Evidence of nonclassical plasma transport in hollow cathodes for electric propulsionIoannis G. Mikellides, Ira Katz, Dan M. Goebel, and Kristina K. Jameson Citation: Journal of Applied Physics 101, 063301 (2007); doi: 10.1063/1.2710763 View online: http://dx.doi.org/10.1063/1.2710763 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modeling of plasma turbulence and transport in the Large Plasma Device Phys. Plasmas 17, 122312 (2010); 10.1063/1.3527987 Internal plasma potential measurements of a Hall thruster using plasma lens focusing Phys. Plasmas 13, 103504 (2006); 10.1063/1.2358331 Internal plasma potential measurements of a Hall thruster using xenon and krypton propellant Phys. Plasmas 13, 093502 (2006); 10.1063/1.2335820 Magnetically filtered Faraday probe for measuring the ion current density profile of a Hall thruster Rev. Sci. Instrum. 77, 013503 (2006); 10.1063/1.2149006 Spectral analysis of Hall-effect thruster plasma oscillations based on the empirical mode decomposition Phys. Plasmas 12, 123506 (2005); 10.1063/1.2145020

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Evidence of nonclassical plasma transport in hollow cathodesfor electric propulsion

Ioannis G. Mikellides,a� Ira Katz, Dan M. Goebel, and Kristina K. JamesonJet Propulsion Laboratory, California Institute of Technology, Pasadena, California, 91109

�Received 24 September 2006; accepted 14 January 2007; published online 16 March 2007�

Measurements, simplified analyses, and two-dimensional numerical simulations with a fluid plasmamodel show that classical resistivity cannot account for the elevated electron temperatures and steepplasma potential gradients measured in a 25–27.5 A electric propulsion hollow cathode. Thecathode consisted of a 1.5 cm hollow tube with an �0.28 cm diameter orifice and was operated with5.5 SCCM �SCCM denotes cubic centimeter per minute at STP� of xenon flow using two differentanode geometries: a segmented cone and a circular flat plate. The numerical simulations show thatclassical resistivity yields as much as four times colder electron temperatures compared to themeasured values in the orifice and near-plume regions of the cathode. Classical transport and Ohm’slaw also predict exceedingly high electron-ion relative drift speeds compared to the electron thermalspeed ��4�. It is found that the addition of anomalous resistivity based on existing growth rateformulas for electron-ion streaming instabilities improves qualitatively the comparison between thenumerical results and the time-averaged measurements. Simplified analyses that have been basedlargely on the axial measurements support the conclusion that additional resistivity is required inOhm’s law to explain the measurements. The combined results from the two-dimensionalsimulations and the analyses bound the range of enhanced resistivity to be 3–100 times the classicalvalue. © 2007 American Institute of Physics. �DOI: 10.1063/1.2710763�

I. INTRODUCTION

Hollow cathodes employed in electric propulsion �EP�systems use porous tungsten inserts similar to the dispensercathode technologies that have been used for decades inmany vacuum technologies such as microwave tubes. How-ever, EP cathodes differ from those used in vacuum applica-tions in that propellant gas flow �usually xenon� is introducedinto the cathode tube. As the gas becomes partially ionizedcurrent is conducted to an anode. The presence of a highconcentration of xenon atoms changes significantly the be-havior of the ionized species throughout the device com-pared to vacuum cathodes. A major issue regarding the use ofEP cathodes in long duration robotic space science missionsconsidered by NASA is the erosion of the keeper electrode.Efforts to understand cathode-keeper erosion intensified afterthe conclusion of the extended life test �ELT� of the NASASolar Electric Propulsion Technology Applications Readiness�NSTAR� engine in 2003 which showed that the dischargecathode keeper was completely eroded after �30 000 h.1

Much of the work on keeper erosion has naturally centeredon measurements of high-energy ions and the identificationof the mechanisms that produce them. Several such mecha-nisms have been proposed, including potential hills,2 chargeexchange between ions and neutrals,3 double ionization,4 andplasma potential oscillations.5

Clearly, the evolution of the plasma in the region thatspans a few keeper orifice diameters downstream of thekeeper exit is a critical aspect of any work that aims at un-derstanding the source�s� of high-energy ions; we call this

region the “near plume.” The two-dimensional structure ofthe plasma potential is of particular interest because it maybe directly associated with attainable ion energies. In thestudies cited above it has been sufficient to regard the exist-ing spatial or temporal profiles of the plasma potential ob-tained by the measurements as de facto for the purpose ofestablishing hypotheses on the generation mechanism�s� anddirection of high-energy ions. The details of how and whythe discharge establishes a spatially rising plasma potentialdownstream of the keeper, while in some cases exhibitinghigh-amplitude temporal fluctuations, or the reason�s� for theabsence of a nonmonotonic profile �potential hill� have onlybeen loosely addressed. For example, it has been suggestedthat the applied magnetic field �as part of their operationprinciple, conventional ion and Hall electric propulsion sys-tems incorporate applied magnetic fields to guide the flow ofelectron� “smoothes out” potential structures on axis,4 butrecent measurements have shown no potential hills even inthe absence of an applied magnetic field.5,6 Moreover, themechanism�s� that heat the relatively cold �1–2 eV� elec-trons inside the cathode to temperatures that in some casesexceed 5 eV in the near-plume region has not been rigor-ously identified. Recent planar anode experiments with a1.5 cm diameter cathode operating at 5.5 SCCM of xenonflow and 27.5 A show that the electron temperature is notaffected significantly by the applied magnetic field.7 Thusmany of the qualitatively distinctive features exhibited by theplasma in these devices exist even in the absence of theapplied magnetic field.

