Phys. Plasmas 27, 022311 (2020); https://doi.org/10.1063/1.5130680 27, 022311

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Non-classical electron transport in thecathode plume of a Hall effect thruster Cite as: Phys. Plasmas 27, 022311 (2020); https://doi.org/10.1063/1.5130680Submitted: 06 October 2019 . Accepted: 13 January 2020 . Published Online: 18 February 2020

Benjamin A. Jorns , Sarah E. Cusson , Zachariah Brown , and Ethan Dale

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Non-classical electron transport in the cathodeplume of a Hall effect thruster

Cite as: Phys. Plasmas 27, 022311 (2020); doi: 10.1063/1.5130680Submitted: 6 October 2019 . Accepted: 13 January 2020 .Published Online: 18 February 2020

Benjamin A. Jorns,a) Sarah E. Cusson, Zachariah Brown, and Ethan Dale

AFFILIATIONS

University of Michigan, Department of Aerospace Engineering, Ann Arbor, Michigan 48105, USA

a)Author to whom correspondence should be addressed: bjorns@umich.edu

ABSTRACT

An experimental investigation is presented into the wave-driven electron transport in the near-field plume of a hollow cathode operating in a300V, 4.5 kW magnetically shielded Hall thruster. Correlational analysis of probe measurements in the cathode plume shows two types ofelectrostatic waves: ion acoustic turbulence propagating along the applied longitudinal magnetic field at frequencies from 500 to 1250 kHzand coherent, azimuthal anti-drift waves with a fundamental frequency of 95 kHz and mode numbers from m ¼ 1� 4. A quasilinear analysisis applied to quantify the impact of each wave on the electron transport in the near-field plume. It is found that the ion acoustic modes giverise to an enhanced effective collision frequency in the direction parallel to the applied magnetic field that exceeds the classical collision fre-quency by two orders of magnitude. The anti-drift waves promote an anisotropic collision frequency that depends on the direction of theelectron drift. While the enhanced collision frequency from these waves is comparable to the classical frequency for motion along the appliedmagnetic field, the effective collision frequency in the azimuthal direction exceeds the classical by three orders of magnitude. These resultsare discussed in the context of their impact on the steady-state plasma gradients in the near-field cathode plume. Closure models for incorpo-rating the effective collision frequencies from both types of waves into fluid-based codes are derived and shown to agree with the measuredwave-driven collision frequencies.

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I. INTRODUCTION

The efficient operation of the Hall effect thruster, a widely usedtype of plasma propulsion, strongly depends on the ability of its elec-tron source, the hollow cathode, to couple electrically to the thruster’splasma discharge.1 These thermionically emitting devices enable theelectrons to both ionize the thruster’s propellant and neutralize thepositively charged ion exhaust after it has been accelerated out ofthe thruster. If changes to the cathode operation or environment impedethe ability of these electrons to connect to the thruster’s main discharge,performance and stability can suffer. Indeed, it has been shown thatvarying key cathode parameters such as flow rate, temperature, orienta-tion,2–10 and test environment11–19 can negatively impact the cathodecoupling and by extension thruster operation. Although these effectshave been well-documented, there are aspects of the underlying physicalprocesses governing the cathode coupling that remain poorly under-stood. This lack of understanding continues to present an obstacle forthe modeling, design, and optimization of Hall thrusters.

The major challenge in building a first-principles description ofthe cathode coupling stems from the existence of non-classical pro-cesses in the plumes of these devices. Most notably, the electron

resistivity is orders of magnitude higher than can be explained by fric-tion from ordinary collisions.20–28 It thus is not possible to model orpredict the local plasma environment using a classical plasmaapproach. In an effort to address this shortcoming, there has been aconcerted effort to identify potential mechanisms that may drive thisenhanced resistivity and apply these to our understanding of hollowcathode operation. These studies have shown that the onset of propa-gating electrostatic waves in the form of current-driven ion acousticturbulence (IAT) is the dominant contributor to an effective,“anomalous” collision frequency on the electrons.21–23 Subsequentefforts have built on these findings to incorporate non-classical, wave-driven effects self-consistently into numerical results. They ultimatelyhave yielded improved predictions both for the cathode plume distri-bution and dynamical evolution.28–31

A major caveat for these previous studies—both numerical andexperimental—is that they have focused primarily on stand-alone con-figurations. These are experimental domains in which the cathode istested in a dedicated facility and not operated as part of a thruster dis-charge. These stand-alone configurations have the advantages of easieraccessibility and controllability to facilitate basic plasma studies, but

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there are substantial differences in their test environments and theenvironment of a hollow cathode operating with an actual Hallthruster. These differences may fundamentally change the electrondynamics in these systems. For example, it is not clear if IAT will havethe same degree of influence on hollow cathode plumes operating inthrusters. A potential reason for this is that cathodes in stand-aloneconfigurations have only one source of neutral gas (through thecathode bore), while cathodes in thrusters are subject to neutral influxfrom the surrounding main discharge.32 Since neutral gas has beenshown to be a major contributor to the damping of IAT,20,25 theenhanced resistivity from turbulence may be curtailed or even non-existent when the cathode is operated in conjunction with a thruster.As another potential discrepancy, while most work on stand-alonedevices to date has been for unmagnetized plumes, hollow cathodes inHall thrusters are subject to strong magnetic fields. As a result, hollowcathode plumes in thrusters have been shown to exhibit gradient-driven, rotational (“anti-drift”) instabilities not present in unmagne-tized devices.33,34 If these modes contribute to electron transport, theycould fundamentally change the non-classical effects governing theelectron dynamics.

In light of the importance of cathode coupling to thruster opera-tion and these pressing questions about the nature of the electrondynamics, the goal of this paper is to investigate experimentally wave-driven, non-classical resistivity when these devices are implemented ina thruster environment. We focus on examining the role of the twoclasses of oscillations that we anticipate should exist in the plume: rota-tional anti-drift waves33 and IAT.23 To this end, this paper is organizedin the following way. In Sec. II, we employ a quasilinear approxima-tion to relate wave amplitude and dispersion of electrostatic waves in acathode plume to an enhanced electron collision frequency. We thendescribe the dispersion relations of the anti-drift wave and ion acousticturbulence and relate these to effective collision frequencies. In Sec. III,we describe the experimental setup and diagnostics employed to char-acterize the wave and plasma properties in the cathode couplingregion. In Sec. IV, we present results for measurements of the waveproperties, calculate an effective collision frequency from the twomodes, and compare these results to the classical electron collision fre-quency. In Sec. V, we discuss our results in the context of previousstudies on cathode coupling dynamics and motivate simple closuremodels for the anomalous collision frequency.

II. WAVE DISPERSION AND TRANSPORTCOEFFICIENTS

The goal of this investigation is to evaluate the role of plasmawaves in driving non-classical transport in the cathode plume of a Hallthruster. In order to establish a metric for assessing these effects, we pre-sent in the following a theoretical framework for relating a key propertygoverning the transport of the electrons, their resistive drag, to the prop-erties of the plasma oscillations (frequency, amplitude, etc.). We thenapply linear dispersion theory to derive expressions for wave-driven“collision frequencies” associated with this enhanced resistive drag.

Figure 1 shows a cylindrical coordinate system as applied to thehollow cathode and Hall thruster we employed in this investigation(discussed further in Sec. III). The Hall thruster is a crossed field(E�B), axisymmetric device in which the main plasma discharge isconfined to an annular channel. The hollow cathode is mounted alongthe thruster centerline and concentric with the discharge channel. As

indicated in Fig. 1, the axial direction, z , is along thruster centerlinewith the origin coincident with the cathode exit plane. For thedomain of interest, the cathode plume is assumed to be subject to astrong and approximately uniform magnetic field oriented in theaxial direction, ~B0 ¼ B0z : Electric fields and pressure gradientscan be both axial and across the magnetic field in the radial direc-tion, r . This crossed-field configuration can give rise to drifts in theh direction.

We adopt the approach of Davidson and Krall35 to relate thewave properties to electron drag in the coordinate frame shown inFig. 1. Following this work, if we assume the waves of interest areelectrostatic (as is consistent with the anti-drift and ion acousticmodes), the Boltzmann equation for the electron distributionfunction can be expanded to second order and averaged over thefast time scale of wave propagation. This yields an effectiveadditional force density on the electrons that arises from wavepropagation,

~RAN ¼ �qhdned~E::i; (1)

where q denotes fundamental charge, dne is the perturbation to theelectron density introduced by the propagating waves, d~E is the per-turbation to the electric field associated with wave propagation, andthe brackets h::i denote an average over the characteristic time scale ofthe plasma waves. Equation (1) shows that if the density and electricfield perturbations associated with a wave have components that are inphase, they can positively couple to act as an additional forcing termon the electrons. This forcing term in turn scales directly with theamplitude of the waves.

