Characteristics and transport effects of the electron drift...

Characteristics and transport effects of theelectron drift instability in Hall-effectthrusters

T Lafleur1,2, S D Baalrud3 and P Chabert1

1 Laboratoire de Physique des Plasmas, CNRS, Sorbonne Universités, UPMC Univ Paris 06, Univ Paris-Sud, Ecole Polytechnique, F-91128 Palaiseau, France2 Centre National d’Etudes Spatiales (CNES), F-31401 Toulouse, France3Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, United States ofAmerica

E-mail: trevor.lafleur@lpp.polytechnique.fr

Received 29 July 2016, revised 16 November 2016Accepted for publication 4 January 2017Published 31 January 2017

AbstractThe large electron E B´ drift (relative to the ions) in the azimuthal direction of Hall-effectthrusters is well known to excite a strong instability. In a recent paper (Lafleur et al 2016 Phys.Plasmas 23 053503) we demonstrated that this instability leads to an enhanced electron–ionfriction force that increases the electron cross-field mobility to levels similar to those seenexperimentally. Here we extend this work by considering in detail the onset criteria for theformation of this instability (both in xenon, and other propellants of interest), and identify anumber of important characteristics that it displays within Hall-effect thrusters (HETs): includingthe appearance of an additional non-dimensionalized scaling parameter (the instability growth-to-convection ratio), which controls the instability evolution and amplitude. We also investigatethe effect that the instability has on electron and ion heating in HETs, and show that it leads to anion rotation in the azimuthal direction that is in agreement with that seen experimentally.

Keywords: Hall-effect thruster, electron drift instability, anomalous electron transport,alternative propellants

(Some figures may appear in colour only in the online journal)

1. Introduction

Hall-effect thrusters (HETs) are one of the most importantplasma technologies used to provide propulsion for a range ofdifferent industrial and research space missions [1]. HETs arecomplex devices that display a surprising richness of physicalphenomena [2, 3], as evidenced by the many different oscil-lations and instabilities present during typical operation.These oscillations span a wide frequency range and includefor example, low frequency (10s of kHz) breathing-mode[1, 4] and rotating-spoke oscillations [5, 6], medium fre-quency (100s of kHz) transit-time oscillations [7], and high-frequency (1–10 MHz) streaming [8] and electron driftinstabilities [9, 10] (see [11] for a more detailed list of knownHET oscillations).

Of the many oscillations, the electron drift instability(EDI) has attracted interest for its possible role in enhancingelectron transport across the radial magnetic field [9, 12]. TheEDI is viewed as a kinetic instability that forms due to thelarge electron drift velocity (relative to the ions) in the azi-muthal E B´ direction, and represents a coupling betweenelectron Bernstein modes and ion acoustic waves. Thisinstability has been observed experimentally [13] and in 1D[14, 15], 2D [9, 16, 17], and 3D [18, 19] particle-in-cell (PIC)simulations. In contrast to other azimuthal instabilities such asfluid gradient-drift modes [20] (10’s of kHz to MHz), or themore commonly known rotating-spoke modes associated withneutral depletion [5, 21] (10’s of kHz), which have largewavelengths (cm’s), the EDI is a short wavelength (mm’s),high-frequency (MHz) kinetic drift instability. Recently [12]it has been shown that the EDI gives rise to an enhanced

Plasma Sources Science and Technology

Plasma Sources Sci. Technol. 26 (2017) 024008 (14pp) https://doi.org/10.1088/1361-6595/aa56e2

electron–ion friction force that can increase the electron cross-field mobility by 1–3 orders of magnitude (compared withclassical electron–neutral or electron–ion Coulomb colli-sions): in agreement with experimental results.

The EDI has previously been studied theoretically withina HET context [10, 22] where the relevant dispersion relationhas been derived and the effect of non-Maxwellian electrondistribution functions and plasma density gradients investi-gated. As a result of the applied magnetic field, the dispersionrelation is quantized and shows narrow wavenumber bandswithin which the growth rate is positive. However, whenconsidering the wavenumbers that can be accommodated bythe finite HET radial channel width, this discrete spectrumtends to be replaced by a continuous spectrum (as verifiedexperimentally [13, 23]) similar to that for ion acousticwaves. This modified ion acoustic type dispersion relation hasnot been studied in detail though, and a number of questionsremain unclear. For example, what are the physical criterialeading to an unstable dispersion relation and hence the for-mation of the instability? Can the instability be damped bycollisions with the residual neutral gas inside and in the near-plume region of the thruster? What is the role of the ion masson the instability, and can different masses influence theinstability threshold? This last point is of particular impor-tance because most HET research has been performed usingxenon, and there is renewed interest in finding alternativepropellants [24–26] that are cheaper and more easily avail-able. Finally, experiments show the presence of periodicerosion patterns [27, 28] on the ceramic walls of HETchannels with wavelengths similar to those for the EDI(∼1 mm). Thus it is natural to ask whether the EDI isresponsible for the appearance of these patterns.

There also remains a worrying apparent discrepancywhen considering the EDI as the cause of the enhancedelectron cross-field transport. All previous fluid or hybridHET simulations require an anomalously high electronmobility in order to get physically meaningful results andagreement with experiment, and this is usually achieved byartificially increasing the electron collisionality [4, 29–36].But if this increased collisionality results from the EDI, thenone would expect that an additional electron heating term(accounting for the power deposition by the azimuthal elec-trostatic wave) would also be needed. Furthermore, if the EDIis strong enough to significantly affect the electron transport,does it have any effect on the ion transport?

In this paper we specifically aim to address the aboveissues by considering in detail the modified dispersion rela-tion for the EDI, as well as the global effects this instabilityhas on the charged particle transport in HETs.

2. Instability characteristics

2.1. Dispersion relation

In [12] we derived the complete dispersion relation of the EDIfor mutually perpendicular electric and magnetic fieldsassuming drifting Maxwellian distributions for the electrons

and ions. To simplify the analysis this was performed using aCartesian coordinate system where the ‘axial’ thruster direc-tion is along the z-axis, the ‘radial’ direction is along the x-axis, and the ‘azimuthal’ E B´ direction is along the y-axis.The resulting dispersion relation agrees with that indepen-dently obtained in [10] (aside from the ion drift that we haveincluded) and is given by

I Zk v n

k v0 1

y ye ce

= - ¢-

´- -b

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( ) ( )

where piq n

0w = is the ion plasma frequency, q∣ ∣ is the

elementary charge, ni is the ion density, 0 is the permittivity

of free space, M is the ion mass, vTiq T

2 i= ∣ ∣ is the ion

thermal velocity, Ti is the ion temperature (in eV),k k k kx y z

2 2 2= + + is the magnitude of the wavevector kwith kx, ky, and kz mutually orthogonal components, ω is thewave frequency, Z is the plasma dispersion function [37],

q nDee

0l =∣ ∣

is the electron Debye length, Te is the electron

temperature (in eV), ne is the electron density, vyeE

x= is the

‘azimuthal’ electron drift velocity with Ez and Bx the ‘axial’

and ‘radial’ magnetic field components, vTeq T

2 e= ∣ ∣ is the

electron thermal velocity, m is the electron mass, k v

k k ky z2 2= +^ , ce

mxw = ∣ ∣ is the electron cyclotron fre-

quency, and In are modified Bessel functions of the first kind.Depending on the plasma parameters, equation (1) can exhibitdiscrete, sharply-peaked, frequency and growth rate bands asa function of wavenumber (see for example [10, 12, 22]).