The inherent two dimensionality of the plasma both in-side and outside the cathode limits the ability of zero-dimensional and one-dimensional theoretical models to rig-a�Electronic mail: ioannis.g.mikellides@jpl.nasa.gov

JOURNAL OF APPLIED PHYSICS 101, 063301 �2007�

0021-8979/2007/101�6�/063301/11/$23.00 © 2007 American Institute of Physics101, 063301-1

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orously explain many of the plasma features alluded toabove. Two, two-dimensional theoretical models of the hol-low cathode have been developed in the past few years bythe authors of this article. The first model, OrCa2D-I �ori-ficed cathode 2D-I�,8 was developed to simulate the emitterregion of the hollow cathode. At that time it was proposedthat anomalous heating by ion acoustic turbulence was pos-sible in view of the elevated temperatures measured insidethe orifice and near-plume regions of the 1.5 cm cathode.The existence of nonclassical heating in the orifice and near-plume regions would require that the appropriate physicsterms appear in the conservation laws which may includeanomalous resistivity and time dependence. OrCa2D-I didnot incorporate such physics terms and its computational re-gion excluded the orifice channel, keeper, and plume regions.Thus, OrCa2-II �Ref. 9� was developed to allow for the glo-bal simulation of the cathode plasma, by augmenting boththe conservation laws and the computational region.OrCa2D-II includes the cathode and keeper orifice channels,and the plume region, and allows theory and experiment tobe compared directly both inside and outside the device.

This article describes measurements, idealized analyses,and two-dimensional numerical simulations of the 1.5 cmhollow cathode mentioned above. The results are mainly re-lated to the effects of classical and nonclassical collisions onthe structure of the plasma in the near-plume region. It isfound that classical resistivity cannot account for the el-evated electron temperatures and steep plasma potential gra-dients measured in the keeper orifice and near-plume regionsof the cathode. Moreover, classical transport and Ohm’s lawpredict exceedingly high electron-ion relative drift speedscompared to the electron thermal speed ��4�, which wouldnormally lead to the excitement of violent streaming insta-bilities. These instabilities would likely disrupt the dischargein this cathode. During nominal operation no such disrup-tions are observed but low-frequency fluctuations by theelectric field have been measured.5 It is therefore possiblethat the instabilities quickly quench into a low-frequency tur-bulent mode that �anomalously� heats the electrons. Prelimi-nary scalings based on existing growth rates that span theoperating plasma conditions of interest have been used toapproximate the effective resistivity as a function of Te /Ti

and the relative drift between electrons and ions. Althoughthese estimates do not yet rigorously account for the transi-tion from the ion acoustic to the Buneman limit on the mac-roscopic properties �e.g., see Refs. 10 and 11�, in the pres-ence of classical �binary� collisions, nonlinear particle-waveinteractions, and plasma inhomogeneities, the comparisonbetween theory and experiment is improved considerablycompared to the classical results. The highest relative driftsare found to occur along the keeper surface facing the anode.Measurements made by Goebel et al.5 revealed plasma oscil-lations with the highest plasma potential amplitudes also oc-curring in this region. The extent of kinetic effects and de-viations from the Maxwellian electron energy distributionfunction �EEDF� in this cathode, possibly due to the micro-instabilities, is not yet known. The effect of a possibly “hot”ion tail on the turbulence12 is also unknown. Therefore, themain intent of this article has been to demonstrate that from

a macroscopic viewpoint a much higher value of the resis-tivity compared to classical value would be needed to ex-plain the measurements.

II. ANOMALOUSLY ENHANCED RESISTIVITYIN THE CATHODE PLASMA

Early theoretical work by the authors8 utilized the two-dimensional fluid model OrCa2D-I to simulate the emitterregion of the same EP hollow cathode that is investigatedhere. It was proposed that nonclassical behavior in theplasma downstream of the orifice entrance is possible. Thehypothesis was based on a comparison between the time-averaged measurements along the axis of symmetry6 and thesteady-state classical result from the OrCa2D-I simulations.The comparison suggested that the classical result could notexplain the steep rise of the plasma potential measureddownstream of the orifice entrance. The hypothesis could notbe substantiated at the time mainly because the model didnot extend into the orifice and plume regions.

Much of the effort to understand the discrepancy be-tween the time-averaged measurements and the numericalresults from classical theory has been driven by the inertia-less electron momentum equation �Ohm’s law� which, afterneglecting the ion contribution and the magnetic field, maybe written as follows:

�� � − �je +��nTe�

n, �1�

where � and � are the plasma potential and resistivity, re-spectively. In all the work presented in this article, quasineu-trality and a charge state of one have been assumed. Theelectron particle density, temperature �in eV�, and currentdensity are denoted by n, Te, and je, respectively. If theplasma obeys the macroscopic law �1� then the rise of theplasma potential in regions of decreasing electron pressurecan only be due to two competing forces: the resistive elec-tric field, which drives a positive potential gradient, and thepressure gradient force, which drives a negative potentialgradient. In the following sections we present measurements,simplified analyses, and two-dimensional numerical simula-tions that support the presence of nonclassical mechanismsin the plasma downstream of the orifice entrance. The nu-merical simulations were performed with OrCa2D-II. Thecomputational region is shown in Fig. 1.