Expanding on the interpretation of the wave propagation as adrag on the electrons, we can define effective wave-driven collisionfrequencies,

�qhdned~E::i ¼ �ANðzÞneme~ueðzÞ þ �ANðhÞneme~ueðhÞ

þ�ANðrÞneme~ueðrÞ; (2)

FIG. 1. Image of the H9 Hall effect thruster with a highlighted region denoting thedomain that was experimentally measured. The diagram shows a zoomed in viewof this region along with the cylindrical coordinate convention. The z axis is alignedwith the thruster centerline.

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where ~ueðÞ and �ANðÞ denote the electron drift and enhanced collisionfrequency in each direction, me is the electron mass, and ne is the time-averaged local electron density. These expressions reflect the anisotropiceffect propagating waves can have on the electron transport in the cath-ode plume. Physically, an enhanced collision frequency along magneticfield lines, �ANðzÞ, will act as an impedance on the electron motion givingrise to higher potential gradients to maintain a set discharge current. Thecollision frequency in the azimuthal direction, �ANðhÞ, however, pro-motes radial, cross field current. This stems from the de-magnetizingeffect of the drag introduced by the waves. While collisions in the radialdirection, �ANðrÞ, also can promote cross field current, this contributionscales inversely with the classical Hall parameter (ratio of cyclotron toclassical collision frequency), which is on the order of 100–1000 for theplasma of interest. It is, therefore, assumed to be negligible compared tothe cross field current induced by the azimuthal drag. For this investiga-tion then, we confine our discussion to the effective collision frequenciesacting in the azimuthal and axial directions.

To express the collision frequencies from Eq. (2) in terms ofwave properties, we represent the electric field and density perturba-tions for the propagating waves as a Fourier composition in the cylin-drical coordinates from Fig. 1:

dne ¼Xkz ;m

neðkz ;mÞei kzzþmh�xtð Þ þ c:c:;

d~E ¼ �Xkz ;m

@/ðkz ;mÞ@r

r þ imr

/ðkz ;mÞh þ ikz/ðkz ;mÞz

� �

� ei kzzþmh�xtð Þ þ c:c:;

(3)

where c:c: denotes the complex conjugate, kz denotes the axial compo-nent of wavenumber, m is the mode number in the azimuthal direc-tion, x is the frequency of each mode, t denotes time, and neðkz ;mÞ and/ðkz ;mÞ are the Fourier amplitudes of the density and the potentialoscillations, respectively. Armed with Eq. (3), we thus can find expres-sions for the collision frequencies from Eq. (2):

�ANðhÞ ¼q

ueðhÞnemeIm

Xkz ;m

mr

� �neðkz ;mÞ/ðkz ;mÞ

" #;

�ANðzÞ ¼q

ueðzÞnemeIm

Xkz ;m

kzneðkz ;mÞ/ðkz ;mÞ� �

;

(4)

where Im½� denotes the imaginary component.Equation (4) provides a prescription for evaluating the non-

classical, wave-driven effects of the two modes of interest in thecathode plume—provided we can measure or estimate the perturbedquantities associated with the wave propagation. To this end, we con-sider the plasma properties and dispersion of both the acoustic waveand the anti-drift wave. In our treatment of these modes, we assume inthe following—as is consistent with the near-field plume of our experi-mental setup (Sec. III)—that the magnetic field is sufficiently strong tomagnetize the electrons but weak enough that ion dynamics are inde-pendent of the field.

A. Ion acoustic waves

The ion acoustic wave is an electrostatic mode that has beenobserved extensively in the plumes of unmagnetized hollow cathodes.It is driven unstable in these devices in the direction of strongest

electron drift. In magnetized plasmas, these waves can appear in thecross field direction provided there is a sufficiently high electron crossfield drift (cf. Ref. 36). However, in the thruster cathode plasma, thehighest drift is along magnetic fields where there is the lowest resis-tance path. Given the azimuthal symmetry of the system shown in Fig.1, these modes thus preferentially will propagate in the z directionwith m¼ 0. Similarly, consistent with the work from Ref. 23, weassume the wavelengths of the excited modes are long compared tothe Debye length, kkde � 1, and that the plasma is not in equilibriumsuch that the electron temperature is not equal to the ion temperature,Te > Ti. We also make the assumption that the sound speed is negligi-ble compared to the electron drift cs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiTe=mi

p� ueðzÞ. Together,

these simplifications allow us to find expressions for the dispersionand the perturbed potential in terms of electron density,29

x ¼ cs þ uiðzÞð Þkz;

q/ðkz ;mÞTe

¼neðkz ;mÞne

1þ i

ffiffiffip2

rueðzÞvte

!;

(5)

where uiðzÞ is the ion drift speed in the axial direction in the laboratoryframe of reference and vte is the electron thermal speed.

Equation (5) illustrates two key expected properties of the ionacoustic wave in the cathode plume. The first is that it has an acousticdispersion in the ion frame of reference—the phase velocity is the ionsound speed. In the laboratory frame, the wave frequency is Dopplershifted by the ion drift in the axial direction. The second property isthat there is a phase delay between the density and potential fluctua-tions introduced by inverse electron Landau damping [the imaginarycomponent in the second line of Eq. (5)]. As noted in Ref. 37, otherparameters can contribute to this phase delay including classical elec-tron collisions, but for a plasma characterized by a strong electrondrift, these other effects generally are negligible compared to the phaseintroduced by this inverse damping. With these results in mind, wecan see from Eqs. (4) and (5) that the IAT will only contribute to thelongitudinal anomalous collision frequency,

�ANðzÞðIATÞ ¼ffiffiffip2

r Xkz

neðkz ;mÞne

� �2

kzvte

" #; (6)

where we implicitly have assumed the wave vector and electron driftare parallel. The physical implication from this result is that the anom-alous collision frequency scales both with the amplitude of the fluctua-tions and the energy distribution across wavenumber.

B. Anti-drift wave

The anti-drift wave is a rotational oscillation that propagates pri-marily in the azimuthal direction with small but finite components inthe axial direction, parallel to the applied magnetic field. It derives itsenergy from both gradient-driven drifts in the azimuthal direction aswell as the electron longitudinal drifts in the near-field magnetizedcathode plume. This energy transfer can result in an effective dragand, therefore, enhanced collisionality on this species in both direc-tions. The key assumptions underlying our treatment of this modeinclude Xi < �e < Xe where Xi;e denotes the species gyrofrequencies;an approximately constant electron temperature and negligible iontemperature, Te > Ti; a wave frequency that satisfies �i < x < �e

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where �i; �e denote the classical ion and electron collision frequencies;and a high phase velocity in the longitudinal direction,x=kz � cs:

The anti-drift waves that were observed in Refs. 33 and 34 in theplume of a Hall thruster hollow cathode exhibited a m¼ 1, bulk prop-agation in the azimuthal direction in the plasma. In the former study,a non-local dispersion relation analysis in cylindrical coordinates wasemployed to compare the measured wave properties to experimentalresults. For this work where our goal is to examine the potentialimpact of these oscillations on the electron dynamics, we forgo a globalanalysis in favor of a simpler plane wave, Cartesian formulation. Thetransformation between cylindrical and Cartesian coordinates can beapproximated with the relation kh ! m=r where kh is the Cartesianrepresentation of the wavenumber in the azimuthal direction,m is themode number, and r is the radial coordinate. Adopting this conven-tion and following the approach of Frias et al.,38 the real dispersionand electron density for the anti-drift wave can be written as

0 ¼ �

mR

� �v� þ i�pl

x� mR

� �vE�B � kzueðzÞ þ i�pl

þ Te

mi

k2

ðx�~k �~uiÞ2

!

q/eðkz ;mÞTe

¼neðkz ;mÞne

x� mr

� �vE�B � kzueðzÞ þ i�pl

mr

� �v� þ i�pl

0BBB@

1CCCA; (7)

where we have introduced the diamagnetic drift v� ¼ Te=ðqB0Þn0eðrÞ=ne, the electric field driven drift vE�B ¼ Er=B0, and a colli-sional term, �pl ¼ k2zTe=me=�e. We note that in a departure from thederivation of Frias et al.,38 we have neglected magnetic field gradient-driven drifts and introduced (following Ref. 33) the contributionsfrom electron motion parallel to the applied magnetic field. Solutionsto the first expression in Eq. (7) yield the relationship between fre-quency and wavenumber of the mode. The second expression showsthat the mechanism for introducing phase delay between the potentialand density fluctuations stems from electron collisions through �pl.This promotes the exchange of energy of the wave and the electrondrifts.