Because the azimuthal direction of a HET is closed, onlydiscrete values of the azimuthal wavenumber (ky in thecoordinate system used here) can meet this closed boundarycondition. However, since the wavenumbers of interestsatisfy, R2

yl p~ (where R is a characteristic radius of

the thruster), the azimuthal spectrum is essentially continuous.Similarly, the radial wavenumber (kx here) is also discretebecause of the finite width of the radial HET walls. As aresult, the minimum value that the radial wavenumber cantake is, kx R

(where RD is the radial width of the chan-

nel). For kx R

it has been shown previously [12, 22] thatthe sharply-peaked frequency and growth rate bands inequation (1) tend to disappear, and that equation (1) simplifiesto a modified ion acoustic type dispersion relation

= - ¢+ -

- ¢+ -

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

· ( )

Plasma Sources Sci. Technol. 26 (2017) 024008 T Lafleur et al

Here we have now also accounted for the effects of ion–neutral, inn , and electron–neutral, enn , collisions using a con-

stant collision frequency model given by ff

t csn s

s n=¶¶( ) for

each species (where fs is the total distribution function), aswell as allowed for an electron drift in all directions. Note thatin this form of the dispersion relation the only role of themagnetic field is to provide an azimuthal electron drift velo-city. Although we will treat the complete expression inequation (2) when investigating criteria for instability onset, itis useful to obtain an approximate dispersion relation to elu-cidate certain important characteristics. To do this we performa large argument expansion ( 1z ) of the plasma dispersionfunction [37] for the ions, Z 2i e1

2z p z¢ » -z

z-( ) , and a

small argument expansion ( 1z ) for the electrons,Z 2 2i e 2 2i

2z p z p z¢ » - - » - -z-( ) . Thusequation (2) becomes

e in di

= + -+ -

´ -+ - ⎪

⎧⎨⎩⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

( · )·

( · ) ( )

Setting iRw w g= + and solving, we obtain an approximatedispersion relation for the EDI

1, 4R di

· ( )

1 1 , 5

2De2 2

»- -+

⎧⎨⎩

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

· ( ) ( )

where csq T

Me= ∣ ∣ is the ion sound speed. By observing

equation (5) we see that both ion–neutral collisions, and aterm consisting of the electron–ion temperature ratio (whicheffectively represents ion Landau damping), act to damp theinstability. In this first order approximation it is also seen thatelectron–neutral collisions do not contribute to the damping.Since the azimuthal electron drift velocity is very high inHETs, any instability is expected to have kk jy» ˆ, and thus

equation (5) can be further simplified in the limit that, 1T

k v vdi de

- ·( ) and 0inn to give

k8 1. 6

De2 3 2

+( )( )

The wavenumber giving maximum growth, kmax, can easily

be found from, 0k

d y=g , which yields

2. 7max

Del= ( )

From this wavenumber we can then obtain simple approx-imate formulas for the wave frequency, maximum growthrate, and instability phase velocity

» ( )

3. 10R

ysphase

w= » ( )

For typical plasma densities ( 1017~ m−3) and electron tem-peratures (∼20 eV), the instability wavelength, frequency,and phase velocity are about 1mm, 3.5MHz, and3000m s−1, respectively.

2.2. Instability modes

From equations (4) and (5) we see that there are two modes,Rw+, g+, and Rw-, g-. If we pick a wavevector, k1, which gives

0g >+ , then we see that automatically the wavevector, k1- ,will give 0g >- and g g=- +. Consider now the groupvelocity of the instability

1. 11g

2De2 3 2

= +( )

Using the wavevectors k1 and k1- again, we can easily seethat

1, 12g di

De2 3 2l

( )( )

De2 3 2

- ( )( )

( )( )

Similarly, the instability phase velocity in the azimuthaldirection, v

kphaseR

y= w , is also equal for both modes. We

therefore conclude that both modes are in fact the same, andhence that only a single unstable mode is present.

2.3. Instability thresholds

In deriving equation (2) (and subsequent approximations) wehave used a Maxwellian distribution function. In our previouswork [12] we showed that the EDI can grow very rapidly andenter a nonlinear regime before eventually saturating. In thisregime the electron distribution is almost certainly non-Maxwellian, and we implicitly accounted for this by con-sidering a generalized dispersion relation coupled to a con-servation equation for the instability energy density. Thismethod however does not allow the complete distributionfunction to be explicitly determined. In the present section weare not interested in this nonlinear regime, but rather in whatconditions lead to instability, for which it is sufficient to

consider a Maxwellian distribution. If ion–neutral collisionsare negligible, then from equation (5) the growth rate for theinstability is positive only if

11 exp

l l++ -

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

∣ · ( )∣( )

This equation shows clearly that the driving force for theinstability is the flow difference, v v vdi deD = - , between theions and electrons. In the radial direction there is not expectedto be any significant flow difference (except perhaps in thethin sheath regions), but we note that flow differences do existin both the axial and azimuthal directions. In the axialdirection ions are moving downstream whereas electrons aremoving upstream towards the anode, thus the flow differenceis effectively increased. Because of the radial magnetic field,the electron mobility is strongly reduced, but in certainregions of the discharge the electron axial velocity can be of asimilar magnitude to the ion axial velocity. In the azimuthaldirection the ion velocity (if any) is very small, but becausethe electrons are magnetized, they experience an E B´ driftapproximately given by, vye

x= . For typical electric

(2× 104 Vm−1) and magnetic (200 G) fields, this gives alarge drift velocity of about 1×106 m s−1, which is of theorder of the electron thermal velocity itself. Thus the EDI isdriven primarily by the electron–ion flow difference in theazimuthal direction.

Although the unstable nature of the EDI has been dis-cussed before in a number of previous works [10, 22, 55, 58],threshold conditions for the onset of such an instability havenot been explicitly studied. Our aim here is to investigate thisonset, and to provide a simple graphical reference that can beused to estimate whether the EDI is expected to be present fora given set of conditions, and which wavenumbers areunstable. Although equation (14) provides a convenientexpression for the threshold conditions of equation (2) in thelimit 1T

i , exact instability thresholds for the collisionless

limit ( 0snn = ) can be computed directly from equation (2)using the Penrose criterion [38]. This states that a necessaryand sufficient condition for instability is provided by

P F uF u F u

u ud 0, 150

¥( ) ( ) ( )

( )( )

mvf uv k vd 16

30 òå d= -( ) ( ) ( ˆ · ) ( )

is a projection of the distribution function along the wave-vector. Here fs0 is the equilibrium component of the dis-tribution function, and u0 is the location of the centralminimum of F(u), which is located between the two peaks ofF(u): u u u1 0 2< < . Figure 1 shows the instability diagramfor a xenon plasma as a function of the temperature ratio, T

and normalized flow speed difference, v

D . Here we have

assumed that the instability is predominantly in the azimuthaldirection so that kk jy» ˆ. Since, c 5000s ~ m s−1, in typical

HETs we have, 100v

cs~D . From figure 1 we see that this falls

in the unstable region of the diagram, indicating that the EDIwill develop. In fact, since v csD > in most of the HETchannel, and for typical electron–ion temperature ratios (seethe dashed lines in figure 1), instability is predicted essentiallythroughout the thruster and near-plume region (i.e. not just atthe maximum electric field location).