A. Probe measurements

Measurements from two different hollow cathode dis-charge tests are described below. Both tests used the samecathode but utilized different anode configurations. The firsttest used a segmented conical anode13 �Fig. 2, top� and thesecond test employed a planar anode �Fig. 2, bottom�.7 Thecathode consisted of a molybdenum tube of 1.5 cm in diam-eter, a thoriated tungsten orifice plate with an inner diameterof �0.28 cm, and a barium-impregnated tungsten insert of2.54 cm in length. A graphite keeper electrode with a keeperorifice of 0.48 cm in diameter fully enclosed the cathode.The tests were conducted in a vacuum chamber of 0.75 m indiameter and 2 m in length. An axial Langmuir probe assem-

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bly with a pneumatic plunger was used to obtain the mea-surements along the axis of symmetry. The pneumatic scanswere performed at speeds up to 2 m/s to ensure that theprobe did not melt while inside the plasma. The test setupand diagnostics are described with greater detail in Ref. 7.The main purpose of the planar anode test was to obtainmeasurements in the same geometry that OrCa2D-II simu-lates. Although some of the measurements referenced below

have included an applied magnetic field, the axial measure-ments used for the comparisons with the theory were ob-tained without an applied magnetic field. The planar anodewas approximately 15 cm in diameter and was placed 8.5 cmaway from the keeper orifice exit. The conical anode had aminimum diameter of 5.4 cm and the entrance to the conicalsection was about 1 cm away from the keeper exit. The op-erating conditions associated with the conical anode mea-surements cited here were as follows: discharge current Id

=25 A, mass flow rate=5.5 SCCM, and discharge voltageVd=26.8 V. The operating conditions for the planar anodetest were Id=27.5 A, mass flow rate=5.5 SCCM, and Vd

=24.4 V. The Langmuir probe time-averaged measurementsalong the axis of symmetry are shown in Fig. 3 �plasmadensity�, Fig. 4 �electron temperature�, and Fig. 5 �plasmapotential�, along with OrCa2D-II results to be discussedlater.

A radially scanning emissive probe was also used in theconical anode test to detect possible plasma oscillations. Aswith the axial Langmuir probe the emissive probe was alsodriven pneumatically into the plasma. The probe tip was atungsten hairpin wire of 0.127 mm in diameter that was fedthrough two adjacent alumina tubes. Each tube was 0.5 mmin diameter. A floating 5 A power supply provided the cur-

FIG. 1. Schematic of the hollow cathode emitter, ori-fice, and plume regions showing the OrCa2D-II compu-tational region �dashed line�.

FIG. 2. Top: Conical anode test setup. Bottom: Planar anode test setup. FIG. 3. Plasma particle density as a function axial location.

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rent to heat the tungsten wire electrode sufficiently to causeelectron emission. The probe signal was fed to a high-impedance, high-frequency circuit and to a buffer amplifierto detect oscillations in the signal. The system was capableof detecting frequencies up to 1 MHz. The plasma potentialas a function of radius outside the 1.5 cm cathode is shownin Fig. 6 �top� when it was operated with 4 SCCM of xenonflow and a small applied magnetic field �Bapp� with a value ofabout 10 G near the orifice. The nominal operation of thiscathode is 5.5 SCCM of xenon flow and Bapp�80 G near theorifice. The plasma potential for the nominal operation isshown in Fig. 6 �middle�. The signals were measured 2 mmdownstream of the keeper front face. The highest amplitudeswere detected in the frequency range of 50–100 kHz, whilesmaller amplitudes were detected at frequencies up to the1 MHz probe limit. Similar temporal behavior was previ-ously detected in a NSTAR-type cathode operated with xe-non. The NSTAR-type cathode is about three times smallerin tube and orifice diameter than the 1.5 cm cathode consid-ered in this study. The plasma potential signal for theNSTAR-type cathode, also measured 2 mm downstream ofthe keeper orifice exit, is shown in Fig. 6 �bottom�. Theoperating conditions for the measurement shown were Id

=13 A, Vd=25 V, and a mass flow rate slightly under5 SCCM. The high-amplitude frequency range for this signalwas 50–500 kHz. Frequencies on the order of 100 kHz are afew orders of magnitude less than the ion plasma frequency��pi�, which suggests that the oscillations may be associated

FIG. 4. Electron temperature as a function of axial location.

FIG. 5. Plasma potential as a function of axial location.

FIG. 6. Plasma potential fluctuations measured 2 mm downstream of thekeeper orifice exit. Top: 1.5 cm cathode �flow rate=4 SCCM, Bapp=10 G,oscillation frequency �100 kHz�. Middle: 1.5 cm cathode �flow rate=5.5 SCCM, Bapp=80 G, oscillation frequency of 200–500 kHz�. Bottom:0.635 cm �NSTAR-type� �flow rate=5 SCCM, Bapp=100 G, oscillation fre-quency of 50–500 kHz�.