Since the anti-drift wave propagates longitudinally and azimuthally,we anticipate there may be contributions to the anomalous collision fre-quency in both of these directions. From Eqs. (4) and (7), we find

�ANðhÞðADÞ ¼ �eXkz ;m

neðkz ;mÞne

� �2 mkzr

� �2

1þkzrueðzÞmueðhÞ

!24

35;

�ANðzÞðADÞ ¼ �eXkz ;m

neðkz ;mÞne

� �2

1þmueðhÞrkzueðzÞ

� �� �" #:

(8)

In order to arrive at these forms, we have made the assumption that

�plr

mv�¼ Xe

�e

� �kzrm

kzLnð Þ� �

� 1; (9)

where Ln ¼ jne=n0eðrÞj denotes the characteristic length scale of thedensity gradient. We have found consistently that this criterion issatisfied for the hollow cathode plasma we investigate in this study(Sec. IV). Physically, the results in Eq. (8) show that with more energy

present in the oscillation (greater the amplitude), the greater the rate atwhich energy is extracted from the electron drift. Similarly, the lineardependence on the electron collision frequency illustrates that it is thisclassical drag that gives rise to a phase delay between potential anddensity oscillations necessary to promote the growth of the waves.

In summary, we have shown in this section from a formulationfor a simplified geometry that the oscillations we anticipate in the hol-low cathode plume can impact the electron transport [Eqs. (6) and(8)]. The onset of ion acoustic waves in the longitudinal direction willlead to a higher effective collision frequency along magnetic field lines.The growth of the anti-drift waves similarly will contribute to higherresistivity along field lines, although they also will promote crossedfield transport through an enhanced collision frequency in the azi-muthal direction. The degree to which these two modes influencetransport will depend on their amplitudes and dispersive properties.The remainder of this work is devoted to measuring these propertiesexperimentally.

III. EXPERIMENTAL SETUP

The test article we employed for this investigation was the H9(Fig. 1), a 9-kW class device designed by the Jet Propulsion Laboratoryin collaboration with the University of Michigan and the Air ForceResearch Laboratory.39,40 Although not a flight unit, the H9 representsthe state of the art in modern flight thrusters41 intended for deep spaceexploration. Key features include a centrally mounted hollow cathode,graphite covers on the magnetic pole pieces, and a magneticallyshielded discharge. Magnetic shielding is a design technique in whichthe magnetic field geometry in the annular discharge is configured toreduce contact of energetic ions with the chamber walls and thereforeprolong thruster life.42,43 The cathode for the H9 employs a LaB6 emit-ter and a graphite keeper that can sustain currents up to 60A. Itsplume is subject to a strong axial magnetic field, consistent with thecoordinate system defined in Sec. II. Electrically, following Ref. 44, thecathode body is tied to the thruster main body. All electrical potentialsin the following work are referenced with respect to this potential.

We performed our experimental investigation of the cathodeplume by operating the H9 on xenon at 4.5 kW power, a dischargevoltage of 300V, and a flow fraction through the cathode of 7%. This isone of the nominal operating conditions of the H9 with measured per-formance metrics that include 294mN thrust, 1906 s specific impulse,and 60.4% efficiency.40 All testing was performed in the Large VacuumTest Facility (LVTF), a 6 m� 9 m cryogenically pumped chamberlocated at the University of Michigan. The base pressure of this systemwas 10�7 Torr-xenon, but during the 4.5 kW operation, the facilitypressure was maintained at 5� 10�6 Torr-xenon as measured in theplane of the thruster.

We employed two sets of diagnostics for this study: a cylindricalLangmuir probe to characterize the background plasma propertiesand an array of ion saturation probes to measure the plasma transi-ents. Both sets of probes were mounted on high-speed translationstages and inserted axially with a reciprocating action into the near-field of the hollow cathode. The region of interrogation (shown graph-ically in Fig. 1) consisted of a 20mm� 20mm rectangle in r-z located25mm downstream of the cathode exit. The measurement resolutionwas 2.5mm in the axial direction and 5mm in the radial.

The Langmuir probe consisted of a cylindrical tungsten tip withan exposed surface area of Ap ¼ 1mm2. During measurements, we

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biased this probe to fixed voltage and monitored current as a functionof probe position as it was inserted into the cathode plume. By varyingthe applied voltage from -25V (ion saturation with respect to thrusterbody) to 25V (in excess of local plasma potential) in 1V incrementsfor each insertion, we constructed spatially resolved current-voltageLangmuir traces. From these measurements, we in turn inferred sev-eral key properties including the plasma density, electron temperature,and plasma potential.

The wave probe array consisted of three cylindrical tungsten tipseach 3.8mm in length and with a 0.38mm radius. The probes werespaced apart such that two were aligned in the axial direction but witha 5.5mm separation and two were aligned in the azimuthal directionwith a 5.1mm gap. During measurements, we biased each probe to�36V to ion saturation and measured the fluctuations in this currentat a rate of 10MHz. The relative magnitude of these fluctuations ascompared to the local mean value of ion saturation served as a proxyfor relative fluctuations in ion density. In our analysis, we performedboth Fourier transforms and cross-correlations between the probe tipsignals to estimate the frequency, amplitude, and local wavenumbersin the axial and azimuthal directions.

IV. RESULTS

We present in this section measurements of both the time-averaged plasma properties and the wave properties in the near-fieldof the hollow cathode. We then use these measurements combinedwith the expressions from Sec. II to evaluate the effective electron colli-sion frequencies from the measured waves.

A. Background plasma properties

Figure 2 shows the time-average electron temperature, plasmapotential, and plasma density in the plume of the thruster hollow cath-ode at the 300V and 4.5 kW condition. We determined the electronproperties through an analysis of the current-voltage Langmuir probetraces generated per the procedure outlined in Sec. III. We estimatedthe plasma potential, /, with the first-derivative technique applied tothese traces, and we calculated the electron temperature, Te, from theslope of the log-linear plot of the IV trace.45 The plasma density, ne,was inferred from the average measured ion saturation current, isat, byapplying the thin sheath approximation—valid for our typical high

density plasma conditions: ne ¼ isat= q0:61Ap

ffiffiffiffiffiffiffiffiffiffiffiffiTe=mi

p� �. In order to

determine the uncertainty for these calculated parameters, we per-formed a bootstrap analysis for each IV trace, re-sampling the datasetand repeating the estimates of the parameters. This yielded typicaluncertainties of 20% in both the electron temperature and plasmapotential and 50% in the plasma density.

The trends in Fig. 2 are consistent with previous work reportedfor centrally-mounted cathodes in Hall thrusters.46,47 The monotonicdecrease in plasma density with position in Fig. 2(a) results from theexpansion of the plasma as it is emitted from the cathode. The nearalignment of the contours of constant density in the axial direction forr > 5mm is an indication of the anisotropy introduced to the plasmaexpansion by the strong confinement from the local axial magneticfield (250G). The approximately constant electron temperature(Te ¼ 4:661 eV) in Fig. 2(b) suggests a high degree of thermal con-ductivity and subsequent isothermality in this area. The existence ofgradients in the potential well shown in Fig. 2(c) is indicative of electric

fields pointing axially toward the cathode that drive electron currentdownstream.

The gradients in potential and density exhibited in Fig. 2 both areconducive to the onset of the waves discussed in Sec. II. The distinctivepotential well can facilitate a strong, axial electron drift, a pre-requisitefor current-driven IAT. Similarly, the marked radial gradients inpotential and density in the region near the cathode centerline canpromote azimuthal electron drifts, one of the requirements for theonset of the anti-drift mode (Sec. II). With this in mind, we next turnto investigating if there is evidence that these waves actually exist inthis region.

FIG. 2. Measurements of time-averaged properties including plasma density (a),electron temperature (b), and plasma potential (c) in the hollow cathode plume forthe thruster operating at 300 V and 4.5 kW. Measurements are referenced withrespect to the exit plane of the cathode and the centerline of the thruster.

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B. Wave measurements

We show in Fig. 3 power spectra and dispersion relations mea-sured in the axial and azimuthal directions at two radial positions,r¼ 0 and r ¼ 20mm, located z ¼ 25mm from the cathode exit. Thepower spectra [Fig. 3(a)] have a frequency resolution of 5 kHz andwere generated from a Fourier analysis of the relative fluctuations inthe ion saturation current,~isat=isat . They thus are a direct indication ofthe magnitude of the density fluctuations in this region. The dispersionrelation [Figs. 3(b)–3(d)] plots were created with a Beall analysis48

applied to the measurements of ion saturation current from the probetips in the array described in Sec. III. They are histograms that repre-sent the probability that a given frequency at which the density oscil-lates is correlated with a wavenumber in the direction defined by theline connecting the set of probes. The reported results consist of theaverage of 1000 correlations applied to the probe signals and have amaximum measurable wavenumber kmax ¼ p=Dx, where Dx denotes

the distance between probes in the array. The frequency resolution forboth figures is 5 kHz, and we have converted the azimuthal compo-nent measurements to mode number, m, with the relation kh

¼ m=r (Sec. II). In all of the histogram plots, we have normalized theintensity to unity but adjusted the color scale to illustrate the salientwave features.

