area between the cathode and anode. Small black bars in Figs. 4–6 denote the keeper geometry on the left-hand side at $z=0$ mm and the anode at $z=40$ mm in the upper and lower right-hand corners.

### A. Measured properties

Figure 4 shows the measured plasma and wave properties for each keeper configuration. The first row shows the measurements for the SO configuration, the second row MO 1, and the third row MO 2. The first column shows the plasma density, the second is the electron temperature, the third is the plasma potential, and the fourth is the anomalous collision frequency. The uncertainty range in each parameter over the experimental domain is noted at the bottom of each subfigure by the value of $\delta $. The driving source of statistical uncertainty is the electron temperature, determined as the standard deviation in the Langmuir probe retarding region curve fit [see Eq. (7)], which propagates to the other plasma parameters. The uncertainty in $\nu eIAT$ is calculated as the standard error across the ensemble of Fourier transforms of the subdivided ion saturation probe traces and found to be approximately 3%–4%. We examine Fig. 4 by column to comment on the trends with increasing keeper hole-pattern radius.

#### 1. Plasma density

The first column of Fig. 4 shows plasma density maps for the three keeper configurations as a function of position with black level curves marking logarithmic changes in density. The red lines show the radial full-width-half-maximum (FWHM) of the density at each axial position. Qualitatively, the plasma density contours appear to have a similar structure; we do not observe high density regions above the holes of the MO keeper configurations. This suggests that the radial and likely azimuthal density gradients associated with the keeper hole-pattern geometry are smoothed out, possibly by diffusion processes, upstream of our measurement domain. This finding justifies the assumption of axisymmetry in our subsequent analysis.

Examining more closely, the increasing hole-pattern radius produces a greater FWHM spread of the plasma, as might be physically expected based on the hole positions. This spread is accompanied by a decreased plasma density, where the SO keeper (top) has the highest peak density and the MO 2 case (bottom) is the lowest by a factor of three. We interpret this result by a current continuity approach. Generously assuming the plasma is beam-like ($ue\u2248uez$) and the discharge current lies entirely within the red FWHM bounds, then for a given discharge current, $Idc$, we can estimate an axial electron velocity averaged over the cross-sectional area as

In this expression, $RFWHM(z)$ represents the outer edge of the assumed beam denoted by the red FWHM boundary. Thus, the increased plasma width ($RFHWM$) of MO 1 and 2 could lead to a lower electron drift velocity, though the slightly lower plasma density may temper this effect. Physically, this is suggestive of a smaller electron velocity and Mach number for the MO cases than the SO case. We investigate this possibility later in Sec. IV B.

#### 2. Electron temperature

The second column of Fig. 4 shows the evolution of the electron temperature in the plume for each keeper configuration. Beginning at the top [Fig. 4(b)], we find typical hollow cathode behavior for a single orifice keeper.^{9,30} The electron temperature rises slowly as a function of axial and radial position, which is linked physically to enhanced Ohmic heating of electrons due to the large wave-driven collision frequency. In the MO 1 configuration [Fig. 4(f)], the electron temperature in the plume tends to rise, globally. Notably, the temperature near the keeper has increased, which may result in a greater erosion of this electrode due to a larger sheath voltage accelerating ions toward the surface. As a function of radial position, the electron temperature rises more rapidly in this configuration than for the single orifice. This culminates in an electron temperature that is over 2 eV higher than the SO case near the anode. These trends persist in the MO 2 configuration [Fig. 4(j)]. The electron temperature rises even more rapidly near the keeper and with increasing radius. Viewing the column of Figs. 4(b), 4(f), and 4(j), we find that increasing the hole-pattern radius leads to greater electron temperature in the plume. In the context of Eq. (3), this result implies greater electron thermal velocity and possibly increased nonlinear ion Landau damping, although limitations in our experimental apparatus did not permit ion temperature measurements to confirm this latter effect on IAT growth.