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with ion acoustic waves. Additional measurements and amore detailed description of the diagnostics used for bothcathodes are reported in Ref. 5.

B. Analytical estimates

In this section the measurements along the axis of sym-metry are used directly in Eq. �1� to determine how muchmore resistivity, if any, would be required to satisfy it. Due toits geometrical simplicity, the analysis is performed hereonly for the planar anode configuration. In one dimension,Eq. �1� may be solved directly to determine the “measured”resistivity

�m �n−1�z�nTe� − �z�

je,z, �2�

with �z� and �z�nTe� determined using analytic functions��z�, n�z�, and Te�z� that fit the experiment data. We clarifythat by measured resistivity we mean the value of the resis-tivity that was derived from the measured quantities, i.e., theelectron density, the temperature, and the plasma potential.The resistivity was not measured directly in the experiment.For a given axial electron current density je,z, the value of �m

may be compared with the “classical” value of the resistivitygiven by

� =me��ei + �en�

e2n, �3�

where �ei and �en denote the average electron-ion �e-i� andelectron-neutral �e-n� classical collision frequencies, respec-tively. The data-fitting functions n�z� and Te�z� were used todetermine the right-hand side of Eq. �3�, including �ei�z� and�en�z�. In addition to its dependence on the electron tempera-ture, �en�z� requires knowledge of the neutral gas density,which has not yet been measured. We have therefore usedcalculated values obtained from the numerical simulations.The simulations are described in Sec. III C.

The only right-hand-side quantity in Eq. �2� that has notbeen obtained by direct measurement is the axial componentof the electron current density along the axis of symmetryje,z�0,z�, and it is therefore estimated as follows. We assumethat je,z�r ,z� follows Gaussian-like profiles whose standarddeviations diverge downstream of the cathode proportionallyto a “bulk-plasma” radius denoted by Rp�z�, as illustrated inFig. 7. Then at each axial location je,z�0,z� satisfies

je,z�0,z� � je,0�z� = − Id/2��0

RA

e−�r/Rp�z��2rdr . �4�

For the planar anode test the upper limit of the integralshould ideally be � but in practice setting it equal to theanode radius RA introduces an error of less than 1%. Theradius Rp�z� may be prescribed by the general expression,

Rp�z� = rK + rK� �z − zK� + c1�z − zK�2 + c2�z − zK�3, �5�

where c1 and c2 are constants that depend on rA, rA� , rK, rK� ,zK, and zA. Subscripts K and A denote locations at the keeperexit and at the anode, respectively, and primed quantitiesdenote local slopes. The slope rA� is taken to be zero. At the

keeper exit Rp�z� is set equal to rK with rK taken to be thekeeper radius. Two quantities remain undefined in Eq. �5�:the slope rK� �a measure of the electron divergence at thekeeper exit� and rA which is the bulk-plasma radius atthe anode. The first is estimated using two cases aimed atassessing the sensitivity of the solution on the assumed elec-tron divergence. Case 1 assumes small electron divergenceand sets rK� such that the computed ratio of the relativee-i drift velocity ud over the electron thermal speed uT,e

= �eTe /me�1/2 is less than or equal to 1.3 at the centerline.This is an ad hoc definition that sets a lower limit on rK�based on the notion that when ud /uT,e�1.3 the plasmawould become Buneman unstable, which would in turn leadto �visible� discharge disruptions. Such disruptions are notobserved during nominal cathode operation. The value of 1.3is discussed further in Sec. III C 2. Case 2 assumes highdivergence and corresponds to the maximum value of rK� forwhich Rp�z� is a monotonically increasing function of z. Thisdepends on the value of rA, which is chosen such that at least99% of the current at the anode �z=zA� is contained within anarea of radius equal to the anode radius RA. The two cases forRp�z� are plotted in Fig. 8.

The resistivity ratio Rs��m /� for the two electron di-vergence cases is shown in Fig. 9. Cases 1 and 2 bound thevalue of the measured resistivity to be somewhere betweenthree and fifty times the classical resistivity. The higher val-ues of the resistivity correspond to case 2 which is morerealistic since case 1 predicts values of the relative driftMach number,

M �uduT,e

, �6�

that near the value of 1.3, as shown in Fig. 9. Larger valuesof Rp�z� lead to higher values of �m. For example, a uniform�rather than Gaussian� profile of je,z�r ,z� in the plume andRp�zA�=RA predicts as much as 300 times the classical resis-

FIG. 7. Representative profile assumed for the �normalized� axial currentdensity of the electrons, je,z�r ,z� / je,z�0,z� as a function of radius, atz=4.5 cm �=zK�.

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tivity, which suggests a high sensitivity of the result on thechoice of je,z�r ,z�. However, although smaller values ofRp�z� lead to smaller �m they also yield unrealistically highvalues of M. Thus, our conclusion from the simplified exer-cise in this section is that regardless of the sensitivity of �m

on the choice of je,z�r ,z�, any realistic electron current den-sity profile predicts resistivity values that are much higherthan those obtained �classically� by Eq. �3�.