The Penrose criterion can also be used to determine therange of unstable wavenumbers, k k kmin

2 2max2< < . Here,

k P Fmax2 = ( ) is calculated from equation (15), and

k kmax 0,min = { ¯}, where [39]

k uF u F u

u ud . 172 2

22òº

¥¯ ( ) ( )

( )( )

Figure 2 shows contours for the threshold wavenumber atseveral values of the electron–ion flow speed difference. Asexpected, the range of unstable wavenumbers broadens as theflow speed difference increases. We also see that instabilityoccurs for wavenumbers of the order of k 1Del ~ (givingwavelengths of the mm level).

The Penrose criterion only applies to collisionless plas-mas and so cannot be used to study the effects of ion–neutralcollisions. Therefore we instead make use of the approximatedispersion relation from equations (4) and (5) to investigatecollisional effects on the EDI instability thresholds. Figure 3shows the pressure thresholds for a xenon plasma, and indi-cates which values of v

D are needed in order to obtain

instability for a given pressure. The vertical dashed lines showthe typical pressures near the thruster exit and anode regionsof HETs (for a mass flow rate of about 5 kg s−1 and an innerand outer channel radius of 2 cm and 4 cm, respectively).Within this pressure range we see that as the plasma density

Figure 1. Stability diagram for Xe+ as a function of the electron–ionflow speed difference, v

D , and electron–ion temperature ratio, T

Parameters above the curve lead to an unstable plasma. The verticaldashed lines mark temperature ratios of 20 and 100 respectively.

increases (contours in figure 3) the flow difference needed isreduced, while as the electron–ion temperature ratio decrea-ses, the required flow difference increases. Nevertheless,since 100v

cs~D , we see that instability is again predicted

essentially throughout the thruster. This indicates that ion–neutral collisions are too weak to damp the instability.

2.4. Alternative propellants

Because of the cost and scarcity of xenon, there is interest ininvestigating alternative propellants for HETs. Testing withsuch alternative propellants has typically involved other noblegases such as krypton [24], although more recently iodine hasbeen identified as a promising candidate because of its lowercost, higher availability, and its ability to be stored as a solid[25, 26]. The influence of the propellant (and hence ion mass)on electron transport has however not seen any significantresearch, and it is not clear if the EDI is enhanced or sup-pressed. Figure 4 shows a stability diagram from the Penrosecriterion for Xe+ (131AMU), Kr+ (84 AMU), Ar+

(40 AMU), and He+ (4 AMU). Here helium is included toinvestigate the effect of a very low ion mass. Note that due todissociation, an iodine plasma is expected to be dominated byI+ ions [26]. Since these ions have a similar mass (127 AMU)to Xe+, we have not explicitly included I+ in figure 4.Figure 4 demonstrates that all propellants show similar sta-bility behavior, but that the onset conditions can be sig-nificantly reduced for lower mass ions.

Figure 5 shows the pressure thresholds for the samepropellants as in figure 4. Here we have assumed that the ionsare moving at the ion sound speed, and have used approx-imate collision cross-sections at this speed taken from [40].Similar to figure 4 we see that the onset conditions forinstability are reduced for lighter ions, and that the unstable

region is broader. This again suggests that the EDI is likely toplay a role in HETs using alternative propellants.

3. Transport effects

3.1. Plasma kinetic equation

In sections 2.3 and 2.4 above we have studied conditionsleading to the onset of the EDI in HETs, but we have notaddressed the macroscopic consequences of this instability. Insections 3.3–3.8 below we will investigate the effect that theEDI has on the macroscopic plasma transport equations byconsidering the velocity moments of the relevant plasmakinetic equation. The macroscopic transport effects of kineticinstabilities have been studied previously by a number ofother authors for fusion applications [41], hollow cathodes[42], magnetoplasmadynamic thrusters [43], and fundamentalplasma physics [44, 45]. Here it is generally found that suchinstabilities lead to increased charged particle heating, as wellas enhanced collisionality. In this section our aim is toinvestigate the macroscopic effect that the EDI has on theparticle transport in HETs, and to provide simple order ofmagnitude estimates for determining the importance of theseeffects. Although the impact of the EDI on the electronmomentum conservation equation has already been pre-viously discussed in a HET context [12, 55], section 3.5briefly highlights some of the main conclusions forcompleteness.

In the presence of electrostatic oscillations, the electricfield and particle distribution functions can be separated intoequilibrium (E and fs) and fluctuating components ( Ed andfsd ), and the kinetic equations for the equilibrium and fluc-tuating components can then be written as [46, 47]

E v Bv

+ + ´¶¶

⎛⎝⎜

⎞⎠⎟

· ( ) ·

· ( )

E v Bv

+ + ´¶¶

= -¶¶

⎡⎣⎢

⎤⎦⎥

· ( ) ·

· · ( )

wheref

s¶¶( ) is a collision operator for collisions between

charged particle species s and neutrals. As can be seen inequation (18), the effect of electrostatic fluctuations is to addan additional term to the right-hand side which represents thecorrelation between any oscillations in the electric field andthe distribution function. This term effectively represents afriction force between charged particle species, and in thepresence of a stable plasma consisting of electrons and ions,leads to the standard Coulomb collision formula. For com-pleteness, and to further emphasis this point, we will rederive

Figure 2. Wavenumbers leading to instability as a function of theelectron–ion temperature ratio, T

i. The curves correspond to different

electron–ion flow speed differences, v

D . Parameters below the

curves lead to an unstable plasma.

this formula in section 3.4 below. In an unstable plasmahowever, this standard Coulomb collision formula no longerapplies and any instabilities can effectively enhance theelectron–ion collisionality by orders of magnitude, as wasrecently demonstrated in a HET context in [12], and as weagain show in section 3.5. In equations (18) and (19) we haveignored the presence of any fluctuating magnetic field in orderto focus on electrostatic modes that have previously beenobserved in PIC simulations. Some studies however havesuggested that fluctuating magnetic fields produced within theplasma might also contribute to an enhanced transport [48].

3.2. Instability convection

Before considering how the EDI affects the macroscopicparticle conservation equations, we briefly discuss convectionof the instability. In HETs the EDI initially grows in timebefore eventually saturating [12, 15]. At the same time it alsocontinuously convects away in space. This growth and con-vection can be studied by considering a conservation equation[49] for the instability energy density (where W E 2dµ ∣ ∣ ),

tW Wv 2 . 20g g

+ =· ( ) ( )

Figure 3. Electron–ion flow speed differences leading to instability as a function of neutral gas pressure for electron–ion temperature ratios of(a) 20T

i= , and (b) 100T

i= . The curves correspond to different plasma densities in units of (m−3). Parameters above the curves lead to an

unstable plasma. The vertical dashed lines mark pressures of 0.03mTorr and 3mTorr, respectively.