The results in Fig. 3 show three non-trivial trends in frequencyand wavenumber. The first, as can be seen from the power spectrumplot in Fig. 3(a), is a low frequency peak at 10 kHz. This is the breath-ing mode, which originates in the thruster channel (cf. Ref. 33). Thisoscillation is pervasive in Hall thruster discharges, but as can be seenfrom this plot, it is not dominant in the cathode plume. The secondoscillation is characterized by a series of discrete peaks with a funda-mental frequency at 95 kHz and three harmonics. The third notablefeature exhibited by the power spectrum is a broadband structurebetween 500 kHz and 1250 kHz that has an inverse decay in amplitude

FIG. 3. Power spectrum (a) and dispersion relation (b-d) measurements at an axial location 25 mm downstream of the cathode exit. The dispersion measurements in (b) and(c) were performed at the radial location r ¼ 20mm and correspond to the azimuthal (b) and axial (c) wavenumbers. The dispersion in (d) was measured in the axial directionon cathode centerline. The color scale of each dispersion plot has been adjusted to better illustrate the correlations.

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with frequency. Along centerline (r¼ 0), this broad spectrum is punc-tuated by a small peak at 1200 kHz that does not persist at largerradii.

We can map these features of the power spectra to wave proper-ties in the cathode plume with the dispersion plots. We first focus onthe lower frequency components in Figs. 3(b) and 3(c) where we showmeasurements from 0 to 500 kHz. The peak at 10 kHz, which we haveidentified with the thruster breathing mode, has no detectable disper-sion in the azimuthal direction and a small but finite component ofwavenumber in the axial direction. This is consistent with the fact thatthe breathing mode is believed to be primarily longitudinal. We alsocan see from Fig. 3(b) that the discrete peaks starting at 95 kHz in thepower spectrum are aligned with mode order. The fundamental corre-sponds to an m¼ 1 mode, the harmonic is aligned with m¼ 2, etc.,until the mode amplitude is no longer detectable at 500kHz. Thenon-local behavior suggests that these oscillations represent bulk rota-tions of the plasma—azimuthally propagating eigenmodes—that areconsistent with the anti-drift wave previously reported in Ref. 33.These rotational modes also exhibit axial wavenumbers [Fig. 3(d)]where the magnitude of the component in the axial directionincreases approximately linearly with the mode order. The smallmagnitude of these wavenumbers indicates that the wavelengths ofthese axial components (>10 cm) are comparable to the character-istic axial length of the cathode plume. We note here as well thatalthough we only have shown dispersion relations at two radiallocations, we found both quantitatively and qualitatively that thisrotational dispersion persisted throughout the plume. Most nota-bly, by employing the transformation kh ¼ m=r (Sec. II), we foundat different radii the same correlation between mode number andfrequency as shown in Fig. 3(b).

We next turn to examining the nature of the higher frequencyoscillations in the power spectrum from Fig. 3(a). To this end, weshow in Fig. 3(d) the dispersion relation measured in the axial direc-tion on cathode centerline. Since the higher frequency, broadbandoscillations (500 kHz–1250 kHz) are an order of magnitude lower inamplitude than the rotating modes [Fig. 3(a)], we have adjusted thecolor scale on this plot to highlight the dispersive nature of the highfrequency content. As a result, the harmonic structure for the0–500 kHz has been obscured. We label this saturated region on theplot as anti-drift waves and draw a dotted line to remind the reader ofthe weak linear trend exhibited in Fig. 3(c). We also have adjusted thewavenumber scale and data depicted in Fig. 3(d) in order to correctfor probe aliasing. This was done following the procedure outlined inRef. 23. As was pointed out in this previous work, if the wavelengths ofthe propagating waves are smaller than the distance between theprobes, they will appear in dispersion analyses as unphysical wave-numbers that have been shifted to negative values by �2p=Dx. This isevidenced by the negative wavenumbers from�500 to 0 rad=m in Fig.3(d). This aliasing can be corrected by shifting the wavenumberdependence of the measured dispersion by one full phase (2p=Dx)and concatenating with the original dataset.

From this result, we immediately can see that the dispersivenature of the higher frequency content is different than the lower fre-quency rotational modes. In particular, there is an evident linear rela-tionship between frequency and wavenumber (emphasized with adrawn line and labeled as “IAT”) that has a shallower slope than therotational modes. The small amplitude high-frequency peak exhibited

in Fig. 3(a) at 1200 kHz also follows the same linear trend as thebroadband content. The range of frequencies where this linear dis-persion exists as well as the slope are consistent in magnitude withthe previous studies of stand-alone cathodes where they were iden-tified as ion acoustic turbulence.23,25 This suggests correlationallythat the power spectral content in this frequency range is the IAT.Moreover, as we would expect for these unmagnetized waves (Sec.II), we have found that the high frequency content is dispersionlessin the azimuthal direction, suggesting these modes are purely in ther-z plane.

Although not shown here, we found that this linear dispersionat high frequency persisted in the axial direction for all measure-ments performed along the cathode centerline—the slope and fre-quency range remained unchanged. We did not, however, observethis dispersion in the Beall analysis for locations at radii greaterthan r > 5mm. While this could be an indication that these modesdo not exist at these larger radii, as can be seen from Fig. 3(a), thepower spectrum shape from 500 to 1250 kHz remained qualitativelythe same throughout the plume. The suggests the IAT may stillpersist, but its dispersion is obscured by the large disparity in themagnitude of the oscillations of the rotating modes and the acousticcontent at larger radii. With this in mind, for the remainder of thisinvestigation, we proceed under the assumption that even thoughwe did not explicitly resolve the dispersion at these off-axis loca-tions, the energy in the spectral range of 500 kHz–1250 kHz isassociated with longitudinal acoustic modes.

We show in Fig. 4 the spatial dependence of the amplitudes ofthe m¼ 1, m¼ 2, and broadband acoustic modes in the plume. Forthe rotational modes, this plot represents the peak from the powerspectrum [e.g., Fig. 3(a)] corresponding to the mode frequency ateach spatial location. For the acoustic-like modes, the plottedquantity is the summation over the power spectrum from 500 kHzto 1250 kHz. The results shown in Fig. 4 illustrate several notablefeatures about the observed oscillations. Both rotational modesexhibit a maximum in amplitude that occurs off-axis from the cen-terline. The m¼ 1 mode, which has the highest amplitude, peaks atapproximately r ¼ 5mm while the m¼ 2 mode peak occursbetween r¼ 5 and 10mm. This non-monotonic dependence onradius is consistent with the results presented in Ref. 33 wherehigh-speed imaging was applied to a comparable plasma plume.The fact that the location of the peak moves radially outward withmode number is also a feature that has been noted for drift-wavesin low-temperature plasmas.49 The amplitude of the broadbandfluctuations for the acoustic oscillations exhibits a peak off center-line as well, although the gradients are more gradual. Indeed, withthe exception of points located at the downstream extremes of thedomain, the amplitude varies less than 50% over the domain. Thisresult is reminiscent of the findings from Ref. 25 that showed thatthe acoustic turbulence in the plume of a stand-alone cathode willsaturate to a thermal limit after growing over a short spatial dis-tance from the cathode exit.

C. Comparison of wave measurements to linear theory

We quantify the observations from Sec. IVB explicitly here byevaluating Eqs. (5) and (7) with the background plasma parametersshown in Fig. 2 and comparing the predicted dispersion to measure-ment. For the ion acoustic waves, we estimate the phase velocity by

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fitting a line in Fig. 3(d) to the linear region of the dispersion relationand multiplying the slope by 2p to account for the conversion to angu-lar frequency. This yields a phase velocity of x=k ¼ 6 61 km=s.Following Eq. (5) and using the average value of electron temperatureof Te 4:6 eV, linear theory indicates that ion acoustic waves in thisplasma environment should propagate with a phase velocity ofx=k 2 km=sþ uiðzÞ. Comparing to the theoretical prediction, thisresult would suggest that the local velocity of ions on centerline isuiðzÞ 461 km=s. This estimate for ion speed is in keeping with thetypical values that have been measured in the plumes of hollow catho-des employed for electric propulsion25,50 and thus provides strong cor-relational evidence that the measured waves in this frequency rangeare ion acoustic in nature. With that said, the discrete peak at1200 kHz we observe in the dispersion on centerline is a notabledeparture from previous works on stand-alone cathodes23 where theacoustic waves exhibited a smooth power law decay with frequency.The mode associated with this peak still appears to be acoustic innature, following the general dispersion shown in Fig. 3(d), yet it is notevident from the simplified linear theory (Sec. II) why energy shouldbe concentrated locally at this frequency. While it is possible thathigher order effects such as reflections off the density gradient in theaxial direction could promote a resonant frequency like the one exhib-ited here,51 exploring these higher order effects ultimately is beyondthe scope of this investigation. Indeed, in the following treatment, thefine details of the structure of the power spectrum become immaterialas we integrate over all the frequencies to assess the average impact[Eq. (6)] of the waves on the electron transport.

