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Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, July 1996A9637288, AIAA Paper 96-3192

Theory of oscillations and conductivity for Hall thruster

Vladimir I. BaranovKeldysh Research Center, Moscow, Russia

Yury S. NazarenkoKeldysh Research Center, Moscow, Russia

Valery A. PetrosovKeldysh Research Center, Moscow, Russia

Anatoly I. VasinKeldysh Research Center, Moscow, Russia

Yury M. YashnovKeldysh Research Center, Moscow, Russia

AIAA, ASME, SAE, and ASEE, Joint Propulsion Conference and Exhibit, 32nd, Lake

Buena Vista, FL, July 1-3, 1996

Plasma oscillations are of primary importance in accelerators with closed electron drift (ACDs) since theydetermine the conductivity and diffusion across the magnetic field and, therefore, the efficiency of ACD operation.Thus, it is impossible to clearly understand ACD operation without having a clear and comprehensive picture ofthe oscillations in them. Plasma oscillations in ACDs usually fall into ionization (10 exp 4 to 10 exp 5) Hz), transit(10 exp 5 to 10 exp 6 Hz), electron-drift (10 exp 7 Hz), electron cyclotronic (10 exp 9 Hz), and Langmuir (10 exp8 to 10 exp 10 Hz) oscillations. The three former types are integral properties of the ACD plasma. Themechanisms of those oscillations, proposed just after the development of initial ACDs in the 1970s, contain anumber of wrong assumptions and essential mistakes in the physical models, so they seriously contradictexperimental data. This work aims to elaborate on a comprehensive universal mechanism of ACD oscillations thatwould account for not only the known types of oscillations, but also the phenomena accompanying them, such aselectron conductivity, abnormal erosion, and plasma jet structure. (AIAA)

AIAA-96-3192

THEORY OF OSCILLATIONS AND CONDUCTIVITY FOR HALL THRUSTER

Vladimir I. Baranov, YuryS. Nazarenko, ValeryA. Petrosov, Anatolyl. Vasin and YuryM. Yashnov*

Keldysh Reserach Center., Moscow, Russia

1. Introduction

Plasma oscillations are of primary importance inaccelerators with closed electron drift (ACDs); afterall, namely they determine the conductivity anddiffusion across the magnetic field, therefore, theefficiency of ACD operation. That is why it isimpossible to clearly understand ACD operationand, therefore, to develop ACDs, not having a clearand comprehensive picture of oscillations in them.

Plasma oscillations in ACDs usually fall intoionization oscillations (104 to 105 Hz), transitoscillations (105 to 106 Hz), electron-driftoscillations (107 Hz), electron cyclotronicoscillations (109 Hz) and Langmuir oscillations (108

to 1010 Hz) 1. The two latter types of oscillationsare characteristic for any magnetized plasma andtheir nature is known. The three former types ofoscillations are integral properties of the ACDplasma, so they are an important subject of studiesto be conducted by ACD developers.

The mechanisms of those oscillations, proposedjust after the development of initial ACDs in the1970s, contain a number of wrong assumptionsand essential mistakes in the physical models, sothey seriously contradict experimental data, inparticular, as follows:

1. In the existing models of ionizationoscillations in the gas discharge, the necessaryconditions for their realization are either thepresence of an electric field as an electron energysource, or the presence of metastable states ofatoms 2. As the model 3 for ionization oscillationsin ACDs, employing unmagnetized ions as theplasma charged component, has neither of thoseconditions, so the assertion that this numericalmodel yields oscillations, what is more, of anecessary frequency, sounds dubious. Naturally,the well known ACD ionization oscillationdependence on the magnetic field, in particular,

the existence of an oscillation threshold, is leftbeyond this model.

2. The comparativity of the ion ACD channeltransit time with the reverse frequency of MHz-frequency oscillations recorded has resulted in adeclaration of a mechanism for transit oscillations4.The oddity of the basic background of the model,that the oscillations are initiated by ions when flyingthrough the ACD channel, consists in the twofollowing factors: a) at the existing plasmaparameters in the ionization channel, the ion flowcannot excite any oscillations (as the ion velocity isless than the phase velocity of any wave) 5; and b)to generate transit oscillations in a system in theclassic meaning of this word, a rather "rigid"connection is required between its boundaries6,which does not exist in ACDs.