Figure 4. Electron–ion flow speed differences leading to instabilityas a function of electron–ion temperature ratio. The curvescorrespond to the different ions indicated: Xe+, Kr+, Ar+, and He+.Parameters above the curves lead to an unstable plasma.

Figure 5. Electron–ion flow speed differences leading to instabilityas a function of neutral gas pressure. The electron–ion temperatureratio is 20T

i= , and the plasma density is 1017m−3. The curves

correspond to the different ions indicated: Xe+, Kr+, Ar+, and He+.Parameters above the curves lead to an unstable plasma.

Here the first term on the left-hand side represents growth ofthe instability in time, while the second term representsconvection in space. In HETs the group velocity of the EDI isdominated by the axial ion drift velocity (see equation (11))so we can approximate, v vg zi» , and thus

zv W W

W2 2 1 .

gaccg g

» -¶¶

» - = -( ) ( )

Here g1

acct = , v

acca = =t

t g, and Lacc is a char-

acteristic length representative of the HET accelerationregion. Consider now what happens when the instability firstforms. If 1a , we see that the instability has a very longtime to grow before it is convected away, indicating that theinstability amplitude can reach very high levels. After a fewgrowth times however the instability will reach a nonlinearregime and will eventually saturate leading to 0W

¶. PIC

simulations suggest that this saturation occurs because of ion-wave trapping in the azimuthal direction [14, 15]. This regimewas recently treated in [12]. For 1a > the instability cannotgrow, while for, 1a = , wave growth is just balanced byconvection so that again we have, 0W

¶(the instability can

still grow in space though), but are now in a quasi-linearregime. This regime has been treated previously in the contextof two-stream instabilities in plasmas with multiple ion spe-cies [50].

Since the parameter α controls the instability regime (i.e.quasi-linear or nonlinear) it also to a large extent determinesthe instability amplitude and subsequent effects on particletransport. It is therefore interesting to consider how this valuedepends on typical HET parameters. We can approximate the

ion drift velocity as, vziqV

2 D» (where VD is the discharge

voltage), and the electron azimuthal drift velocity as,vye

x acc= » . Then using equation (9) for the maximum

growth rate, and simplifying, we obtain

27. 22e

» ( )

Interestingly, this does not depend directly on the ion mass(although the choice of VD, Te, and ne, are likely to dependindirectly on the mass) or the system geometry to zerothorder. A common rule of thumb [1] for HETs is that 0.1T

and thus equation (22) becomes

. 23ce

Since ce pew w in most HETs, we have that 1a , and thusthat the EDI can be expected to grow to very large amplitudesand hence enter the nonlinear regime. In fact we see fromequation (23) that only the magnetic field strength and plasmadensity play a role in this transition. If the magnetic field isincreased too far, then 1a and the strength of theinstability is expected to be significantly reduced. If thisoccurs, the enhanced electron–ion collisionality associatedwith the instability will also be strongly diminished, as well asthe consequent electron cross-field transport. Depending on

the operating conditions, it is possible that this could lead tointermittent/unstable thruster operation because of insuffi-cient electron current to sustain the discharge. This cascade ofevents effectively occurs because the azimuthal electron driftvelocity (and hence instability growth rate) is reduced withhigher magnetic fields. Interestingly though, as long as theratio 0.1T

D~ is maintained, increasing the discharge voltage

to compensate does not seem to be an option. Fundamentallythis is because increasing the discharge voltage will alsoincrease the axial ion velocity, and hence the instabilityconvection rate. Note that as the instability evolves in time,both the electron and ion distribution functions are expectedto be modified, which would in turn change the growth rate.Thus the analysis above is only an approximate estimate, butone which nonetheless highlights the importance of instabilityconvection and nonlinear effects.

In deriving equation (23) we have chosen the accelera-tion region as a convenient characteristic location, and used arepresentative ion velocity that most likely overestimates thetrue value in this region. Since the initial growth rate of theEDI is also high outside of the acceleration region, and sincethe ion velocity there could be lower, there may well beanother location in the thruster where the growth-to-convec-tion ratio (i.e. α) is in fact lower. This would imply that theinstability has an even longer relative time to grow beforebeing convected away, which further emphasizes our con-clusion in this section.

It is interesting to note that the scaling in equation (23)has been observed numerically to play a role in HET opera-tion. In [51] 2D PIC simulations of the z–θ directions of aHET were conducted with a scaling factor to artificiallyincrease the permittivity of free space (so as to reduce com-putational requirements). As the scaling factor was reducedbelow a certain value, a transition was observed leading tolarge changes in the plasma properties and electron mobility.This transition was found to depend on the ratio, ce

, which is

identical to our expression for α. Since changing thepermittivity of free space alters pew , this serves to againhighlight the importance of the ratio of instability growth-to-convection.

3.3. Electron and ion continuity equations

The particle continuity equations are obtained by integratingequation (18) over all velocities to obtain

xv , 24s

s ds sn¶¶

=· ( ) ( )

where ns and vds are the particle density and flow/driftvelocities respectively and Ssn is a source term accounting forcollisions with neutral particles only (i.e. such as ionization).The correlation term on the right-hand side of equation (18)vanishes during integration because the distribution functionmust go to zero at infinite velocities. The result inequation (24) is trivial and simply demonstrates the fact thatelectron–ion collisions do not create or destroy particles, nordo they cause an interchange between particle species. Thusthe EDI has no effect on the particle continuity equations.

3.4. Electron momentum conservation equation: stable plasma

The electron momentum conservation equation is found bymultiplying equation (18) by mv and again integrating overall velocities

tmn mn

E v B R R 25

e de e de de

e de e en eiP

= + ´ - + +

( ) · ( )

q nR E 26ei ed d= á ñ ( )

and where eP is the electron pressure tensor, and Ren is themomentum loss due to electron–neutral collisions. The elec-tron–ion friction force density, Rei

LB, for a stable plasma (i.e.where the EDI is not present) can be determined by directlycomputing the correlation of density and electric field fluc-tuations associated with individual particles using the meanfield response functions. This leads to the momentum momentof the Lenard–Balescu equation [52, 53]:

q v f vm C f fR E vd d ,ei e e iLB 3 3

LBò òd d= á ñ = ( ), where

C f f v

LBò=- ¢

¢ -¢¢

⎛⎝⎜

⎞⎠⎟

( ) ·

( )( )

( ) ( )

is the Lenard–Balescu collision operator, and

kk k uk k v

( )( · )

∣ˆ ( · )∣

( )( )

Here e q= ∣ ∣, Z is the ion charge number, u v v= - ¢ and thesecond form follows from taking the dielectric response to bethe adiabatic value for a stable plasma: 1

De2e » +

Assuming the electron and ion distribution functions areshifted Maxwellians, and carrying out the six velocity inte-grals, leads to (see appendix A of [54]):

4, 29ei e ei

py= - D

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

3 430ei

( ) ¯( )

is the standard electron–ion collision frequency. Here,v v v vei Te Ti Te

2 2 2 2= + »¯ , m meimM

m M= »

+is the reduced mass,

lnL is the Coulomb logarithm, andx t t t x x xd exp erf exp

2òy = - = - -

p p( ) ( ) ( ) ( )

is the Maxwell integral.Typically, one is concerned with the slow flow limit

, in which case the term in square brackets in

equation (29) asymptotes to 1, returning the familiar expres-sion for friction in a plasma. However, in the azimuthaldirection of a HET this is not an appropriate expansionbecause 1v

Te» »D

¯. In this case, the term in square

brackets in equation (29) is approximately 0.57 (see figure 1of [54]). As a consequence, an estimate for the magnitude ofthe stable plasma electron–ion friction force density in theazimuthal direction is

R n m vn q

2.7 ln

4, 31ei y e ei Te

» »L∣ ∣

( )( )

where we have used quasi-neutrality so that n ni e» , andassumed singly charged ions so that Z=1.