#### 3. Plasma potential

The third column of Fig. 4 shows the plasma potential for the three keeper configurations. Starting with Fig. 4(c), we again find a typical potential structure for a cathode with a SO keeper.^{30} The potential rises with axial and radial position from the cathode to the anode by around $\Delta Vp=18$ V, leveling off near the discharge voltage (29.2 V for this condition). This potential gradient indicates the presence of an electric field in the plume, which previous work has shown is a response of the plasma to the wave-driven collision frequency that inhibits the flow of electrons toward the anode.^{31} Figure 4(g) shows that pushing the plasma slightly off axis with the MO 1 keeper maintains the same qualitative potential structure but the rise from the cathode to the anode is now slightly reduced to $\Delta Vp=12$ V. The larger pattern radius MO 2 case is qualitatively different. Near the keeper, we observe the formation of a local anode with the potential falling toward the true anode. This finding implies a reversal of the electric field in the plume region in the MO 2 configuration, pulling electrons back toward the cathode. Physically, this phenomenon does not clearly fit into the previous framework of IAT-driven collision frequency, as we physically expect the electric field to be induced by the effective drag that is preventing the conservation of current. Rather, it points to the more typical picture of an electropositive plasma, where the highly mobile electrons are restrained by the ambipolar electric field to maintain charge neutrality. We provide a more detailed discussion of this result in Sec. V.

As a final point of discussion for Figs. 4(c), 4(g), and 4(k), we note that the plasma potential near the keeper exit tends to increase with greater pattern radius. This finding is difficult to explain when only considering the cathode plume. To understand this behavior, we likely need to consider processes inside the cathode and the interaction between the plasma and the keeper surface. These more complex mechanisms are outside the scope of this experimental work because this region of the plasma was not accessible to our diagnostics. Potentially, great physical insight could be gained from numerical modeling of cathodes with MO keepers. Complementary optical plasma diagnostics, such as incoherent Thomson scattering^{32} and laser induced fluorescence,^{33} to directly capture the velocity distribution functions may also be key to understanding the rise in plasma potential.

#### 4. Anomalous collision frequency

Using the curve fits to the Fourier transforms of the ion saturation current oscillations, we calculate an effective collision frequency through Eq. (4) at each position in the plume for all three keeper configurations. The statistical uncertainty determined by calculating the standard error in the sum over modes is approximately 0.1% of the measured value. We also assessed the weak-growth assumption of the linear theory and found that it is met throughout the plume for all keeper configurations, with $(\gamma /\omega 0<0.09\u226a1)$.

The results of this analysis are shown in Figs. 4(d), 4(h), and 4(l). For all three cases, we find that the effective collision frequency rises as a function of axial and radial position. This observation is consistent with similar (but not identical configuration) numerical simulations of these devices that include the growth and convection of IAT wave energy,^{17} previous 1D on axis measurements,^{9} and 2D measurements.^{34} We find that the MO hole-patterns globally decrease the collision frequency in magnitude. Physically, this explains why we find smaller plasma potential gradients with increasing hole-pattern radius. Electrons are experiencing fewer effective collisions as they transit the plume to the anode. We note that there is a caveat to this interpretation, because of the systematic uncertainty in our collision frequency calculation, which is discussed in greater detail in Sec. V and Appendix A. While the trend in reduced IAT collision frequency largely remains, it becomes less pronounced (see Fig. 8). The question remains, why has the keeper geometry reduced this effective collision frequency? We devote the rest of this work to better understanding how the keeper geometry is influencing plasma and wave properties.

### B. Electron flow field

Using the measurements in Fig. 4, we calculate the net current flux using Eq. (6) for each of the keeper configurations. Under the assumption that electrons are primarily responsible for the current in this discharge ($ue\u2212ui\u223cue$), we can estimate the electron drift velocity and, therefore, the electron Mach number (the critical parameter for IAT growth). Figure 5 shows this parameter with electron streamlines overlaid in black. The uncertainty in the Mach number and the streamlines is propagated through Eq. (6) using standard error propagation. The results of this uncertainty analysis are the plots in the first and third columns of Fig. 5. Column one represents the most radial electron streamlines, i.e., the largest radial component, within experimental uncertainty ($uer\u2192u\xafer+\delta uer$, adopting the bar notation to imply average value), and the smallest possible axial component ($uez\u2192u\xafez\u2212\delta uez$). Column three represents the most axial electron streamlines, which are calculated using the largest axial velocity and smallest radial velocity uncertainty bounds. The average value (most probable) of the Mach number and streamlines is shown in the second column [Figs. 5(b), 5(e), and 5(h)].