3. The mechanism of an azimuthal electron-driftwave1 running along the whole channel at avelocity, several times lower than the drift onecE0/B 7, found only slight justification only for theacceleration zone where £„ ^ 0, and did not yield agood quantitative agreement with experimentaldata. Until recently, the mechanism of waveexcitation in the ionization zone (£„ = 0) wasunclear. So an interesting situation has appearedrelating to the understanding of the mechanism ofthe electron transfer to the anode: they said thisprocess is stimulated by oscillations 1, but theiradvent and, all the more, influence on the electrontransfer were still a puzzle.

The investigations recently conducted at theKeRC (NIITP) Electric Thruster Laboratory allowedelaboration and substantiation mechanism of thoseoscillations 8'11 that adequately describe processesoccurring in ACDs.

This work is aimed to elaborate^ Orcomprehensive universal mechanism of ACDoscillations that would account for not only the

Copyright © 1996 by Petrosov. Published by the American Institute of Aeronautics and Astronautics, Inc. withpermission.

known types of oscillations, but also phenomenaaccompanying them, such as electron conductivity,abnormal erosion, plasma jet structurization, etc.

2. Nomenclature

n. A . ". > v, > ̂ '<•> v.

"»,M,u

E,Ea,B

e,c

Tc

v,£>

U

oa

G,m

W

- cylindric coordinates;

- electron, ion and atomconcentrations and velocities,respectively;

- plasma and Larmourfrequencies;

- electron and ion masses andtheir ratio;

- general electric andstationary electric andmagnetic fields;

- frequency and its real andimaginery components andwave vector;

- charge and light speed;

- electron temperature;

- collision frequency anddiffusion factor;

- acceleration and ionizationzone lengths;

- electric potential;

- ionization parameter;

- flow rate parameter and flowrate;

- wave velocity.

3. Initial Equations and Methods for TheirSolution

The equation system, describing the dynamicsof three-component plasma (containing electrons,ions and atoms) in the ACD channel, includescontinuity and motion equations for eachcomponent and Poisson's equation:

dt m mc m n

otV. = const;

It is assumed therewith that the ions are "cool"and unmagnetized.

The system is solved in a standard way. First,the stationary system is solved, i.e., all d/dt = 0,which yields A° (A° is any variable). Then thesolution of the initial non-stationary system ispresented in the form of A = A° + A', where A' is asmall deviation. After that, the solution of the non-stationary system for A' is found in the form ofA' a exp {-i((at- kr)}, with the assumption of slightnon-uniformity (Vn = 0).

As a result, a dispersion equation F(co,/fc) = 0 isderived in which the required condition of theoscillations build-up is ca, > 0. To find theoscillation thresholds an assumption of weak non-uniformity is quite reasonable. However, if that isnot true, then k(r] should be substituted fork = const.

Let us show the effectiveness of such system asapplied to processes in ACD channel plasma, in sodoing we will consider separately the accelerationzone (Ea&Q) and the ionization zone (E0 = 0),analyzing all frequencies from high to low ones,observing the hierarchy: a>,,nt,<ol,v,cai,,Sll.

We shall consider the movement only along thecoordinates "q>" and "z".

The following important aspects ofapproach being stated should be marked out.

the

1. In spite of the advantages in thecomprehensive description of the phenomenaoccurring in plasma of the kinetic model incomparison with the hydrodynamic one, thepreference is given to the latter because:

- in the kinetic model requires a detailedknowtedge-of-L the type- ef-th&eieetrort<Jistrrbatiorrfunction, and as applied to ACD this problem isonly now is finding the right interpretation;

American Institute of Aeronautics and Astronautics

for qualitative understanding of thephenomena, which is the purpose of this work, thehydrodynamic description of magnetized plasma inthe form of a two-liquid model is undoubtedly moreillustrative;

- the oscillations with frequencies less than Q,are investigated.

2. To avoid the unjustified losses of solutionsduring consideration of a specific frequency range,the restrictions, related to this range, are mainlyintroduced not into the initial equations but to thederived solution.