3.5. Electron momentum conservation equation: unstableplasma

In the case of an unstable plasma, equations (25) and (26) stillequally apply, but the resulting final expression forequation (26) is different. Since we have previously treatedthis case in detail in [12] we quote only the final result here,which is obtained by considering the dispersion relation forthe EDI, the Vlasov equation for the fluctuating electrondistribution component (i.e. fed ), and the conservationequation for the instability energy density,

cn TR j v j

Te y sdi e e

p b d= - = -

∣ ˜∣ ˆ ∣ ∣ ∣ · ( )∣ ˆ

Here the superscript IE refers to instability-enhanced, β is avariable representing the portion of the electron distributionfunction at the wave resonance (see [12] and/or section 3.6below), and Eyd∣ ˜ ∣ is the amplitude of the instability electricfield (assuming a single dominant sinusoidal mode at thewavevector giving the maximum growth rate). We havewritten two expressions in equation (32) because both will beneeded in the sections below. The direction and sign of Rei

depends on the coordinate system used, as well as thedirection of the electron azimuthal drift. In the coordinatesystem adopted here (see section 2.1) the friction force is inthe azimuthal direction, and acts in the opposite direction tothe electron drift, since it represents an electron–ion frictionforce (i.e. a momentum loss).

To demonstrate the importance of the EDI, we comparethe magnitude of the instability enhanced electron–ion fric-tion force (Equation (32) with R Rei ei

IE IE= ∣ ∣) to the standardvalue for a stable plasma (equation (31)). To do this weapproximate the derivative in equation (32) as

n T v n Tvdi e e z iz e ev n T

diz e e

acc » »· ( ) ( ) , and viz

2 D» ∣ ∣ , and

we then obtain

q n L8.4

ln. 33ei

For n 5 10e17= ´ m−3, T 20 eVe = , L 1 cmacc = , and

V 300 VD = , we obtain 500R

,LB ~ . Thus the instability

enhanced electron–ion friction force is orders of magnitudehigher than that due to standard electron–ion Coulombcollisions.

3.6. Electron energy conservation equation: stable plasma

Multiplying equation (18) by mv1

22 and integrating over all

velocities gives the electron energy conservation equation

tn T mn v

n T mn v

qn P P

e e e de

e e e e de de de

e de en ei

+ + + +

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥· ·

where Qe is the heat flux, Pen is the energy loss from electron–neutral collisions, and Pei is given by

P q vvf

2d . 35ei

e3 2ò dd

= -¶¶

· ( )

Like the friction force calculated in section 3.4, Pei can becomputed from the appropriate correlation of fluctuatingquantities associated with individual particles in the meanfield response approximation. Here, this leads to the energymoment of the Lenard–Balescu equation [52, 53]:P v mv C f fd ,ei e i

LB 3 1

LBò= ( ). Applying equations (27) and(28) and integrating by parts twice, first to remove the leadingvelocity-space gradient and second to remove the velocity-space gradients of fe and fi, leads to

v u u v v

d . 36

¢ ¢ + + ¢ ¢¢⎜ ⎟⎛⎝

⎞⎠· ( ) ( ) ( )

Assuming the electron and ion distribution functions areshifted Maxwellians and carrying out the six velocity inte-grals leads to

Mn T T

erf , 37

eiTi de Te di

eie ei e i

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

( ) ¯¯

where ReiLB is given in equation (29). Typically, one is con-

cerned with the slow flow limit, 1v

, in which case the

first term in equation (37) is negligible and the term in squarebrackets asymptotes to 1, returning the familiar expression fortemperature relaxation in a plasma. Here, we are interested inthe situation where 1v

vei»D

¯. In this case the term in square

brackets takes a values of approximately 0.75, and Rei isgiven by equation (31). Choosing the reference frame to bethe ion fluid (so v 0di = ), and the electron flow speed to bev vde Te» (note that we show in section 3.8 that the ions flowat a fraction of cs, which slightly reduces this estimate for theelectron flow in the ion rest frame, but this has a negligiblecorrection to the results of this section), and using v vei Te»¯ ,the first term in equation (37) is found to be 1T

e smaller

than the second term (see section 3.9). With these, we have

MR v4 38ei ei y Te

LB» ( )

as an estimate for the electron–ion energy exchange rate. Notethat this returns the familiar result from Coulomb collisiontheory that electron–ion energy exchange occurs more slowlythan momentum exchange (i.e. friction) by a factor of theelectron–ion mass ratio: m

M sn n~ . Accounting for the largedrift only influences the numerical factor in front of thisexpression.

3.7. Electron energy conservation equation: unstable plasma

Assuming that the instability is predominantly in the azi-muthal direction, we can use integration by parts (noting thatthe boundary terms go to zero at infinity) with equation (35)to obtain

P q E vv f J Ed . 39ei y y e y yIE 3òd d d d= á ñ = á ñ ( )

Assuming that the instability is dominated by a single mode atthe wavenumber of maximum growth rate, we can use sinu-soidally varying quantities in time such that,E ERe ey y

k y ti y Rd d= w-{ ˜ }( ) , f fRe ee ek y ti y Rd d= w-{ ˜ }( ) , and

hence P J E J EReei y y y y1

2*d d d d= á ñ = { ˜ ˜ }. Using these expres-

sions in equation (19) [12] and (39) we get

i. 40e

˜ ∣ ∣ ˜( )

i. 41ei

⎧⎨⎩⎫⎬⎭

∣ ˜ ∣( )

Because the EDI is in the nonlinear regime, the equilibriumdistribution function is unlikely to be Maxwellian. In [12] weaccounted for this by using a generalized distribution givenby

v u gg

d, 42e

Te3 3ò

=¢ ¢

-⎛⎝⎜

⎞⎠⎟( )

where g is some function. Defining uv

Te= - , using

equation (42), and simplifying, equation (41) becomes

Pq n E

d. 43ei

⎧⎨⎩⎫⎬⎭

∣ ˜ ∣ ( )( )

Here Gug u

ò ¢ ¢

( )with uP parallel to ky and u perpend-

icular to ky,v

Teh = , and

k vy de

y Tez = w-

. Applying the Plemelj

relation to the integral in equation (43) gives

Pq n E

mv kP u

⎧⎨⎩

⎡⎣⎢

⎦⎥⎥

⎫⎬⎪⎭⎪

∣ ˜ ∣

( ) ( )

where P represents the principal value of the integral. Taking

the real part yields

Pq n E

p d z h

p d z h b

∣ ˜ ∣ ( )

∣ ∣ ( )( )

where we have introduced the definition (see also [12]),G

z. Substituting the magnitude of the first expression

in equation (32) into (45) we have

P R v . 46ei ei TeIE IE z h= - +( ) ( )

Here R R 0ei eiIE IE = ∣ ∣ . Now

k v k v

v k v k vy de

y Te y Te

Te y Te

y Tez h h= = - = - » -w w w w-

. Thus

equation (46) becomes

R v . 47ei eiR

IE IE IEphase

w= - = - ( )

Thus we have the satisfying result that the power deposition isgiven by the enhanced friction force multiplied by theinstability phase velocity. In addition this power deposition isnegative indicating that the instability takes energy from theelectrons, which intuitively makes sense since the electronslose momentum in the azimuthal direction.