We next compare the measured wave properties to the linear the-ory predictions for the rotational, anti-drift mode. To this end, we firstoutline a range of values for the background plasma properties in Eq.(7) based on the measurements in the near-field region of the cathodein the H9 Hall thruster where we observed the rotational waves tohave the highest amplitude (r < 10mm). We then evaluate the lineardispersion relation numerically over this range of values. As shown inTable I, we use the range of Te ¼ 4� 6 eV for the electron tempera-ture and a local axial value of the magnetic field of B0 250G.We assume there is negligible ion swirl in the azimuthal direction asthey are unmagnetized and approximate the ion velocity as strictlyaxial with a range informed by the ion acoustic measurements,uiðzÞ ¼ 3–5km=s. For the electron drift in the axial direction, weassume that since this is a highly magnetized environment, the

TABLE I. Range of values for evaluating the theoretical dispersion of the anti-driftwaves.

Plasma property Range of values

Te 4–6 eVB0 250 GuiðzÞ 3–5 km/sueðzÞ 1000–3000 km/s�e ð2� 15Þ � 106 s�1

vE�B 10–40 km/sv� 10–80 km/sr 5mm

FIG. 4. Spatial dependence of the amplitude of relative fluctuations in ion saturationcurrent for the m¼ 1 mode (a), the m¼ 2 mode (b), and integrated from thepower spectrum from 500 kHz to 1250 kHz (c). Note that each plot has a differentscale.

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electron current primarily will be confined to motion along magneticfield lines. We thus can calculate an average axial drift by relating it tothe total current, ueðzÞ I=ðq

ÐnerdrÞ, where the integral is applied at

fixed axial location and I ¼ 15A denotes the electron current. Usingour measured density profiles from Fig. 2(a), this yields a range of pos-sible local speeds of ueðzÞ ¼ 1000–3000 km=s.

For the classical electron collision frequency, we consider bothCoulomb and electron-neutral interactions (cf. Ref. 52),

�ei ¼ 2:9� 10�12nelnK

T3=2e

;

�en ¼ 6:6� 10�19Te

4� 0:1

1þ Te

4

� �1:6

26664

37775nn

ffiffiffiffiffiffiffiffi8Te

pme

r;

(10)

where K denotes the Coulomb logarithm, nn is the local neutral den-sity, and Te is given units of energy. We have all the required plasmameasurements to estimate the range of these collision frequenciesin the near-field cathode plume with the exception of the neutraldensity. To approximate this parameter, we follow the prescriptionof Goebel and Katz52 and later Spektor et al.24 in assuming that theneutral plume from the cathode expands at a fixed angle, a, fromthe orifice. The neutral gas at axial location z thus can be approxi-mated as

nn ¼_m

mivthðnÞp rd þ tan azð Þ2; (11)

where _m denotes the mass flow rate through the cathode, vtðnÞ is theneutral gas thermal temperature, and rd denotes the diameter of thecathode keeper orifice. Assuming a typical neutral temperature of1000K exiting the cathode (consistent with the assumption that theneutrals are at equilibrium with the cathode’s emitter temperature) andallowing for possible expansion angles ranging from a ¼ 30�–70�, wefind at the axial location where the measurements were performed arange of neutral densities of nn ¼ 4–30� 1018 m�3. With these valuesand the plasma properties in Fig. 2, we evaluate the expressions inEq. (10) to find a range for total electron collision frequencies of�e ¼ �ei þ �en ¼ ð2� 15Þ � 106 s�1.

For the azimuthal drift velocities, we use the background proper-ties in the near-field region (Fig. 2) at r < 10mm to establish therange vE�B ¼ 10� 40 km=s and v� ¼ 10� 80 km=s. Finally, we notethat in evaluating Eq. (7), we need to make an assumption about thevalue of the radius, r. This free parameter was introduced by our deci-sion to employ a Cartesian formulation for the dispersion instead of afull cylindrical analysis (recall kh ¼ m=r). We follow the precedent ofRef. 49 in evaluating the dispersion at the radial location co-locatedwith the maximum amplitude of the dominant, m¼ 1 rotationalmode, r ¼ 5mm.

Armed with these experimentally informed estimates for thebackground plasma parameters, we explicitly can evaluate the pre-dicted frequency and wavenumbers of the excited waves from Eq. (7).To illustrate the approach, we first show in Fig. 5 an example of thenumerical predictions from Eq. (7) for a set of representative driftsand plasma parameters. Here we have plotted both the frequency ofoscillation and the predicted growth rate as a function of axial

wavenumber, kz, for four different mode numbers. As this plot shows,the expected frequency increases linearly with mode number.Moreover, as an encouraging preliminary result, we see that the mag-nitude of the frequency at each mode number is comparable with themeasured values shown in Fig. 3. The growth rate, on the other hand,exhibits a non-monotonic dependence, showing an axial wavenumberthat corresponds to maximum growth in each case. Physically, theexistence of a maximum growth rate stems from the fact (Sec. II) thatit is electron collisions in the longitudinal direction that provide thephase delay necessary for the anti-drift waves to extract energy fromthe electron flow. As kz !1, the characteristic length scale of theoscillation in the longitudinal direction becomes negligible comparedto the mean free path such that collisions have a vanishingly smalleffect on the wave dynamics. This drives the growth to zero.Conversely, at asymptotically long wavelengths (kz ! 0), there is nowave motion along the longitudinal direction to facilitate the necessaryphase delay between density and potential for growth. Between theseextremes in wavenumber, there is an optimal phasing between density

FIG. 5. Predicted dispersion (a) and growth rate (b) for the rotational mode from Eq.(7) as a function of axial wavenumber. Results are shown for m¼ 1 to m¼ 4.Background plasma parameters employed in this evaluation include Te ¼ 4:6eV;vE�B ¼ 30km=s; v� ¼ 30km=s; �en ¼ 107 s�1; uiðzÞ ¼ 4km=s; B0 ¼ 250G; ueðzÞ¼ 1000km=s, and r ¼ 5mm.

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and potential oscillations that promotes maximal energy exchangewith the background drift. With this in mind, in order to map the lin-ear predictions to experiment, we assume that the mode frequencyand axial wavenumber that will actually appear in the plasma will cor-respond to this maximum growth rate. For example, in Fig. 5 the fast-est growing m¼ 1 mode would have an axial wavenumber ofkz 10m�1 with a corresponding real component of frequency of70 kHz. In order to account for the variance of experimental measure-ments, we sampled 600 times randomly from the range of valuesshown in Table I and generated dispersion relations similar to Fig. 5for each case. This yielded a dataset of predicted axial wavelength andfrequency for each mode number. We report the average values fromthis dataset as well as confidence intervals that represent the variance.

Figure 6 shows the theoretical predictions against experimentalmeasurements of the wave properties. The measured values are drawnfrom Figs. 3(b) and 3(c) where we have assigned error bars based onthe full-width half maximum in the dispersion plots. Comparing theseresults, we see both qualitative and quantitative agreement withinuncertainty of both the frequency and wavenumber. The frequencies

exhibit particularly good quantitative agreement within error bars,increasing as a function of mode number. Similarly, the axial compo-nent for both the predicted and measured values increases linearlywith mode number, although, quantitatively, the predicted values arelower. This discrepancy may stem from a number of non-ideal factorin the actual plume not included in the theoretical formulation. Theseinclude the existence of axial gradients, non-uniform temperaturesand drifts, the spatial dependence of the plasma properties, and thefact that our analysis is inherently Cartesian, while the actual oscilla-tion is non-local. In spite of these discrepancies, the agreement withprediction is still marked and provides additional quantitative supportthat these modes are described by the simple dispersion presented inEq. (7) for drift-driven waves in the plasma. This is in keeping withthe more detailed non-local analysis applied in Ref. 33 (though only am¼ 1 mode was considered in the previous work).

In summary, we have shown in the preceding evidence that bothion acoustic waves and anti-drift waves exist in the near-field plume ofa hollow cathode operating in a Hall thruster. This is the first direct con-firmation of these ion acoustic modes for a cathode actually operatingin this environment. And, it is the first quantitative measurement ofgreater than m¼ 1 modes for the drift oscillation. With this in mind,we next consider the central question as to the degree to which theseoscillations may impact the electron dynamics in the near-field plume.

D. Contributions to non-classical transport

We leverage in this section the formulation from Sec. II as pre-sented in Eqs. (6) and (8) to quantify the impact of the IAT and anti-drift waves on electron transport. Before proceeding, however, we notethat in order to estimate the amplitude of the fluctuations in plasmadensity in both of these expressions, we make the assumption that wecan linearly relate this quantity to the ion saturation current:ðneðkz ;mÞ=neÞ

2 ð~isatðkz ;mÞ=�isatÞ2. This is a valid approximation pro-

vided two key conditions are satisfied. The first, which is consistentwith the derivations in Sec. II, is that quasineutrality applies on thetime scale of both oscillations of interest such that fluctuations in theion density mirror fluctuations in the electron density, ðneðkz ;mÞ=neÞ¼ ðniðkz ;mÞ=niÞ. The second assumption is that the relative fluctuationsin ion saturation current are proportional to relative fluctuations inthe ion density: ð~isatðkz ;mÞ=�isatÞ ¼ ðniðkz ;mÞ=niÞ. The validity of thisequivalence is contingent on the probe sheath being able to adjustfaster than the characteristic time scale of the plasma oscillation andthe electron temperature remaining constant during the oscillations.Given that the formation of the Langmuir probe sheath is assumed tooccur on the ion plasma frequency time scale, we assume the formercriterion is met. For the latter criterion, we did not have the capabilityto resolve the temperature explicitly at these high frequencies.However, we note that since the dispersion relations for both theacoustic and anti-drift modes are contingent on the assumption ofconstant electron temperature and we have found close agreementbetween measurement and the linear dispersion, we proceedunder the assumption that the temperature is also constant asthese waves propagate. With that said, we recognize that this rela-tionship between density and ion saturation measurements is anapproximation and the results presented should be understoodwith this caveat.