3. The absence of correlation between thecoordinate and the electron velocity in ACD plasmapermits averaging over the ensemble of particles,but not over the time, in finding the electron flowvelocity, which allows the concept of the driftvelocity cE0/B to be introduced for the time

4. Acceleration Zone

We consider the plasma to be fully ionized anduniform, and E0 and Bto be constant with "z".

4.1, Foro), comparable with co; and higher thanco, , and on the assumption that the drift azimuthalelectron velocity is higher than the thermal one,Eqs. (1) are transformed as follows:

m, (2)

A<D = -4 ne( «,-«,).

Not changing the essence of the phenomenawe assume that the ions pass through theacceleration zone with a constant velocity

Then we obtain a dispersion equation from (2):

r+,———4-T-1- (3)CO

where V°=cEa/B.

4.1.1. For co > ky* in the coordinate system,which is moving with the velocity V°, taking intoaccount that co,2/co* = u = 10"!, from (3) we obtain atrivial solution in the form of the upper hybridfrequency:

(4)

The consideration of the non-uniformity of Bwith z results in the advent in the dispersionequation of an additional term («VFe), which yieldsformation of an instability area in the ACD channel,i.e., the area of a decrease in the magnetic fieldaside the anode. This effect has been validatedexperimentally 14: the ACD exit area is an intensivesource of electron cyclotronic oscillations.

For co < k^°, it is more convenient to presentsolution (3) in the form:

co = kff ± (5)

The build-up of oscillations in (5) is determinedby the condition that:

^<,<V^L (6)

or, for Q, = 2-10V,co,=(3-5)-10I°-s-!, which arecharacteristic for ACDs 0.3 < \ < 3 (mm).

As &9 > kz, it is easy to understand that thedisturbance front will be propagated basically alongthe axis "z", the inclination angle of the disturbancefront to the "z" axis being of arctg kjk^ » 15°.

So, a stable spatial periodic plasma structure isformed in the acceleration zone, the structureappearing in experiments to be similar to theerosion structure on channel walls from the verybeginning of ACD operation (Fig. 1a). Themechanism of this phenomena is described in 8i 9.

It should be noted that Eqs. (4) and (5) presentall the four solutions of Eq.(3) (in (4), the sign ± isomitted).


Fig. 1. Erosion structure on acceleration channel inner wall, HTT-100

4.2. For co <Q,. Eqs. (1) change to:

0 = —

dt

(7)

=-E;n.=n.' m, '

4.2.1. For azimuthal movement, Eqs. (7) yieldthe following dispersion equation:

(8)

The condition of the oscillation amplification in

co'

(8) is V° > — and for characteristic conditionsW

of ACD is always true, that is, an electron-driftwave propagates along the azimuth in theacceleration zone. The frequency of the wave lies

in the range u — -<co<£F°. For 5 = 102 Gs,

v, = 10V,£0 = 300 V/cm from (8) we findco < 10V , and the wave velocity fF= 107 cm/s.

That is in agreement with the experimental factthat the velocity of electron-drift wave is less thanthe drift velocity with £„/ B by several times andthis cannot be otherwise, because the condition of

cits generation is Fe° = c— - > W .

B

4.2.2. Now let us show the mechanism ofelectron conductivity amplification across themagnetic field to the anode, which is initiated bythe electron-drift wave, intentionally complicatingthe situation by considering the plasmacollisionless.

As the frequency of theelectron-drift wave is muchlower than the electronLarmour frequency, theelectrons in the Ev waveaccelerating phase will drift inthe crossing constant fields Ev

and B away from the anode,and the electrons in thedecelerating phase, towardsthe anode. This movement ofthe electrons along the axis "z"will cause the advent of the

electric field EJt adding the field £0 and havingsuch a direction, that the azimuthal electron driftvelocity c(E0 + Elt)/B will increase in theaccelerating phase and decrease in thedecelerating phase. As a result, the electron willgroup in the area where the wave field Ev = 0 justin zero, where the field switches from acceleratingto decelerating. And as the condition for excitationis V*>W, the grouping will take place in thedecelerating phase of the wave, i.e., the number ofthe electrons moving against the E0 field (i.e.,towards the anode) will exceed the number ofthose moving along the Ea field at any time.