We can ask how the magnitude of this new powerdeposition term compares with the power absorption due tothe equilibrium electric field: P qn v Ee deabs = . Since the elec-tric field is largest in the axial direction, this power depositionterm simplifies to P qn v Ee ez zabs » . From [12] we showed thatthe enhanced electron–ion friction force gives rise to anincreased electron mobility that can approximately be writtenas

q n E11 . 48ez

e zeff

⎡⎣⎢

⎤⎦⎥∣ ∣

( )∣ ∣

If we assume that the electron–neutral collision frequency isnegligible, this reduces to

q n E B. 49ei

e z xeff

m »∣ ∣

BR v . 50ei

xei yeabs

IE IE» » ( )

Therefore, as a result of the instability the axial electronpower deposition is enhanced and is given by the enhancedfriction force multiplied by the azimuthal electron driftvelocity. But

v. 51ei

phase= ( )

Since vphase is of the order of cs (see section 2.1), we therefore

see that 1v

phase , and hence 1P

abs . This demonstrates

that the extra term in the electron energy conservationequation is expected to be negligible, and so long as the effect

of the instability is included in fluid simulations via anenhanced mobility/collisionality/friction force in the electronmomentum conservation equation, the effects of theinstability appear to be ‘fully’ captured. This explains whyprevious fluid simulations have been able to reproduceexperimental results by simply adding an anomalous electronmobility [36, 55, 56].

The above result is interesting because it shows thatdespite the instability being in the azimuthal direction, themain power deposition is in the axial direction. This occursbecause in the azimuthal direction the electric field and per-turbed electron current density are oscillating in time, so atsome points in the cycle there is an energy gain, and at othersthere is an energy loss (overall though there is still a net loss).By contrast, because of the magnetic field, the electronmotion in the azimuthal direction is coupled to the axialdirection, so the enhanced scattering in the azimuthal direc-tion leads to an enhanced drift velocity in the axial direction.This, together with the temporally constant (on instabilitytime scales) axial electric field, means that the electron energygain in this direction is always positive, and so can be sig-nificantly larger than that in the azimuthal direction.

Finally, we compare the power transferred betweenelectrons and ions via the wave-mediated interaction with thestandard Coulomb collision rate. Equations (38) and (47)provide

R4. 52ei

M ei Te

IEphase

LB» ~ ( )

This shows that although the energy transferred by standardelectron–ion collisions is m

Msmaller than the momentum

transferred, energy transferred via interaction with the wavesis not reduced by this scaling. However, the relevant wavespeed is v vm

M Tephase » , compared to vTe, so the overall mass

scaling of the ratio of the power exchanged from each

mechanism is M

m. Equation (33) showed that 500R

LB ~ , so

2.5 10P

LB ~ ´ . Since PeiIE is already small compared to the

power absorbed by the equilibrium electric field, this showsthat power exchanged between electrons and ions by standardCoulomb collisions is truly negligible.

3.8. Ion momentum conservation equation

Because the EDI leads to an enhanced electron–ion frictionforce, by momentum conservation this should also lead to anenhanced ion–electron friction force which is equal in mag-nitude but opposite in direction (i.e. R Rie ei= - ). The ionmomentum conservation equation is

tMn Mn qn

v v v E v B

v R R , 53

i di i di di i di

i m i di in ienP

+ = + ´

- - + +

( ) · ( ) ( )

· ( )

where Rin is the momentum loss due to ion–neutral collisions.In order to get a simple idea of the effect of the enhanced ion–electron collisions, we ignore time variations in equation (53),as well as ion–neutral collisions, and assume unmagnetized

ions with an isotropic pressure tensor. Since the instability ismainly in the azimuthal direction, equation (53) becomes

xMn v v

zMn v v

yR j . 54

i xi yi i yi i yi zi

= -¶¶

( ) ( ) ( )

· ˆ ( )

Here we have also substituted, R Rie eiIE IE= - . If there are no

significant large scale spatial gradients in the radial and azi-muthal direction, equation (54) can be simplified to give

zMn v v R j

d. 55i yi zi ei

IE= -( ) · ˆ ( )

Substituting the second expression for the electron–ion fric-tion force in equation (26), while using quasi-neutrality sothat n ne i» and using vv kdi zi» ˆ we obtain

zMn v v

c zv n T

d. 56i yi zi

szi i e=( ) ∣ ∣ ( ) ( )

Now along the length of the thruster the ion flux towards theexit increases from zero, and thus we can approximate thederivatives as: n v v

z i zi yin v v

di zi yi

acc»( ) , and n v T

z i zi en v T

di zi e

acc»( ) .

Thus we can solve for the azimuthal ion velocity fromequation (56) to obtain

4 6. 57yi

s» ( )

As a result of the instability, ions undergo a rotation in theazimuthal direction with a velocity about ten times smallerthan the ion acoustic speed. Since the electron–ion frictionforce acts to oppose the electron azimuthal drift, and since theion–electron friction force is opposite to the electron–ionfriction force, the ions drift in the same direction as theelectrons. For a typical electron temperature of, T 20 eVe = ,xenon ions would rotate with a velocity of about 400m s−1.In [57] ion rotation has indeed been measured in a HET. Themagnitude of this ion rotation was found to be between about200 and 600m s−1, which is consistent with what is predictedhere, and which is much higher than can be explained due toany residual magnetic field effects. Thus the presentinstability theory offers a possible reason for this ‘anomalous’ion rotation.