C. Two-dimensional numerical simulations

1. General description of the computational two-fluidplasma and neutral gas models

The two-dimensional numerical model solves the con-servation laws for the plasma and neutral gas self-consistently as deemed necessary by the close coupling thatexists between them through ionization. With the two-fluidplasma model we seek time-averaged solutions with no in-tent to resolve the dynamics from possible particle-wave in-teractions. The main goal of the fluid simulations performedhere is to support the conclusions of our analytical estimateswhich have been based largely on the data. The region ofspace simulated by OrCa2D-II is illustrated in Fig. 1. Theconservation laws and related boundary conditions have beenpresented in detail in previous articles8,9 and will only bedescribed briefly here.

In OrCa2D-II the continuity and momentum equationsfor the ions and electrons are solved directly to yield thefollowing main plasma variables: plasma particle density, ionand electron current densities, and the plasma potential. Thecontinuity equations account for ionization of the neutral gas.In formulating the momentum equations for the plasma, weassume that if wave motion is present the waves quicklyquench into low-frequency plasma turbulence, an assumptionthat is based largely on the emissive probe measurements

�see Sec. II A�. The main assumption that follows is that theturbulence acts simply as an effective enhancer of thecharged particle collision cross section. If the collision meanfree path �s between s species scales with the product oftheir thermal velocity uT,s and the time between collisions s,then for ions and electrons �i��Ti /Te�2�e, so the ion-ion�i-i� mean free path �mfp� can be smaller than the electron-electron �e-e� mfp when the ions are cold relative to theelectrons. It is common for plasma turbulence collisionalmodels to yield effective collision times for the electrons thatare at least one to two orders of magnitude lower than theclassical �Coulomb� values. In our simulations we find thatdownstream of the orifice conical section the classical mfp�e is �0.1–50 cm while the turbulent model yields�0.01–0.8 cm. Since Ti /Te�0.01 then the i-i mpf associ-ated with the turbulence model is �i5�10−3 cm, with thesmaller values occurring closer to the keeper exit. For refer-ence, at the exit of the keeper channel �rK=0.238 cm� thecharge-exchange mfp between ions and neutrals is�0.04 cm. In the plume the scale length L associated withthe electron number density gradient �n �the steepest of allother plasma gradients� is found to range between 0.1 and10 cm, where L has been defined as n / �n. Since �s is muchsmaller than the characteristic lengths of the problem, weseek steady-state transport-dominated solutions using mo-mentum conservation laws that exclude the inertia terms��nmu�s /�t+� · �nmuu�s for both electrons �s=e� and ions�s= i�.

The electron temperature Te is obtained directly from theelectron energy equation. The energy equation includes ther-mal diffusion, energy losses due to ionization, and the workdone on the electrons by the electric field. It can easily beshown �e.g., see Spitzer14� that when two groups of particles1 and 2 are at different temperatures but each group is char-acterized by a Maxwellian velocity distribution, the time toreach thermal equilibrium is proportional to eq

��A1A2 /n2 ln ���T1 /A1+T2 /A2�3/2 where A�m /mp �mp

=proton mass� and ln � is the Coulomb logarithm. When thetwo groups are ions and electrons then �e-i�eq= �mi /2me�e

FIG. 8. Two cases for the effective radius of the bulk plume plasma in theplanar anode test used to estimate analytically �to order of magnitude� theeffective rise of the resistivity according to the measurements. The self-consistent result from the two-dimensional numerical simulations is alsoshown.

FIG. 9. Analytical estimates of Rs and M using the planar anode experimentmeasurements along the axis of symmetry and the assumed electron currentdensity profiles.

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which is much longer than the equilibration time between theheavy species �ions and neutrals� �i-n�eq�in if classicaltransport is assumed. At the keeper exit, for example,�e-i�eq/�i-n�eq�100. So the ions and neutrals are assumedto be in thermal equilibrium at temperature T�=Tn=Ti�, and asingle equation is employed for the conservation of energy ofthe heavy species. It is noted here that the approximation onthe temperature equilibration is based on classical �Coulomb�collision times, with no drift between the two particlegroups. If the effect of turbulence is to reduce these collisiontimes, then thermal equilibration between ions and electronswill occur faster and Ti /Te will not be as small. A largerTi /Te would in turn increase Landau damping of the waves.On the other hand the relative drift between ions and elec-trons will tend to drive Ti /Te to lower values because the twogroups do not stay at the same location long enough toachieve thermal equilibrium. The drift would also tend tosustain the turbulence which drives the higher electron tem-peratures through Joule heating. These competing mecha-nisms have not yet been taken into account by the two-fluidmodel which would require �in part� that the ions are de-scribed by their own energy equation. Thus, the present nu-merical results give an upper bound of the effect of the as-sumed turbulence on the local plasma properties.

The magnetic field has been excluded from the simula-tions. Therefore, the numerical results are compared onlywith those measurements that have been obtained in the ab-sence of an applied magnetic field. Sheath boundary condi-tions are applied at the cathode wall boundaries and includeemission from the emitter.8 It is assumed that all availablecurrent is collected by the anode, so no ion and no electronfluxes are allowed out of the free-flow boundaries in theplume. Both the electron pressure and temperature gradientsare also assumed to be zero at these boundaries. Simulationswith smaller computational regions for the plume �by a fewcentimeters�, which in turn reduced the anode diameter andproximity of the free-flow boundaries to the keeper exit,have shown that the solution in the near plume is relativelyinsensitive to such changes. All simulation results presentedin this paper have used a planar anode and have assumed thatthe plasma density is uniform along the anode boundary. Thevalue of the plasma density at the center of the anode hasbeen specified according to the measurement. Additional cal-culations that employed a uniform electron current densityboundary condition at the anode have shown little change inthe near-plume results.