5. lonization ZoneWe consider the plasma to be nonuniform along

"z" and £„ = 0.

5.1.1. For co < 00, system (7) dispersion equation(8), following from the system are valid, becausethe plasma is uniform in the azimuthal direction.But now

m. r. i! + v/ (9)

AH computations for the acceleration zone arevalid here, too, but with the following exceptions:

- the electron-drift wave is generated by theazimuthal current of electrons, moving in thenonuniform magnetized plasma (Vn° is ananalogue of £0 - field);

- although the current flows along the azimuth,there is no electron motion in this direction;

- co »co, «107 s"1 .


5.1.2. The mechanism of electron diffusionamplification towards the anode across themagnetic field, stimulated by the electron-driftwave, is the same as the conductivity mechanismin Section 4.2.2. So, excluding the qualitativeaspect of the mechanism, we shall show itsquantitative estimation.

The energy, required for wave drive, isextracted from the energy of the azimuthal current.Let us introduce an average effective intensity Etf

of the wave field, which characterizes the processof energy extraction, and find it.

As cor * co,, the energy being extracted for theperiod of oscillation is comparable with themagnitude of the energy proper and is equal tom.'VT'• Setting the spatial period /„ of oscillationsequal to lf = V° I Q€

3, we shall obtain

where F,° is taken from Eq. (9).

We shall consider, that an electron drifts to theanode in the crossed E and S-fields, so its

velocity Vd = -c-^ —f-, oreB rf

D=cTeB. (10)

Vd is estimated to be Vd = 5- 1 06 cm/s.

Therefore, even in weakly-nonuniformmagnetized plasma the electron moves across themagnetic field in the direction V«° at a velocityproportional to 1/B.

5.2. For frequencies co < <zna, the ions andelectrons move as a whole and system (1) takesthe following form:

—-+V(noFa) = G - a.n,na; Ve = const.(11)

So the condition of excitation of ionizingoscillations is written as an inequality 11

Following 11 , analysis of (12) allows one todetermine a non-oscillation region in a three-dimensional configurational space in the followingcoordinates: magnetic induction B, externalpotential U, flow rate m, see Rg. 2. Just in thisregion the ACD will be of the optimum operatingconditions.

(12)

Rg. 2. Area without large-amplitude oscillations

6. Plasma Jet

The objective of this small section is to showthat oscillations in ACDs can cause new effectsthat would manifest themselves far away from theACD.

When recording an angular distribution of theion current in the ACD-generated plasma jet,azimuthal nonuniformity in the distribution wasdetected (Fig. 3 15). When the vacuum conditionswere deteriorated to pressures at which the ionmean free path becomes less than the distancebetween the thruster and sensor (60 cm), thenonuniformity in the distribution of the ion currentdisappeared (Fig. 3).

The processing of ion current nonuniformityparameters performed by us revealed a distinctperiodic structure (Fig. 4) with a period of about1 cm, which corresponds to the increased periodof the stationary plasma structure (6) occurring inthe acceleration zone and responsible for abnormalerosion. The increase of period is a result of jetangular divergency in which one can easily beconvinced.

Qualitatively, the mechanism of jet structuring isas follows.


1000Discharge Voltage 300 VDischarge Current 16 AAnode Row 12.6 mg/sCathode Row 0.7 mg/sMagnet Current 4.5 A

3x10 TorrSx10'5 Ton-

Cathode on sidecorrespondingto positive angles

.01-100 -75 -25 0 25

Angle, degrees100

Rg. 3. Effect of facility pressure on HTT-160 ion current profile 15

i.o

I

<5§

L = 60cm

Angle, degreesRg. 4. Ion current distribution for large angles HTT-160.

Pressure 3-10"6 TorrDistance from a thruster 60 cm 1S

loo

The ions involved in formation of a plasmastructure in the acceleration zone (see Section2.2.1), when leaving it, gain an azimutal velocitycomponent in the structure fields. During thesubsequent free motion of the ions whose chargeis instantly compensated by electrons for the timeof about co~', as the ions are moving, the obtainedvelocity modulation will change over to densitygrouping.