Figure 5(b) shows that electrons drift out of the SO keeper at approximately Mach $0.9$ and rapidly decelerate to $\u223c$Mach $0.1$ as a result of the drag induced by the turbulence and the expansion of the plasma. We find that electrons originating off-axis experience significant radial acceleration. This result suggests that few electrons originating at the keeper exit flow to the inner surface of the anode, contradicting the predictions of similar (not identical) numerical models.^{31} A more detailed analysis of this result and comparison to ongoing modeling efforts is discussed in Ref. 19. To briefly summarize this earlier work, the highly divergent flow field is the result of large electric fields pulling electrons out radially. Analyzing the electron momentum through Ohm's law, leads to predicted current densities that do not properly conserve current. It was suggested that the underlying assumptions in Ohm’s law equation (6) may explain the potential issues with current conservation noted in the earlier work.

Next, in Fig. 5(e), we show the flow of electrons for the MO 1 keeper in which there are some notable differences from the SO case. We find a large calculated electron Mach number near the keeper surface $Me\u22486$ but which rapidly drops off to $Me\u22480.1$ near the anode. This upstream value is unreasonably high as this indicates a velocity of $ue\u223c0.01c$, where $c$ is the speed of light. We do not expect the electrons to be approaching relativistic conditions in this cathode system. More likely, probe perturbations have systematically affected the calculation either by suppressing the collision frequency or that measurement noise propagated through numerical derivatives has artificially enhanced the contribution of the electric and pressure forces. The second notable difference between the MO 1 and SO cases is the shape of the streamlines connecting the cathode to the downstream boundary of the experimental domain. The streamlines are more collimated (flowing along the axial, $z^$, direction) with a greater number flowing toward the anode, suggesting that a greater proportion of the electrons exiting the cathode connected to the anode.

Last, we show in Fig. 5(h) the electron streamlines and Mach number for the MO 2 keeper configuration. The spatial distribution of the Mach number is similar to the previous two conditions with an off-axis peak of $Me\u22484$ near the keeper and decreases toward the anode to $Me\u223c0.04$. Examining the streamlines in Fig. 5(h), we see that the electrons coming from the cathode propagate primarily in the axial direction toward the anode. This finding contrasts the other two cases where a significant number of cathode-born electron streamlines do not exit the axial boundary of the experimental domain on the anode side. Ultimately, these streamlines suggest a more efficient use of the current extracted from the cathode with the larger spread in the keeper hole-pattern. A major caveat to this conclusion, however, is that the uncertainty in the streamlines for this case is relatively large (a result of the stronger dependence of the streamlines on the pressure gradient, which carries a greater uncertainty since it is the product of the two measured quantities). Within experimental error, the electrons may still be streaming away from the anode as in the SO case. To further support our conclusion, we would require access to higher-fidelity plasma diagnostics such as incoherent Thomson scattering, which requires significant investment to develop, to directly measure the electron velocity vectors.^{32}

Examining these results together, we find that the MO cathodes tend to produce larger electron Mach numbers in the cathode plume than the SO case. Physically, this observation agrees with the notion that the electrons are moving more freely in the plume because of reduced IAT effective collisions found in Fig. 4.

### C. Summary of key experimental findings

The plasma potential gradients in the plume are reduced with greater MO hole pattern spread. The flatter potential surface is likely tied to the reduced anomalous collision frequency in the plume.

Near the keeper boundary, we find a greater electron temperature and plasma potential for the MO cases. These suggest greater heating within the device that may lead to greater sheath voltages at the keeper and enhance erosion.

The measurement of the plasma density in Fig. 4, along with a current continuity argument, suggested that the enhanced spreading of the plasma achieved by the MO cathodes might reduce the electron Mach number and that this mechanism might be responsible for the reduction in IAT growth and, therefore, anomalous collision frequency. However, we found in Fig. 5 that the Mach number tends to increase in the MO keeper configurations, contradicting this earlier explanation. Instead, we conclude that rather than affecting the growth of IAT, the MO keeper geometry may be changing the damping factors and, therefore, allowing electrons to more freely transit the plume to the anode. We discuss the plausibility of this mechanism to explain our findings in Sec. V.

## V. DISCUSSION

In this section, we discuss our experimental results with the goal of understanding how keeper pattern radius influences the transport of electrons. First, we overview the role of systematic uncertainty in our measurements to contextualize our conclusions. We then examine the contributions of electric and pressure forces on the flow of electrons and discuss the physical implication of our results.

### A. The role of systematic uncertainty

In this section, we discuss the role of systematic uncertainty in our analysis of the measurements and how they might influence our conclusions.