The neutral gas density is determined by the neutral gascontinuity equation, which includes the ionization sourceterm. Inside the cathode the neutral gas satisfies the con-tinuum assumption so the fluid momentum equation issolved to yield the neutral gas flux. However, for the cathodestudied here the mfps for neutral collisions are found to becomparable to the characteristic dimensions inside the orificeand can be many times the keeper diameter in the keeperregion. For the simulation of the rarefied regions, it is cus-tomary to implement particle methods such as direct MonteCarlo simulation or particle-in-cell. In the highly collisionalregion of the high-current ��25 A� cathodes studied here,particle methods would require an excessive number of par-

ticles and consequently long computational times. Due to thelarge system of equations that must be solved in the hollowcathode, computationally inexpensive approaches must bedevised which sacrifice some accuracy for the sake of com-putational speed and smoothness of the solution. InOrCa2D-II this is done by assuming that the gas particlesexpand freely in straight-line trajectories from a predeter-mined boundary, the “transition line,” which is chosen in thepresent simulations to be close to the exit of the cylindricalorifice section. Beyond this line a collisionless region is as-sumed for the neutrals. The fundamental assumption in thisregion is that the gas emanates from surfaces with a positivenormal velocity and a thermal spread perpendicular to thatsurface. Then at large distances from the surface the perpen-dicular velocity spread is reduced due to geometrical selec-tion. The flux of particles is thus only altered by either anionization event or an encounter with walls. Particles impact-ing the walls are allowed to return back to the computationalregion with a thermal speed that is determined based on thelocal wall temperature. Under these assumptions the problemthen becomes one of computing all the geometrical viewfactors �but only once� and keeping track of particle fluxesassociated with the various wall boundaries. The collision-less region is “fed” with the solution of the fluid momentumequation inside the cathode. The two regions comprisingfluid and collisionless neutrals are coupled at the transitionline, which in the present simulations is located inside theorifice channel. It is found that the classical e-i collisionfrequency is approximately one order of magnitude higherthan the e-n collision frequency inside the keeper channeland several inner diameters downstream of it. Moreover, theadditional collision frequency needed to account for theplume measurements always exceeds the classical e-i colli-sion frequency by about two orders of magnitude in someregions of the plume. It is therefore postulated that our con-clusions regarding the anomalously enhanced resitivity willbe negligibly altered by a more rigorous neutral gas model.Past comparisons with a full-fluid approach for the neutralgas model have shown only factor-of-two differences in theneutral gas density.

The conservation equations are descretized using finitevolumes. All vectors are defined at cell edges and all scalarsare defined at cell centers. The system is solved in a time-split manner with the plasma equations strongly implicitized.The neutral gas continuity and momentum equations aresolved explicitly. The numerical approach for the neutral gasfluid momentum equation uses an upwind finite volumescheme by applying the Godunov first-order upwind fluxesacross each edge with no flux limiting.

2. Numerical simulation results and the effectsof anomalous resistivity

Wave motion and loss of collisionality between electronsin the plume region can lead to deviations from the Maxwell-ian EEDF and to “runaway” electrons. If the waves becometurbulent and the deviations from the Maxwellian EEDF aresmall, however, it may be possible to model the electron-wave effects by appropriate forms of the effective collisionfrequency that are based on the turbulence spectra.15 The

063301-7 Mikellides et al. J. Appl. Phys. 101, 063301 �2007�

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effective or “anomalous” collision frequency can then be in-corporated into the fluid conservation laws to assess the mac-roscopic effects of the wave motion on the plasma. The workpresented here considers only the current-driven two-stream�electrostatic� instability, but this should not imply that otherinstabilities are not possible in this device. The reason for thechoice of the two-stream instability in the present study islargely based on our estimates of the e-i relative drift andTe /Ti ratio in the vicinity of the keeper exit. We find valuesthat would be high enough to excite the instability in a ho-mogeneous, collisionless, and nonmagnetized plasma. More-over, the emissive probe measurements show the presence oflow-frequency plasma oscillations �in the range of�100 kHz�, which seems to persist within the same low-frequency range both in the presence and the absence of an�mainly axial� applied magnetic field. Although such fre-quencies are many times associated with ion acoustic waves,a range of other electrostatic instabilities may exist in thisdevice. An in-depth study to identify all possible modes hasnot yet taken place. Thus, the main goal of the present nu-merical study has been to confirm the order-of-magnitudeenhancements in the resistivity needed to explain the mea-surements, in support of the conclusions made in Sec. II B.