The structure is hard to be observed at thecenter of the jet as the ion trajectories cross eachother. Grouping in a kinetic tube is a processsimilar to the above-described one.

Determination in the vacuum because of ioncollisions (see the ion motion equation in (1)) willcause chaotization (disappearance) of thestructure.

ConclusionThe comprehensive ACD oscillation theory

suggested pioneered the following:- adequate description of the known types of

oscillation;

- determination of their role in transferphenomena;

- suggest mechanism of such phenomena asabnormal erosion and jet structurization;

- prediction of unknown types of oscillation;

- definition of technical routes for ACDimprovement.


References

1. Bugrova A. I., Kim V. P. Modern State ofPhysical Studies in Accelerators with ClosedElectrons Drift and Extended Zone of Acceleration.- In: Plasma Accelerators and Ion Injectors. M.:Nauka, 1984, p. 107-129.

2. Raizer Yu. P. Physics of Gas Discharge. - M.:Nauka, 1987. - 587 pp.

3. Fedotov A. P., Sveshnikov A. G., Yakunin S.A. Numerical Investigations of NonstationaryProcesses in Plasma Accelerator. MoscowUniversity prepr., Ns 20, 1981, pp. 4 (in Russian).

4. Bugrova A. I., Morozov A. I. Peculiaritiez ofPhysical Processes in Accelerators with ClosedElectron Drift and Extended Acceleration Zone. In:Ionic Injectors and Plasma Accelerators. M.:Energoatomizdat, 1990, p. 42-56 (in Russian).

5. Plasma Electrodynamics/Ed, by A.M.: Nauka, 1974. - 720 pp.

Ahiezer.

6. Nezlin M. V. Dynamics of Beams in Plasma.M.: Energoizdat, 1982. - 264 pp.

7. Janes G., Dotson J. ExperimentalInvestigations of Oscillation and Related AbnormalElectron Diffusion in Hall Accelerators. In: AppliedMagnetic Hydrodynamics. M.: Mir, 1965, p. 235 -259 (in Russian).

8. Baranov V. I., Nazarenko Yu. S., Petrosov V.A., Vasin A. I., Yashnov Yu. M. Anomalous Erosionin Accelerators with Closed Drift of Electrons -IEPC-95-59. Moscow, Russia, 1995.

9. Baranov V. I., Nazarenko Yu. S., Petrosov V.A., Vasin A. I., Yashnov Yu. M. Mechanism ofDielectric Anomalous Erosion on Exposure toPlasma Flow. - Letters to Journal of TechnicalPhysics. Vol. 20, N° 5, 1994 (in Russian).

10. Baranov V. I., Nazarenko Yu. S., Petrosov V.A., Vasin A. I., Yashnov Yu. M. Electron DriftOscillations Outside the Accelerator with ClosedDrift of Electrons. - IEPC-95-062. Moscow, Russia,1995.

11. Baranov V. I., Nazarenko Yu. S., Petrosov V.A., Vasin A. I., Yashnov Yu. M. Mechanism oflonization Oscillations in Accelerators with ClosedDrift of Electrons. - IEPC-95-059. Moscow, Russia,1995.

12. Baranov V. I., Nazarenko Yu. S., Petrosov V.A., Vasin A. I., Yashnov Yu. M. Function ofElectrons Distribution in Accelerators with ClosedDrift of Electrons. - IEPC-95-061. Moscow, Russia,1995.

13. Baranov V. I., Nazarenko Yu. S., Petrosov V.A., Vasin A. I., Yashnov Yu. M. Mechanism forFormation of Electron Distribution Function inPlasma ACD. Letters to Journal of TechnicalPhysics. Vol. 21, Issue 24, p. 38 - 41, 1995 (inRussian).

14. Kirdyshev K. P. High-Frequency Processesin Plasma Dynamic Systems. M.: Energoatomizdat,1982, pp. 144 (in Russian).

15. Sankovic J. M., Haag T. W. PerformanceEvaluation of 4.5 kW SPT Thruster. - IEPC-95-30,Moscow, Russia, 1995.


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