With this in mind, we use our power spectral measurements forthe ion saturation fluctuations as well as our measurements of the

FIG. 6. (a) Comparison of the measured and theoretical value for the real compo-nent of the anti-drift wave as a function of mode number. (b) Comparison of themeasured axial number and the predicted axial wavenumber corresponding to max-imum growth.

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background and wave properties presented in Secs. IVA and IVB toestimate the anomalous collision frequencies for the propagatingwaves. For the ion acoustic contribution [Eq. (6)], we evaluate thesummation over frequency by making the substitution (informed bydispersion relation measurements) that kz x=ð6000Þm�1 and sum-ming over the measured power spectrum from 500 to 1250 kHz. Forthe contributions to the anomalous collision frequency from the anti-drift waves [Eq. (8)], we use the measurements of the backgroundplasma parameters shown in Fig. 2 to estimate the drifts, and we evalu-ate at each location in the plume the summations over mode number.The amplitude of density fluctuations is inferred from the power spec-tra, and the measured axial wavenumber associated with each mode isdetermined from the Beall plots [e.g., Fig. 3(c)].

We show in Fig. 7 the results for the anomalous collision fre-quency from the ion acoustic waves in the axial direction,�ANðzÞðIATÞ, and for the anti-drift waves in the axial, �ANðzÞðADÞ, andazimuthal, �ANðhÞðADÞ, directions. The calculations are shown as afunction of axial location at a position r¼ 5mm from the cathode cen-terline. This is a region where we unambiguously detected the IATand where the strong plasma gradients suggest the presence of non-classical effects. For comparison on this plot, we also show the esti-mated electron-neutral collision frequency, �en, electron-ion Coulombcollision frequency, �ei, and the electron-cyclotron frequency, Xce.Axially, along the direction of applied magnetic field, the collision fre-quency from IAT is almost three orders of magnitude higher than theelectron-ion collision frequency and up to an order of magnitudelarger than the electron-neutral collisions. On the other hand, theanomalous collision frequency in the axial direction from the anti-driftmode, while higher than the electron-coulomb frequency for theregion closer to the cathode, is less than the classical electron-neutralcollision frequency. Taken together, these results suggest that the elec-tron resistivity in the longitudinal direction in the near-field cathode

plume is non-classical and is dominated by the action of the ion acous-tic turbulence. In the azimuthal direction, the IAT collision frequencydoes not have a contribution (as this mode was assumed and latershown to propagate primarily in the longitudinal direction). However,we do find that the rotational mode collision frequency, �ANhðADÞ, isthree orders of magnitude higher than the Coulomb collision fre-quency and two orders of magnitude higher than the electron-neutral.This indicates that the electron dynamics in this direction (which hasdirect bearing on cross field transport) are non-classical and differentin magnitude and scaling when compared to the longitudinal IAT-dominated motion. In both cases, longitudinal and azimuthal, wenote that the electron cyclotron frequency still exceeds the collisionfrequency by an order of magnitude, suggesting that despite theenhanced collisionality, the plasma remains magnetized.

In summary, the above findings confirm experimentally for thefirst time that the transport in the strongly magnetized plasma of ahollow cathode operating in a Hall thruster is non-classical, driven bythe presence of waves. However, the magnitude of this non-classicaltransport is anisotropic, exhibiting a different magnitude and spatialdependence in the azimuthal and longitudinal directions. Althoughnot shown here, we also note that these trends in non-classical trans-port, as captured by these effective collision frequencies, persistedeverywhere in the measurement domain. In Sec. V, we discuss theseresults in the context of previous work, their physical significance, andon-going attempts to model these non-classical effects self-consistently.

V. DISCUSSIONA. Implications for previous studies

Our experimental finding that wave-driven drag acts on the elec-trons in the cathode plume of a Hall thruster extends the conclusionsfrom previous studies in two critical ways. First, although it has beenshown that IAT exists in the plumes of stand-alone cathodes,23,25 itwas an open question as to if the IAT would persist if the cathodeswere operated in a thruster environment. This is because of the differ-ences in magnetic field and background neutral density. Our study isthe first demonstration to our knowledge that not only does IAT existin a cathode operating in conjunction with a Hall thruster but that itcan dominate the electron resistivity in the longitudinal direction.Second, we have shown that in addition to IAT, a second wave, theanti-drift mode, enhances the electron drag anisotropically and non-classically in both the azimuthal and axial directions. This suggeststhat in a treatment of a magnetized hollow cathode plume, the dispa-rate effects of both the IAT and anti-drift waves must be considered.

In addition to this discussion of the cross field transport, we alsocan relate our work to the first experimental study of the anti-driftmode.33 It was noted in this previous investigation that even thoughthe detected anti-drift wave in the cathode plume was propagating azi-muthally, there was an oscillation in the total discharge current to thethruster at this same frequency. This was unexpected because the dis-charge current is an azimuthally integrated measurement. Any crossfield driven current would be exhibited as a constant offset in the mea-sured current instead of a varying value at the frequency of propaga-tion. It was postulated in this previous work that this effect could beexplained if the rotational mode somehow lost its azimuthal characterand converted to a longitudinal oscillation as it propagated from thecathode into the thruster discharge chamber. Our results here,

FIG. 7. Electron collision frequencies including wave-driven and classical contribu-tions as a function of axial distance from cathode exit at a radial location ofr ¼ 5mm.

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however, offer an alternative explanation as we have shown that therotational wave has a finite axial component. This can give rise to atime-varying longitudinal current on the time scale of the oscillation.Given the magnitude of the oscillation (approaching 100% of the back-ground in Fig. 4), it thus may not be surprising that this mode oscilla-tion does appear in total current measurements.

B. Implications for modeling non-classical transport

The prediction of the two-dimensional plasma state that resultsfrom the anomalous collision frequency effects would require a self-consistent model of this region. While this is beyond the scope of thisinvestigation, we discuss here qualitatively how the measured plasmastate is consistent with the presence of non-classical effects and howwe can use our results to guide future modeling efforts. To this end,we first note that qualitatively we anticipate that the longitudinalenhanced resistivity will promote larger electric fields in the axialdirection. This is what drives the steep gradients shown in Fig. 2(c)from z ¼ 25� 35mm. The azimuthal collision frequency, on theother hand facilitates enhanced cross field transport radially, relaxingthe gradients in this direction. This in turn can drive the plasma toequipotential—as exhibited downstream of z ¼ 35mm in Fig. 2(c).The interplay between the longitudinal and cross field transport ulti-mately is what gives rise to the observed two-dimensional plasmastate.

While we do not attempt to model this interplay directly here, wenote that our results can be used to help guide on-going numericalefforts. Indeed, most modeling approaches for hollow cathodes to datehave been fluid-based and formulated in the r-z plane. They thus can-not model the growth of kinetically driven IAT, the propagation of azi-muthal modes, or the direct influence of these waves on the electrondynamics. These codes instead rely on the use of ad hoc transport coef-ficients, such as an anomalous collision frequency, to represent thekinetic and three-dimensional effects.21,26,27,30 While the use of thesead hoc terms may be effective in yielding agreement with experimentalresults, the challenge in adopting this approach stems from identifyingclosure models, i.e., expressions for the enhanced wave-driven collisionfrequency that accurately represent the non-classical processes. It alsomust be possible to solve for these coefficients self-consistently in afluid framework. Specifically, the new, ad hoc terms must depend onfluid properties, e.g., �ANðTe; ne;…Þ.