The largest source of systematic uncertainty in our experiment is our method for curve fitting the IAT spectra as is shown in Fig. 3. We note that all three cases show some form of peaked structure in their power spectra at frequencies greater than 1 MHz. In particular, the MO 2 keeper geometry exhibits the largest resonances that may or may not be related to the IAT propagating in the plasma and can contribute significantly to the total energy of the potential oscillations. While we argue that these resonances are most likely the result of spurious experimental noise (see Appendix A), we cannot unambiguously rule out the possibility that they may influence the electron transport in the plume. If we model these resonances as IAT, we can apply the expressions in Sec. II A and include these modes in the calculation of the anomalous collision frequency, yielding an upper bound on IAT-driven collision frequency. Globally, the trends in collision frequency remain the same between designs although less stark. We discuss the results of this calculation more thoroughly in Appendix A with Fig. 8.

This adjustment in collision frequency most notably propagates to the calculation of the electron Mach number (although note that it does not affect our prediction of the electron streamlines). The more detailed plots resulting from this analysis are discussed in Appendix A and shown in Fig. 10 but we provide a summary here. For the SO and MO 1 cases, we observed at most a 20% reduction in the derived Mach number when including resonance features. For the MO 2 case, we find that the Mach number is reduced by up to an order of magnitude in certain regions of the plasma such that the peak Mach number is $Me\u223c0.4$. This would affect our conclusions in Sec. IV C. If the Mach number is indeed being reduced by the MO geometry, then our earlier plasma spreading hypothesis established by the density measurements in Fig. 4 holds.

Whether the keeper design is affecting the growth or damping factors cannot be unambiguously resolved by our measurements; however, it is clear that the effect of IAT in the plume of the cathode is reduced. Both conclusions must be borne in mind as we continue our analysis of our experimental measurements in Secs. V B–V E under the assumption that the resonances are not contributing to the wave-driven collision frequency.

### B. Transport properties

Armed with the experimental measurements from Figs. 4(a)–4(l), we combine these quantities and compute gradients to determine the contributions of the electric force $FE=\u2212E$ and the pressure force $FP=\u2212\u2207(nTe)/n$ per unit charge. These values can then be used to infer the effective drag force $FD=\u2212meq(ue\u2212ui)\nu e$ on the electrons through Eq. (1). In Fig. 6, we show the results of these calculations. We organize the plots as follows: each row represents a keeper configuration, and each major column contains three subcolumns showing the electric, pressure, and drag forces for that configuration. The left major column shows the radial force contributions, while the right major column shows the axial contributions.

Figure 6 shows several trends with increasing pattern radius. Both the radial and axial electric forces tend to decrease in magnitude, while both radial and axial pressure forces increase in magnitude, eventually overtaking the electric force as the dominant contributor. Together, these trends result in a reduced effective drag force, making it easier for electrons to transit the plume with increasing pattern radius.

Recall that Ohm’s law [Eq. (6)] implies that the electron motion is determined by the electric and pressure forces. With this in mind, we examine their contributions and those of drag to electron transport in Figs. 6(a)–6(r). First, we note that the radial electric field dominates over the axial in the SO and MO 1 cases [Figs. 6(a), 6(d), 6(g), and 6(j)], tending to pull electrons out to higher radii as observed in the streamlines in Figs. 5(b) and 5(e). For the MO 2 case, the $r$ and $z$ components of the electric field are relatively equal in magnitude [see Figs. 6(m) and 6(p)] and negative indicating that this force is preventing the radially outward motion of electrons. For the pressure force, the magnitude of the axial and radial components of the SO [Figs. 6(b) and 6(d)] and MO 1 cases [Figs. 6(g) and 6(k)] are of the same order and small compared to the electric force. In comparison, we find that the axial pressure gradient is the largest force in the plume of the MO 2 design [Figs. 6(n) and 6(q)], which pushes the electrons toward the anode.

Synthesizing these results, we find that increasing the pattern radius suppresses electric fields, allowing the pressure force to dominate the transport of electrons. In the context of Fig. 5, we find that this transition tends to collimate the streamlines toward the anode. Physically, this finding has interesting implications. By changing the keeper hole-pattern radius, we have exchanged field-dominant transport for pressure-dominant transport. Thus, we can think of the MO 2 cathode as generating sufficient internal pressure to push electrons through the turbulence to the anode without requiring a significant electric field. In fact, an electric field is established to slow the electrons down [see Fig. 6(p)].