3.9. Ion energy conservation equation

Although we saw above that the enhanced friction force doesnot produce any significant additional electron powerdeposition, this occurred in large part because the electronsare magnetized and so the azimuthal and axial directions arecoupled. For the ions which are unmagnetized, this is nolonger the case and so we can ask whether the instability leadsto ion heating. The ion energy conservation equation is

tn T m n v n T

m n v qn P

v v v E

i i i i di i i i

i i di di di i di ie

+ + = +

⎜ ⎟ ⎜

⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

· ( )

If we use a similar analysis to that in section 3.6 for theenhanced electron power deposition term Pei

IE, we will obtain

P P R v . 59ie ei eiIE IE IE

phase= - = ( )

This result logically makes sense, because the friction forceterm previously calculated in [12] was derived for a saturatedinstability indicating that the wave energy density no longerincreases. Thus any energy loss by electrons due to theinstability must equal the energy gain by ions. To get anapproximate idea of the effect of this energy gain we simplifyequation (58) by ignoring any time variation, neutral colli-sional-energy losses, large scale gradients in the radial andazimuthal directions, the ion heat flux, and the stress tensor.Thus we obtain

zq n T m n v v q n v E R v

i i i i di zi i zi z ei2 IE

phase+ = +⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥∣ ∣ ∣ ∣

Since the ion drift energy is mainly in the axial direction weset v vdi zi

M2 2 2 D» » ∣ ∣ , and also approximate the derivative as

acc» . Similarly, Ez

acc» . Substituting for Rei

IE and sim-

plifying, we get

2 4 6. 61i

phase= ( )

But since vkphase

y= w , we have from equations (7) and (8) that

v csphase2

3= . Thus equation (61) becomes

30. 62i

e= ( )

Since the electron temperature in HETs is about 20–30 eV,we see that the instability leads to ion heating with a temp-erature of almost 1eV. We note that the above is just anapproximate estimate of the heating effect due to theinstability itself. Ion heating in the axial direction is alreadywell known to occur because of the broad overlap betweenthe ionization and acceleration zones [57].

4. Wall erosion

Life-time tests of a number of HETs have shown theappearance of periodic erosion patterns on both the inner andouter ceramic channel walls (see for example [27]). Theseperiodic patterns have a wavelength of about 1 mm as well asa very slight azimuthal inclination, but the reason for theirpresence remains unknown. It is interesting to note that thewavelength of the EDI studied here is also of the order of1mm. Furthermore, at saturation the EDI is characterized byoscillations in the plasma density of about 20%–30%, as wellas the appearance of a population of very high energy ionswhich become trapped by the electrostatic wave [9, 12, 14–16]; factors that could both conceivably lead to an enhancederosion rate on the channel walls.

Although the EDI is predominately in the azimuthaldirection, it undergoes convection in the axial directionbecause of the large ion drift due to acceleration by the

applied discharge voltage (see sections 2.2 and 3.2). For an

ion with a velocity of, vziq V

2 D» ∣ ∣ , and an acceleration

length inside the thruster of L 1 cmacc » , the time taken forthe instability to convect out of the thruster is, z

acct ~ .

Because the speed of the instability in the azimuthal directionis about cs (see equation (10)), in a time zt it will have pro-pagated a distance, L cy z st= . For an electron temperature of20eV and a discharge voltage of 300V, this gives a distanceof only 1.8mm (and hence an azimuthal inclination of about10°). Thus the instability only travels a very short distance inthe azimuthal direction before leaving the thruster, and wouldgive the appearance of a standing wave on these time scales(consistent with the erosion patterns observed experimen-tally). However it is unclear yet what determines the initialphase of the instability. If the phase is determined by thethruster operating conditions/geometry, then the apparentstanding wave pattern would persist over longer time scalesand lead to a definite time-averaged structure. But if the phaseis random, then successive waves would be expected tocancel any coherent structure so that the time-averaged ero-sion would be uniform. Furthermore, in our discussion abovewe have assumed a single mode. If multiple non-coherentmodes are present then an erosion pattern would only beexpected if one of the modes is dominant (with the largestamplitude occurring at the wavenumber giving the maximumgrowth rate for example). Although a dominant mode hasbeen observed in PIC simulations [9, 14–17], it is not clearyet if such a mode leads to a coherent time-averaged structure.Further work is needed to clarify these issues.

5. Discussion and conclusions

5.1. Instability onset

In section 2 we have studied in detail the onset conditionsleading to the formation of the EDI in HETs. The large azi-muthal electron drift velocity provides the driving force forthe instability which develops essentially throughout thethruster and near-plume regions. Furthermore, ion–neutralcollisions are too infrequent to damp the instability, whichsuggests that the EDI is expected to be present in most HETs.Since the use of lighter ions reduces the required onset con-ditions for this instability, alternative propellants are unlikelyto change this conclusion. In performing these stability ana-lyses we have made use of the modified ion acoustic dis-persion relation (equation (2)) instead of the full discretedispersion relation (equation (1)), which is justified based onprevious theoretical/numerical studies [12, 22]. Even ifconditions are encountered where this is no longer strictlytrue, the maximum growth rate for the full dispersion relationalways appears to be higher than that for the ion acousticrelation [12, 22], hence suggesting that the full dispersionrelation will be unstable if the ion acoustic dispersion relationis. Although we have not directly considered secondaryelectron emission in our present analysis, 2D PIC simulations

[16] have shown that such emission does not appear to sig-nificantly damp the EDI.

5.2. Instability convection

By considering the balance between instability growth andconvection in section 3, we have identified an additional non-dimensionalized scaling parameter for HETs, ce

pea ~ w

1a , either because of an increased magnetic field or adecreased electron density, the effect of the instability on themacroscopic plasma transport is reduced. If the magnetic fieldis increased too far, the instability becomes too weak and theassociated electron–ion friction can no longer enhance theelectron cross-field transport. Since α is a function of both theelectron density, and the permittivity of free space, theintroduction of scaling factors in PIC simulations (in order toreduce computational times) will alter the instabilityevolution.

5.3. Plasma transport

In section 3 we investigated the effect of the EDI on themacroscopic plasma transport. In addition to an electron–ionfriction force enhancing the electron cross-field mobility, theinstability also leads to an ion rotation in the azimuthaldirection that is in good agreement with that seen experi-mentally. Because of the magnetic field, the additional elec-tron power deposition by the instability in the azimuthaldirection is insignificant compared with that in the axialdirection. From this we deduce that the anomalous electronenergy loses used in the hybrid simulations in [4, 29, 30] mustoccur due to a different physical reason, and is most likelyassociated with electron energy losses and secondary electronemission at the radial thruster walls. For the ions which areunmagnetized, the power deposition by the instability can beimportant, and we predict that this can lead to ion heating. Wehighlight though that the effects of the instability on the iontransport (both ion rotation and heating) have been deduced ina very simple way and serve only as estimates. For example,although an anomalous ion rotation was experimentallymeasured in [57], the location of the hollow cathode wasfound to affect some of the results, thus possibly indicating aresidual radial electric field effect. Furthermore, large scalegradients in the azimuthal direction, such as due to rotatingspokes, could also very likely lead to an ion rotation.

5.4. Landau damping

Recent work [58] has suggested that an apparent discrepancyexists when considering the EDI as the cause of the anom-alous electron transport in HETs. Since the azimuthal electrondrift velocity (and hence instability growth rate) is largest inthe acceleration region, one might also expect a very highanomalous scattering rate in this region. However, exper-imental measurements suggest that it is in fact the lowest inthis region. Since relatively high ion temperatures [57, 59] (ofa few eV) have been measured in HETs, [58] has made theinteresting proposal that ion Landau damping from the main

ion beam reduces the instability growth and scattering, andthat it is instead an instability associated with the presence ofa second, lower energy, ion population that is responsible.Since this second population is mainly important outside ofthe acceleration region, anomalous scattering is small insidethe thruster and largest downstream.