The closure problem as it relates to the IAT and anti-drift waves isillustrated by the forms of Eqs. (6) and (8). In particular, while theseexpressions do depend on properties such as temperature and drift thatcan be solved for in a fluid model, they also scale explicitly with thewave amplitude, neðkz ;mÞ; and depend on key wave properties such aswavenumber and frequency. Since these properties are not solved for instandard r-z fluid models, the closure problem thus becomes one ofidentifying a functional relationship between the amplitude of the wavesand the plasma properties, e.g., neðkz ;mÞðTe; ne; ::Þ. This problem hasbeen examined in detail for the IAT in cathode plumes in stand-aloneconfigurations. Solutions have included adopting simplified models forthe wave energy25 as well as implementing expressions for the waveamplitude based on the assumption that IAT growth has been saturatedby nonlinear effects.21,26 This latter approach also has been applied tomodeling the hollow cathode plume in a Hall thruster where the follow-ing expression was adopted:

�cANðzÞðIATÞ ¼ aueðzÞvte

xpi; (12)

where we have introduced the superscript “c” to denote closure, xpi isthe ion plasma frequency, and a is a constant of order unity. It shouldbe noted that this expression was applied without experimental confir-mation that IAT actually existed in the thruster/cathode environment.However, its use did yield predictions in qualitative agreement withthe measured spatial profiles of the plasma properties in the plume ofa centrally mounted cathode in a Hall thruster.26

Now that we have explicit measurements of the effective collisionfrequency driven by the IAT in the hollow cathode plume (Fig. 7) andmeasurements of the local plasma properties, we can compare them tothe closure model, Eq. (12). Figure 8 shows this comparison as a func-tion of distance from cathode exit plane at a location r¼ 5mm fromthe thruster centerline. Here we have adjusted the coefficient, a ¼ 0:5,to yield the best fit with the data. The resulting trend shows both quan-titative and qualitative agreement in this region of the cathode sugges-ting that this closure, if not exact, is a reasonable approximation to beused in this class of device. This helps explain in part the relative suc-cess found in Ref. 26 in using this closure to model the cathode plume.It further suggests that this model may be applied for approximatingthe IAT in future fluid simulations.

With that said, while closure models have been proposed andapplied in hollow cathode plume simulations to represent the influ-ence of IAT, there has yet to be an attempt at closure based on theaction of the anti-drift waves in the azimuthal direction. To motivatean expression, we re-consider Eq. (8) in light of our experimental mea-surements. First, as we note from Fig. 3, there is an evident linear rela-tionship between frequency and mode number, m, and axialwavenumber, kz. This suggests that the ratio of these quantities should

FIG. 8. Closure models for wave-driven collision frequency compared to measurementsat a location r ¼ 5 mm from centerline. Best fit coefficients include a ¼ 0:5 for the IATclosure, d ¼ 2 for the closure model for the anti-drift wave assuming thermal saturationof the waves and d ¼ 60 for the closure model assuming wave-driven saturation.

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be constant ðm=rÞ=kz ¼ d > 1, where the inequality is greater thanunity because the waves propagate primarily in the azimuthal direc-tion. With this new variable, we can simplify Eq. (8) to

�cANðhÞðADÞ ¼ d�e dþueðhÞueðzÞ

� �Xkz ;m

neðkz ;mÞne

� �2

: (13)

The summation over the wave amplitudes in Eq. (13) is not known apriori. However, we do note from Fig. 4 that the relative amplitude offluctuations of these modes with the background is approaching thesteady-state value, i.e., neðkz ;mÞ=ne ! O½1�. Physically, this large ampli-tude suggests that the wave growth may be approaching a nonlinearlimit. Operating under this assumption, we consider two physicallyplausible mechanisms for the saturation of these drift-driven waves(cf. Ref. 35). The first is that the upper bound on the growth is limitedby the amount of thermal energy available in the plasma such thatneðkz ;mÞ=ne q/eðkz ;mÞ=Te O½1�. The second limits stems from theassumption that the upper bound on growth is limited by the amountof energy available in the drift that drives the mode unstable: neðkz ;mÞ=ne q/eðkz ;mÞ=Te meu2eðhÞ=Te. Substituting these expressions in Eq.(13), we find

�cANðhÞðADÞjthermal ¼ d�e dþueðhÞueðzÞ

� �;

�cANðhÞðADÞjdrift ¼ d�e dþueðhÞueðzÞ

� � u2eðhÞv2te

;

(14)

where we have introduced the labels of “thermal” and “drift” to denotethe two proposed saturation mechanisms. Using the plasma measure-ments from Sec. IV as well as the measured collision frequency fromFig. 7, we show in Fig. 8 a comparison of Eq. (14) with the measuredresult. We have used values of d ¼ 2 and d ¼ 60 for the thermal anddrift closures, respectively, in order to yield the best match with data.From this plot, we see that although both closures can be adjusted toagree in magnitude with the measured collision frequency, the�cANðhÞðADÞjdrift closure yields better agreement over the measurementdomain. This latter result in particular provides at least correlationalevidence that simplified closure may be appropriate and accurate influid based models. As a final comment, although we treated d as afree parameter to match the data, we can use the measured ratio ofwavenumbers (Fig. 6) to calculate a theoretical value. The result,d 10, is relatively close—within an order of magnitude—to the d¼ 60 that yielded the best fit to the data in Fig. 8. The lack of exactquantitative agreement may be attributed to the several simplifyingassumptions we employed in our analysis such as the adoption of asimplified canonical geometry and the use of simplified scaling law forthe saturation energy.

C. Implications for susceptibility of hollow cathodesto facility effects

The form of Eq. (14) has an implication in the context of one ofthe major outstanding challenges related to Hall thruster development:the problem of facility effects. Indeed, it is well known that Hallthruster operation will change as properties of the test environment,such as the ambient background pressure, vary. One of the most well-documented of these effects is the cathode coupling voltage, i.e., thepotential increase from the cathode to the adjoining thruster plume

will decrease with increased facility pressure. In an attempt to explainand predict the response of this parameter to facility pressure, Spektoret al.24 developed a quasi-1D model where he postulated that increas-ing neutral density would increase the electron-neutral collision fre-quency and, therefore, facilitate more cross field transport. Thisrelaxes the potential and leads to lower coupling voltages. He showedthat while his model did correctly capture trends in the coupling volt-age with pressure, it was necessary to employ Hall parameters thatwere an order of magnitude lower than the actual values in the mod-eled thrusters. Recognizing that this was an unphysical assumption,Spektor suggested that there may be non-classical effects present con-tributing to an enhanced collision frequency. We have shown herethat not only does such a non-classical mechanism exist but that itscales linearly with the electron-neutral collision frequency. This leadsto an effective collision frequency that has the same dependencies onthe background plasma as the electron-neutral collision frequency butwith an order of magnitude higher amplitude (and therefore an orderof magnitude lower Hall parameter). Coupled with Spektor’s 1Dmodel, our result suggests that the cathode’s response to facility effectsmay indeed be explained by the enhanced non-classical transport thatresults from the presence of addition neutrals.

D. Implications for ion energy in cathode plume

As the focus of this study has been on plasma waves in the nearcathode plume and their impact on the electron dynamics, we havenot performed measurements on the ion energies or attempted to cor-relate the wave properties with the ion dynamics. With that said, giventhat aspects of the ion energy distribution in stand-alone cathodeexperiments have been shown to be non-classical and wave-driven,20,22,50 we include here a qualitative discussion of the implica-tions of our findings for the ion dynamics for a cathode in an actualHall thruster.

In stand-alone experiments, the IAT has been shown to lead toenhanced heating of ions while the presence of low frequency, largeamplitude waves has been correlated with transient potential struc-tures that accelerate high energy ions back to the cathode surface. Inboth cases, these processes have been linked to enhanced erosion ofsurfaces in the near vicinity of the cathode. The presence of both IATand large amplitude waves as measured in our cathode suggests thatsimilar ion energization may be occurring. In correlational support ofthis hypothesis, two studies to date have indicated that non-classicalion acceleration does occur near cathodes operating as part of athruster discharge.53,54 These investigations focused on the pole (themagnetic surface surrounding the cathode) in a magnetically shieldedthruster with a centrally mounted cathode. It was found in both ofthese works that the ion energy distributions in the vicinity of the cath-ode were characterized by high energy tails (larger than the localplasma potential). It was hypothesized in Ref. 53 that these tails maybe attributed to the presence of large amplitude oscillations—just aswe have reported in this work. Our findings thus offer a possible expla-nation for non-classical ion properties that have been noted in Hallthruster cathode plumes.

Leaving aside non-classical ion effects, we also comment here onthe strong assumption we have made that the cathode plasma is domi-nated by singly charged ions. This is justified by the observation thatthe electron temperature is relatively low in this region. With that said,previous investigations (cf. Ref. 55) have shown that there may be

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multiple charge states—though in small fraction—in stand-alone cath-ode plumes. If present, these ions will gain even more energy frompotential gradients, further enhancing erosion of surfaces they impact.In sufficient quantity, the presence of multiply charged species mayalso alter the dispersion of the predicted waves. Exploring this effectultimately is beyond the scope of this paper, but we remark that under-standing the influence of higher charge states as well as how instabil-ities impact these species are critical questions for future investigationsof these devices.

E. Impact of magnetic field topology and cathodelocation

The measurements we have presented here were performed on aHall thruster that is magnetically shielded with a cathode mounted onthe axis of symmetry of the thruster. However, many conventionalHall thrusters currently in service rely on external cathodes (mountedoutside the annular discharge) with a magnetic field geometry that isnot shielded. This invites the question as to whether the same oscilla-tions that have been reported here exist and impact the cathodedynamics in the same way for these other systems. We discuss herequalitatively the expected influence of both changing magnetic fieldconfiguration and cathode position.