### C. Wave properties

Our measurements of collision frequency and electron Mach number suggest that the MO cathodes affect the growth of IAT by increasing the damping factors in Eq. (3). We can evaluate this hypothesis by examining the conservation of wave energy in the plume through Eq. (5) and solving for the growth rate, under the assumption of slowly drifting ions such that the group velocity is $vg=ui+cs\u223ccs$. Armed with the measured Mach number and the analytical expression for the growth rate in Eq. (3), we can solve for the damping rate, i.e., the rate at which energy is extracted from the waves through ion-neutral collisions and ion Landau damping. The result of this analysis is shown in Fig. 7, which shows a few key trends that help us understand the effect of the keeper geometry on the damping of IAT.

First, we find that the highest damping occurs near the cathode in all three cases. The two sources of damping for IAT are collisions with neutrals and ion Landau damping. We physically expect the neutral density, and, therefore, the ion-neutral collision frequency, to be highest near the cathode since the cathode is the source of neutral gas in the system. This expectation is consistent with the peaked structure of the damping rate found in Fig. 7. On the other hand, ion Landau damping depends strongly on the temperature ratio $Te/Ti$. Previous work in SO cathodes using laser induced fluorescence and Langmuir probes has shown that this ratio typically decreases as a function of distance from the cathode.^{9,34} For $Te/Ti\u223cO(1\u201310)$, the damping rate increases with decreasing $Te/Ti$. Armed with these qualitative trends, we conclude that the damping rate surfaces in Fig. 7 are more consistent with collision damping of the IAT; however, confirming this requires a measurement of the ion temperature and neutral density, which we were unable to make with our experimental apparatus.

Second, we find that the damping rate rises consistently with a greater MO hole pattern radius. This result supports the notion that the MO configurations are hindering the growth of IAT, likely through collisional damping of the waves and allowing electrons to flow more freely to the anode. Thus, a smaller voltage drop across the plume is required to conserve the discharge current. Interestingly, our findings point to the performance of the MO cathodes being driven by the same collisional mechanism leveraged by neutral gas injection techniques in high-current cathodes. MO cathodes may be achieving the same effect in two ways. First, the MO geometry reduces the gas conductance, increasing the neutral pressure within the device and possibly leading to slightly higher pressures at the keeper exit. Second, the MO configuration increases plasma–surface contact that may mediate recombination near the keeper. Confirming this mechanism would require more detailed neutral density measurements. In addition, we note that this neutral damping effect on the IAT reduces the ion energy and keeper erosion rates in standard cathodes.^{14} This phenomenon may also occur in the MO devices and could be detected either with LIF or a retarding potential analyzer; however, these diagnostics were unavailable for this study.

### D. Physical description of plume transport processes

In this section, we summarize the qualitative physical picture emerging for how a MO keeper affects the transport of electrons in the plume, subject to the experimental uncertainty and analysis limitations discussed previously.

For a fixed total extraction area, employing a multiple orifice keeper reduces the gas conductance, which increases the pressure within the cathode and near the keeper. This greater neutral density enhances the ion-neutral collision frequency that is critical for damping the growth of IAT as found in Fig. 8. As a consequence, the wave-driven collision frequency, shown in Figs. 4(d), 4(h), and 4(l), is suppressed in the MO cases. This reduction in collisionality (resistivity) has several repercussions on electron transport properties. Notably, the lower collision frequency results in a lower drag force, as shown in Fig. 6. Thus, to conserve the discharge current, a smaller electric field and thus total voltage drop across the plume is required. This effect likely contributes to the lower discharge voltages found when using certain MO keepers. In addition to the drag and electric forces, the MO configuration also affects the plasma pressure gradient such that it becomes the dominant force in the plume, dictating the evolution of the electron flow field. This effect likely stems from a greater internal neutral pressure that raises the ionization rate within the cathode and creates a larger plasma pressure. Together, these phenomena result in a more collimated electron flow field and larger electron drift velocity as found in Fig. 5.

### E. Implications for global discharge properties

The above picture suggests that we may be able to tune the balance of pressure, drag, and fields in the plume by changing the keeper geometry. Notably, pushing the discharge toward a pressure-drag balance (rather than a field-drag balance) could be advantageous for reducing the discharge voltage as exhibited by the flatter plasma potential topography of the MO 2 configuration. However, the apparent rise in overall potential [see Figs. 4(c), 4(g), and 4(k)] and temperature [see Figs. 4(b), 4(f), and 4(j)] is indicative of changes in the effective resistivity of the internal plasma. This is possibly brought on by a greater interaction of the plasma with the keeper as the pattern radius is increased. Although beyond the scope of the work, this finding points to plasma–surface interactions with the keeper being a possible fundamental limit for improving/optimizing the electrical efficiency of the cathode through the spreading of the plasma.