The current-driven two-stream instability may be excitedunder a wide range of Ti /Te and drift Mach numbers M. Thewell-known Stringer diagram16 shown in Fig. 10 illustratesthe M −Ti /Te− relationship �Fig. 10� for the full wavespectrum in hydrogen �where � max/�pi�. In the near-plume region of the cathode �operating with xenon�, two-dimensional numerical simulations clearly show that classi-cal resistivity yields Ti /Te ratios lower than 0.04 �Fig. 11,top� and M ratios that can be as high as 4 �Fig. 11, bottom�.Compared to maximum values of the linear growth rate for

streaming type of instabilities in a homogeneous plasma, it isalso found that electron �classical� collisions can become suf-ficiently rare in the plume region to allow for the growth ofsuch instabilities.

The current-driven ion acoustic waves become unstablewhen the relative e-i drift velocity exceeds the ion acousticspeed Cs��2e�Te+Ti� /mi�1/2, but is less than the electronthermal speed uT,e. This instability also requires that Te�Ti.Treuman17 proposed that the well-known Sagdeev ion acous-tic anomalous collision frequency,

�A/S � 10−2Te

Ti

ud

Cs�pi, �7�

applies under the condition ud /Cs6 to avoid Landau damp-ing of the waves. The plasma is subject to the electron-iontwo-stream �Buneman� instability if M �1.3.18 It is notedthat for xenon, ud /Cs=6 implies M =0.017 so the ion acous-tic instability can be excited at much lower electron driftswhen the condition Te�Ti is satisfied. When M �1.3 theanomalous frequency �A/B is approximated here based on themaximum growth rate which is proportional to19

�A/B � �peme

mi�1/3

, �8�

assuming single ions �more accurately the maximum growthrate includes a multiplication factor of 31/2 /24/3=0.687�.Various authors have also proposed different exponents forthe mass ratio, namely, 0.5 �Ref. 17� and 2/3.20 The lattervalue was proposed by Hirose to represent the saturationvalue of the Buneman instability �later updated to 0.61 byIshihara and Hirose21�. The frequencies above express twospecific limits of the electron-drift-driven wave spectrum.Based on the measured electron temperature and the com-puted heavy-species temperature, the cathode studied hereoperates in the approximate range of 0.01Ti /Te0.1 �seeregion outlined in red in Fig. 10, left�. A wide range of driftMach numbers is possible in the orifice and plume regionsdepending on the approach used to estimate them. The purelyclassical results from the two-dimensional simulations indi-cate that the ratio could be as high as 4, which is an unrea-sonably high value since the discharge is nominally sustainedin a steady state. The analyses in Sec. II B, which used theaxial data and a range of assumed current density profiles,suggest that M may be as low as 0.2, a more reasonablevalue. For the purpose of assessing the impact of the anoma-lous resistivity effects in the full spectrum of the M −Ti /Te

− diagram it is assumed that Ti /Te=0.01. In this work theanomalous frequency for xenon has been scaled usingStringer’s diagram as follows:

�A � me

mi�1/3

f�M��2�pe,

f�M� � �Meb, M 1.3,

f�M� � 1, M � 1.3. �9�

where �=0.171 and b=1.25, as determined by the variationof as a function of M in Fig. 10 �i.e., for hydrogen� at

FIG. 10. Growth rate of the fastest-growing wavelength in a current-carrying plasma �mi /me=1836� as a function of M �ud /uT,e and the ion-to-electron temperature ratio Ti /Te � is the ratio of the maximum growth rateto ion plasma frequency�. �After Ref. 16�. The outlined region indicatesapproximate range of operation of the hollow cathode orifice and plumeregion plasma.

063301-8 Mikellides et al. J. Appl. Phys. 101, 063301 �2007�

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Ti /Te=0.01; this �� ,b� set is the first of two sets used in thenumerical simulations. It is intended to assess the effects ofenhanced resistivity assuming that the effective collision fre-quency scales with the linear growth rate.

If all conditions for the growth of the instability are metthen in many real cases the waves will saturate into a turbu-lent mode. Therefore, a second set is used in the simulationsin an ad hoc attempt to include a “turbulent” collision fre-quency at low Mach numbers. Specifically, we note thatcontrary to Stringer’s formulation for the maximum growthrate, Sagdeev’s anomalous frequency includes nonlinear ef-fects that are associated with particle-wave interactions.Thus, Eq. �7� predicts a much higher effective frequencycompared to Stringer’s predictions for the maximum growthrate in the ion acoustic range. For example, according toFig. 10, at M �0.06 and Ti /Te�0.01, the maximum growthrate is �0.01 while Sagdeev’s anomalous formula predicts�A/S /�pi��0.01/21/2��Te /Ti��M��mi /me�1/2=1.8, which ismore than two orders of magnitude higher than . In theiroriginal monograph, Sagdeev and Galeev acknowledge thatthe effective collision frequency is higher than the growthrate.22 By contrast, in the Buneman limit Stringer’s diagrampredicts �2.2, and Eq. �8� gives �A/B /�pi��mi /me�1/6

=3.5, which are much more comparable values. The litera-ture includes evidence of several experimental verificationsof Sagdeev’s formula �e.g., see Refs. 19 and 23�. We there-

fore proceed to present results using Eq. �9� for two cases, asplotted in Fig. 12. The first case uses �=0.171 and b=1.25based on Stringer’s diagram for the growth rate and the sec-ond uses �=0.171 and b=0.25. The value of b for the secondcase is chosen such that �A approaches �to order of magni-tude� the values predicted by Eq. �7� at low Mach numbers.