As to the influence of the magnetic field topology, we note that thedifferences between shielded and conventional thrusters occurs primar-ily in the annular discharge region. Exterior to this location and at themagnetic poles, the field structure remains relatively unaltered. Thissimilarity was remarked upon in Ref. 33. We, therefore, do not antici-pate qualitatively that the differences that arise in a shielded configura-tion versus unshielded will fundamentally impact the conclusions wehave drawn about the near-field environment of the cathode.

The cathode placement, on the other hand, can lead to more sub-stantial differences in the local plasma environment. For externallymounted cathodes, axisymmetry is no longer valid, and the local mag-netic field is no longer purely axial with respect to the cathode body.With that said, despite these differences, there is some correlational evi-dence to suggest that non-classical effects may still persist. For example,we anticipate that at least the same criteria for the growth of both insta-bilities will be met at external locations. There are still cross field gra-dients in the plasma properties in externally mounted cathodes7 thatcould drive drift-waves, and there is still a low impedance path alongthe magnetic field lines for electrons to acquire the high drift speedsthat provide the energy source for acoustic modes. Similarly, the cath-ode coupling voltage in external cathodes has been shown to be evenmore susceptible to facility effects (Sec. VC) than cathodes mountedon thruster centerline.3,43,56 It has been suggested this is because theneutral density in the near-field of externally mounted cathodes islower than for internal cathodes. Variations in background neutraldensity that result from changes in facility pressure thus producehigher fractional changes in the neutral environment for external cath-odes. This high sensitivity to facility pressure is consistent with ourfinding that non-classical electron resistivity is directly linked to theneutral density environment and provides at least a qualitative indica-tion that the electron dynamics in external cathodes are dominated bysimilar processes as the ones we have found in this work.

As an additional note, we have remarked how the potential gra-dients in both the cross field and longitudinal directions in the cathodeplume can be attributed to non-classical effects. These potential

structures in turn [Fig. 2(c)] lead to an electric field directed inwardtoward the cathode. Intriguingly, a previous study on a Hall thrusterwith an externally mounted cathode showed that there is a transversedrift in the ion population in the direction of the cathode57 that intro-duced a slight asymmetry to the discharge. It was suggested in thiswork that this drift may be the result of the presence of a weak electricfield directed toward the cathode. Our findings support this conclu-sion, and as extension, we would anticipate that with decreasing facil-ity pressure (and therefore a higher resistive path for electrons), thistype of transverse drift may become even more pronounced.

F. Relationship of findings to stand-alone tests

The question of whether stand-alone cathodes can faithfullyrecreate the behavior of hollow cathodes operating in Hall dischargeshas been the subject of extensive recent investigations (cf. Ref. 58)where different boundary conditions and flow environments havebeen explored. Our results suggest that in addition to the global oper-ating conditions (current and flow), there are at least two critical ele-ments that must be duplicated to attempt to map stand-aloneexperiments to cathodes operating in a thruster environment: themagnetic field and the neutral density. The former is critical as wehave shown that the anti-drift wave (which only exists in the presenceof a magnetic field) is a dominant oscillation in the plume driving theelectron dynamics and the subsequent distribution of plasma back-ground properties. The latter is also important as locally it is responsi-ble for either damping (for the IAT) or driving unstable (for theanti-drift wave) the propagating waves. While recent efforts havefocused on developing higher fidelity re-creations of the magnetic fieldenvironment, these same studies have also shown that the neutral den-sity in stand-alone configurations can be 5–50 times higher than theneutral density experienced in Hall thruster environments.58 Thisstems from the fact that the stand-alone configurations employ adownstream anode for completing the electrical circuit. This structurehas the unintended effect of blocking the flowing gas, artificially raisingthe neutral density in the near field. In light of our findings outlinedabove, we anticipate this higher neutral density environment may arti-ficially damp the IAT through collisions while enhancing the effectivecollision frequency from the growth of the anti-drift waves. Botheffects could reduce the electrical impedance along and across mag-netic field lines, relaxing the potential gradients in the stand-alone con-figuration when compared to a real thruster environment. The majorimplication is that in the absence of a faithfully reproduced neutraland magnetic field environment, it may not be possible to re-create ina stand-alone configuration the behavior exhibited by hollow cathodesoperating in conjunction with a thruster.

G. Extensions beyond linear theory

As a final discussion point, we remark here on the validity of andextensions to our linearized analysis of the cathode waves and non-classical transport. To this end, one of the key tenets of the theory wehave outlined for both the IAT and anti-drift waves is that the electrontemperature remains constant on the time scale of the oscillations.Although there are techniques available for probing this quantity (orproxies for this quantity) in real time,59,60 we did not have these capa-bilities in our setup. If this plasma property does in fact vary, it wouldalter the dispersion and growth relations of both waves. However, the

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close correspondence we have found between our linear theory andexperiment suggests that at least in the treatment of the waves, theapproximation of constant temperature is appropriate.

Similarly, we note that although we have treated the IAT andanti-drift waves as separate oscillations in our analysis, it is possiblethat the two are coupled. Indeed, it is implicit in the formulation of theanti-drift wave in Sec. II that the electron collisions along magneticfield lines are dominated by classical effects. However, we have shownthat the collisions in the longitudinal direction are non-classical. Onthe time scale of the slower frequency rotating oscillations, it thus ispossible that the electron drag from the IAT along magnetic fieldcould promote additional phase delay for the electron motion, therebyallowing further growth of this oscillation. Functionally, this mightsuggest that the transport coefficients for the IAT and drift waves aredirectly linked. For example, �e ! �ANðzÞðIATÞ þ �en in Eqs. (7), (8),and (14). With that said, this type of link between lower frequency,coherent oscillations and incoherent turbulence remains an active areaof investigation.61

VI. CONCLUSION

In this work, we have investigated experimentally the role of elec-trostatic plasma oscillations in driving electron transport in the near-field plume of a hollow cathode operating with a Hall thruster. Byemploying a combination of ion saturation and Langmuir probes, wehave measured the properties of the propagating waves and comparedthem to the predictions from linear dispersion relations. We haveshown that both ion acoustic turbulence (IAT) (measured from 500 to1250 kHz) and anti-drift waves (measured from 50 to 400 kHz) propa-gate simultaneously in the cathode plume. The IAT is primarily longi-tudinal and driven unstable by the strong electron drift along themagnetic field. The anti-drift waves propagate in the direction of dia-magnetic drift, are rotational in nature, and derive their energy fromazimuthal electron currents. In both cases, we have derived expres-sions for how the growth of these instabilities can impact the electrondynamics. Specifically, we have represented both effects with a trans-port coefficient, an anomalous collision frequency. By employing mea-surements of the wave dispersion and background plasma properties,we have demonstrated that both instabilities can give rise to effectivecollision frequencies that exceed the classical collision frequencies by1–3 orders of magnitude. Significantly, we have found that thisenhanced electron collision frequency is anisotropic. The contributionfrom the IAT collisions is dominant in the longitudinal direction whilethe collision frequency resulting from the rotational waves is dominantin the azimuthal direction. Whereas the IAT thus effects promotesteeper gradients as they enhance the resistivity along magnetic lines,the rotational waves can facilitate transport across magnetic field lines,giving rise to more relaxed potential gradients. Both of these effectsmust be considered in arriving at a self-consistent description of thenear-field plasma. We have expanded on this conclusion by proposingand validating simplified algebraic closure models that could be incor-porated into self-consistent fluid-based models for the near-field cath-ode region. We also have discussed how our findings may help explainkey aspects of cathode operation including the role of facility pressurein changing coupling voltage, the presence of anomalously high ionenergies in cathode plumes, and the differences that exist betweencathodes that are externally versus internally mounted in thrusters.We outlined as well how the transport in the thruster environment is

fundamentally different and more nuanced than the non-classicaltransport that has been documented for stand-alone cathodes. And wenoted that these differences pose a substantial challenge for re-creatingthruster-like operation in a stand-alone configuration.

Taken together, the above results provide a comprehensive and,to our knowledge, first detailed experimental description of the wave-driven transport in the near-field of a hollow cathode operating in aHall thruster. While we anticipate that these findings may pose a newchallenge for the self-consistent modeling and prediction of the plasmadynamics in this region, the understanding of the non-classical effectsand proposed closure models we have presented in this study ulti-mately may help guide the development and validation of these criti-cal, higher fidelity simulations.

ACKNOWLEDGMENTS

The authors also would like to acknowledge the technicalsupport of Mr. Eric Viges in the operation and setup of the testfacility. This work was in part supported through an Air ForceOffice of Scientific Research (AFOSR) Grant No. FA9550-17-1-0035.S. E. Cusson’s effort was supported by the NASA fellowshipNNX15AQ43H. Z. Brown’s contribution was supported by theNational Science Foundation Graduate Research Fellowship ProgramGrant No. DGE 1256260. Any opinions, findings, and conclusions orrecommendations expressed in this material are those of the authorsand do not necessarily reflect the views of the National ScienceFoundation.

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