Recall from Sec. III and Fig. 1 that the SO configuration had the lowest discharge voltage, the MO 1 keeper had the highest, and the MO 2 was $\u223c3$ V above the SO case. In the context of the preceding discussion, we can potentially explain the relatively poor performance of the MO 1 configuration as the worst of both worlds. The central blockage of the design has raised the internal plasma potential through interaction with the surface but has not significantly lowered the gas conductance. Therefore, a greater neutral pressure and ionization rate were not achieved to suppress wave-driven drag or increase the plasma pressure gradient in the plume. Combined, this results in a larger electric field and discharge voltage. A comparison of the MO 1 and 2 cases suggests that there may be some optimal pattern radius to take advantage of the plasma spreading without excessively raising the internal plasma potential. Alternatively, a more complex pattern could potentially be used to achieve a similar result. With that said, a better understanding of how the keeper configuration affects the internal plasma potential structure through numerical simulation of these devices could eventually help optimize the keeper shape for electrical efficiency.

## VI. CONCLUSION

In conclusion, we experimentally investigated the influence of the keeper geometry on the transport processes that dictate the flow of electrons in the hollow cathode plume, assuming a fluid picture. Three keeper configurations (one single orifice and two multiple orifice) were studied where the radius of the hole-pattern was varied. For each case, the electron density, temperature, and plasma potential were inferred from electrostatic probe measurements. The results indicated that a larger pattern radius tends to increase the electron temperature, decrease the density, increase the plasma potential, and decrease in anomalous collision frequency. These properties were then combined with Ohm’s law to infer the electron (net current) flow field. The results showed that the electron Mach number tends to increase when using a multiple orifice keeper and that the streamlines are better directed toward the anode.

Following these results, we examined contributions of pressure, fields, and drag to the transport of electrons through an analysis of the generalized Ohm’s law for electron momentum. This more detailed analysis showed that a greater keeper pattern radius leads to a greater pressure force, a lower electric field, and lower drag. We then showed that the lower drag force was a result of increased damping of the IAT, consistent with greater neutral damping of the waves. Notably, this is the mechanism used to stabilize high-current cathodes with auxiliary external gas injectors. Combined, these effects shift the balance of forces on the electrons from an electric-drag force balance to a pressure-drag force balance thereby mitigating the potential rise in the plume. This advantageous feature, however, comes at a cost. The plasma potential within the keeper tends to rise with increasing pattern radius, raising the overall discharge voltage. Our findings suggest that there may be an optimal pattern radius to take advantage of the neutral flow effect of a multiple orifice keeper while mitigating the losses that are thought to be the result of plasma–surface interactions with the keeper.

## ACKNOWLEDGMENTS

This work was supported by U.S. Naval Research Laboratory Base Program (No. WU 6B86).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: DETAILED ANALYSIS OF SYSTEMATIC UNCERTAINTY IN IAT COLLISION FREQUENCY CALCULATION

The high frequency resonances in the Fourier transform of the high-speed ion saturation probe measurement (see Fig. 3) were found at every location in the plasma plume. We eliminated these resonances by assuming they are some external noise introduced into the measurement and curve fitting the underlying IAT spectrum; however, more generally, these resonances could be:

IAT waves,

ionization instabilities, in the plume or within the cathode, and

azimuthal asymmetry driven instabilities.

#### 1. IAT related phenomenon

If we assume these resonances are part of the IAT spectrum, then they may be related to some nonlinear wave phenomenon like wave-wave or wave-particle coupling. In that case, the linear theory of IAT growth and wave energy conservation would not strictly apply. We can estimate the IAT contribution to electron collision frequency by including all the modes in the spectrum, i.e., summing over the raw spectrum from 200 kHz to 7 MHz and calculating an upper bound. This will increase its overall value of the collision frequency. Figure 8 shows the estimated wave-driven collision frequency using this upper bound technique. Comparing with the results in Fig. 4, we find that the SO and MO1 cases are increased by no more than 20% and 5%, respectively. The MO2 case, where the resonances are particularly strong increases the collision frequency by about an order of magnitude. Although there is this significant increase in collision frequency for the MO2, we still find that it is generally reduced for the multiple orifice cases.