The results with anomalous resistivity for the two afore-mentioned cases �b=1.25,0.25� are compared with the mea-

FIG. 11. Results from two-dimensional numerical simulationsthat assumed fluid �inertialess� elec-trons and classical resistivity. Top:Ti /Te. Bottom: M.

FIG. 12. Models of the anomalous resistivity dependence on the relativeelectron drift Mach number M �ud /uT,e assumed in the two-dimensional�fluid� numerical simulations.

063301-9 Mikellides et al. J. Appl. Phys. 101, 063301 �2007�

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surements �at Bapp=0� in Fig. 3 �density�, Fig. 4 �electrontemperature�, and Fig. 5 �plasma potential�. Several qualita-tive and quantitative points may be made based on thecomparisons. Most evident is the fact that by comparison tothe classical result, the addition of anomalous resistivity isthe only means by which the time-averaged electrontemperatures—measured to be much higher in the plumecompared to the cathode interior in both anode geometries—may be qualitatively reproduced by the fluid model. Thesame argument applies to the steep plasma potential risemeasured inside the orifice and near-plume regions of theconical anode configuration. The anomalous model predictsthe same qualitative behavior of the plasma potential for bothchoices of the parameter b, namely, a steep rise of the poten-tial in the orifice and near-plume regions. The measurementsshow a more gradual increase of the potential with distancefrom the keeper in the case of the planar anode compared tothe conical anode. Since the present capabilities of the nu-merical model do not allow for the simulation of the seg-mented conical anode geometry, it has not been possible toassess the impact of that geometry on the near-plume plasmavia numerical simulation. It is possible that the proximity ofthe first few anode rings to the keeper exit may have led tohigher divergence of the electrons, equivalent to a highervalue of Rp�z� in the analysis of Sec. II B. Thus, although thesimulations self-consistently compute the electron currentdensity in two dimensional the numerical results must beviewed within the context of the aforementioned limitationsassociated with the anode geometry. As a response to theselimitations the analyses of Sec. II B assumed Gaussian pro-files with a relatively small divergence �e.g., Rp�z=zA��3.5 cm compared to RA=7.5 cm�, chosen in an attempt toidentify a range for the value of the effective resistivity. Spe-cifically, the profiles in Fig. 13 show that the numerical simu-lation predicts a near-Gaussian axial current density profileclose to keeper exit, similarly to what was assumed in theanalyses of Sec. II B. Farther downstream, however, the nu-merical simulation predicts a steeper divergence of the cur-

rent density, “opening” up to an almost uniform profile be-yond z�8 cm. The radius Rp�z� is compared to the analyticalprofiles of Sec. II B in Fig. 8. The evolution to a uniformprofile near the anode is a consequence of the condition im-posed on the plasma density. The density is set to be uniformat the anode boundary. The numerical results for the resistiv-ity ratio Rs and drift Mach number M are shown in Fig. 14.Together the results from Secs. II B and II C bound the tworatios: �100�Rs�3 and 1.3�M � �0.05.

III. CONCLUSIONS

Analyses of axial probe measurements and numericalsimulations of a 1.5 cm diameter hollow cathode operatingbetween 25–27.5 A and 5.5 SCCM of xenon flow suggestthat the heating of the electrons in the cathode-keeper orificechannel and near-plume regions cannot be explained by clas-sical transport and Ohm’s law. The simulations with classicalresistivity show that the relative e-i drift is several times theelectron thermal speed. The latter result supports the hypoth-esis that electrostatic instabilities may be excited in the ori-fice channel and/or near-plume regions of the hollow cath-ode. Emissive probe measurements capable of detectingfrequencies up to 1 MHz show plasma potential oscillationswith the highest amplitudes attained in the range of 100 kHz,although oscillations with lower amplitudes were observedwith frequencies up to the probe limit. Simulations assumingfluid electrons with approximate scalings of the anomalousresistivity that have been based largely on existing growthrate and turbulence models predict parts of the measurementssuch as the elevated electron temperatures and plasma poten-tials. The density comparison is also greatly improved com-pared to the classical result. However, kinetic effects havenot yet been quantified in this environment by direct mea-surement of the EEDF, and thus the deductions from the fluidsimulations must be viewed in combination with the mea-sured plasma fluctuations and the simplified analyses. Thecombined results from the simplified analyses and numericalsimulations bound the effective resistivity to be between 3and 100 times the classical value in the plume region of thecathode.

FIG. 13. Radial profiles of the axial electron current densityje,z�r ,z� / je,z�0,z�, as computed by OrCa2D-II. Also shown are the profilesassumed in the analyses of Sec. II B. The 1/e �dotted� line defines thebulk-plasma radius Rp.

FIG. 14. Numerical results for Rs and M from the OrCa2D-II simulationsfor two cases of the anomalous resistivity.

063301-10 Mikellides et al. J. Appl. Phys. 101, 063301 �2007�

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ACKNOWLEDGMENTS

The research described in this paper was carried out bythe Jet Propulsion Laboratory, California Institute of Tech-nology, under a contract with the National Aeronautics andSpace Administration.

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063301-11 Mikellides et al. J. Appl. Phys. 101, 063301 �2007�

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