The collision frequency in turn feeds into the electron Mach number calculation and ultimately into the damping. These results are shown in Figs. 9 and 10, respectively. In these plots, we see the increase in collision frequency for the MO 2 case has reduced the electron Mach number and damping rate, compared to Figs. 5 and 7. Note that the electron streamlines remain unchanged, because they strictly depend on the electric and pressure forces, which were not affected.

The major conclusions of the work are still supported when using this alternative method. The MO keeper

reduces the wave-driven collision frequency,

reduces the drag and electric forces, and

increases the pressure force.

However, the more detailed conclusions about the connection of increased neutral pressure to the improved performance become more tenuous. Rather than the MO keeper increasing the collisional damping rate of the IAT and leading to the above conclusions, it appears that the greatest effect is in electron Mach number, particularly in the MO 2 case. This could potentially be explained by a spreading effect due to the keeper geometry. The increased spread in the plasma resulting from the greater pattern radius in MO 2 design reduces the current density by the continuity argument examined earlier in Sec. IV A 1.

#### 2. Ionization related phenomena

If the resonances are ionization phenomena, then we cannot attribute them to the IAT collision frequency and removing the resonances by curve fitting to the underlying broadband waves is likely the most accurate method for calculating the IAT contribution to the electron collision frequency. To investigate the possibility that these resonances are ionization related, we develop a criterion for the minimum neutral pressure for our findings to be consistent with this phenomenon.

From our measurements of electron temperature and density in the plume, we can estimate the ionization rate coefficient by integrating the ionization cross section over a Maxwellian energy distribution function.^{35} If we assume the frequency of an ionization oscillation scales as $f\u223c\nu iz\u2261nnkiz$, where $\nu iz$ is the ionization frequency and $kiz$ is the ionization rate coefficient, then we can estimate the critical neutral gas pressure required for this scaling in oscillation frequency,

where $Tn$ is the neutral temperature, which is assumed to be the wall temperature (300 K).

Before evaluating Eq. (A1), it is important to establish bounds on realistic neutral pressures in the plume, although we note that, in general, this quantity is not well known. From upstream measurements with a capacitance manometer of our system, the internal pressure to the cathode is typically $O$(5 Torr). Experimental measurements in a similarly configured SO cathode testbed showed that the local neutral pressure near the exit is approximately 0.0001 Torr at 20 sccm of Xe,^{36} although we note that their vacuum facility has a greater pumping speed than the one used in our experiments. As a reasonable upper bound, we do not expect the neutral pressure to exceed this previous measurement by more than an order of magnitude (0.001 Torr) over our domain.

The critical pressure for all three keeper geometries is plotted in Fig. 11. For all three conditions, we find that, for the majority of the plume, the minimum pressure required for the resonances to be ionization related is several orders of magnitude too high. The only exception is very close to the MO 2 cathode where this pressure is maybe exceeded by a single order of magnitude. From these plots, we conclude that the neutral pressure in the plume is likely insufficient to support the generation of an ionization-type instability.

While the internal plasma properties were not accessible to our experimental apparatus, we note that previous measurements in similarly configured SO cathodes operating on Xe showed that ionization instabilities in the plume are not typically observed inside the device because of the high gas and plasma pressure.^{37} With the MO cathodes, we expect higher gas and plasma pressure within the device. Thus, the increasing strength of the resonances in the MO 2 case, which should have the highest internal pressure of all our test conditions, is not consistent with the typical behavior of ionization instabilities in cathodes.

#### 3. Azimuthal asymmetry

The multiple orifice design introduces an azimuthal non-uniformity in the cathode plume. The potential energy stored in the azimuthal gradients may be capable of driving plasma instabilities. We cannot rule out the possibility that these modes are related to azimuthal nonuniformity; however, our measurements suggest that this effect is likely weak in our experimental domain for the following reasons. First, our probe (which is physically smaller than the hole size) did not show any evidence of radial plasma jets emanating from the keeper as a function of radius. Physically, this is likely related to diffusion smoothing out density gradients in the plasma. It follows that azimuthal non-uniformity is also likely weak in our plasma. Thus, we would expect these modes to start damping with increasing distance from the cathode as the density gradient diminishes. Experimentally, we find that these resonances exist throughout the plume with increasing magnitude. Last, we note that some form of high frequency resonance was found in all three conditions, suggesting that this effect may not be strongly linked to the geometry of the keeper.

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