It is interesting to note though that the enhanced electronscattering found in section 3.5 (and discussed in more detailin [12]) already offers a second possible explanation for theabove mentioned discrepancy. By considering the nonlineareffect of the instability on the electron distribution function[12] (which is significantly different from the Maxwellianused in [58]), the electron scattering in the acceleration regionis naturally found to be low. This can be seen directly fromequation (32). In the acceleration region the ion flux n vi zi

(n ne i» ) begins to saturate as most of the propellant isionized, while the electron temperature reaches a maximum.Thus the derivative v n T

z zi i ed

d( ) is low in this region, and hence

so too is the electron scattering force (i.e. RieIE). Physically, in

this region of the discharge, although the instability amplitudeis high, the electron density oscillations are almost completelyout-of-phase with the electric field oscillations, resulting in avery low time-averaged scattering force [12]. Because of thetypical spatial variation of the plasma properties though, thederivative v n T

z zi i ed

d( ) is not expected to be low/zero down-

stream of the thruster, and so as long as the instability is ableto grow to large amplitudes (which would occur if 1a ),enhanced electron transport should occur in this downstreamregion.

Although it is still true that high ion temperatures havebeen measured in HETs, these measurements are of the axialtemperature. Since the instability is predominantly in theazimuthal direction, it is the azimuthal ion temperature whichwould be expected to determine the Landau damping rate.Both the present work, and that in [58], have howeverassumed isotropic ion distributions. Further work is needed toclarify the role of anisotropy in the ion distribution.

Acknowledgments

TL acknowledges financial support from a CNES post-doctoral research award. The work of SDB was supported byUS Department of Energy Grant No. DE-SC0016473.

References

[1] Goebel D M and Katz I 2008 Fundamentals of ElectricPropulsion: Ion and Hall Thrusters (New Jersey: Wiley)

[2] Zhurin V V, Kaufman H R and Robinson R S 1999 PlasmaSources Sci. Technol. 8 R1

[3] Morozov A I and Savelyev V V 2000 Reviews of PlasmaPhysics ed B B Kadomtsev and V D Shafranov (New York:Springer)

[4] Boeuf J P and Garrigues L 1998 J. Appl. Phys. 84 3541[5] McDonald M S 2012 Electron transport in Hall thrusters PhD

Thesis University of Michigan

[6] Ellison C L, Raitses Y and Fisch N J 2012 Phys. Plasmas 19013503

[7] Barral S, Makowski K, Peradzyński Z and Dudeck M 2005Phys. Plasmas 12 073504

[8] Tsikata S, Cavalier J, Héron A, Honoré C, Lemoine N,Grésillon D and Coulette D 2014 Phys. Plasmas 21 072116

[9] Adam J C, Héron A and Laval G 2004 Phys. Plasmas 11 295[10] Ducrocq A, Adam J C, Héron A and Laval G 2006 Phys.

Plasmas 13 102111[11] Choueiri E Y 2001 Phys. Plasmas 8 1411[12] Lafleur T, Baalrud S D and Chabert P 2016 Phys. Plasmas 23

053503[13] Tsikata S, Lemoine N, Pisarev V and Grésillon D M 2009

Phys. Plasmas 16 033506[14] Boeuf J P 2014 Front. Phys. 2 74[15] Lafleur T, Baalrud S D and Chabert P 2016 Phys. Plasmas 23

053502[16] Héron A and Adam J C 2013 Phys. Plasmas 20 082313[17] Coche P and Garrigues L 2014 Phys. Plasmas 21 023503[18] Taccogna F, Minelli P, Capitelli M and Longo S 2012 AIP

Conf. Proc. 1501 1390[19] Taccogna F and Minelli P 2015 Joint Conf. of 30th Int. Symp.

on Space Technology and Science 34th Int. ElectricPropulsion Conf. and 6th Nano-satellite Symp. (Hyogo-Kobe, Japan: The Electric Rocket Propulsion Society) IEPC-2015-418

[20] Frias W, Smolyakov A I, Kaganovich I D and Raitses Y 2013Phys. Plasmas 20 052108

[21] Matyash K, Schneider R, Mazouffre S, Tsikata S,Raitses Y and Diallo A 2013 33rd Int. Electric PropulsionConf. (Washington, DC: The Electric Rocket PropulsionSociety) IEPC-2013-307

[22] Cavalier J, Lemoine N, Bonhomme G, Tsikata S,Honoré C and Grésillon D 2013 Phys. Plasmas 20 082107

[23] Lazurenko A, Coduti G, Mazouffre S and Bonhomme G 2008Phys. Plasmas 15 034502

[24] Linnell J A and Gallimore A D 2006 J. Propul. Power 22 1402[25] Szabo J, Pote B, Paintal S, Robin M, Hillier A,

Branam R D and Huffman R E 2012 J. Propul. Power28 848

[26] Grondein P, Lafleur T, Chabert P and Aanesland A 2016 Phys.Plasmas 23 033514

[27] Dudeck M, Doveil F, Arcis N and Zurbach S 2012 IOP Conf.Ser.: Mater. Sci. Eng. 29 012010

[28] de Grys K, Mathers A, Welander B and Khayms V 2008 46thAIAA/ASME/SAE/ASEE Joint Propulsion Conf. & Exhibit(Nashville, TN: American Institute of Aeronautics andAstronautics) AIAA-2010-6698

[29] Hagelaar G J M, Bareilles J, Garrigues L and Boeuf J P 2002J. Appl. Phys. 91 5592

[30] Bareilles J, Hagelaar G J M, Garrigues L, Boniface C,Boeuf J P and Gascon N 2004 Phys. Plasmas 11 3035

[31] Adam J C et al 2008 Plasma Phys. Control. Fusion 50 124041[32] Garrigues L, Pérez-Luna J, Lo J, Hagelaar G J M,

Boeuf J P and Mazouffre S 2009 Appl. Phys. Lett. 95141501

[33] Scharfe M K, Gascon N, Cappelli M A and Fernandez E 2006Phys. Plasmas 13 083505

[34] Hofer R R, Katz I, Mikellides I G, Goebel D M, Jameson K K,Sullivan R M and Johnson L K 2008 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conf. & Exhibit (Hartford, CT:American Institute of Aeronautics and Astronautics) AIAA-2008-4924

[35] Mikellides I G and Katz I 2012 Phys. Rev. E 86 046703[36] Ortega A L and Mikellides I G 2016 Phys. Plasmas 23 043515[37] Fried B D and Conte S D 1961 The Plasma Dispersion

Function (New York: Academic)[38] Penrose O 1960 Phys. Fluids 3 258

[39] Davidson R C 1972 Methods in Nonlinear Plasma Theory(New York: Academic)

[40] Phelps A V 2005 Compilation of Atomic and Molecular Datasee (http://jila.colorado.edu/∼avp/collision_data/ionneutral/IONATOM.TXT)

[41] Davidson R C and Krall N A 1977 Nucl. Fusion 17 1313[42] Jorns B A, Ortega A L and Mikellides I 2016 52nd AIAA/SAE/

ASEE Joint Propulsion Conf., AIAA Propulsion and EnergyForum (Salt Lake City, UT: American Institute ofAeronautics and Astronautics) AIAA 2016-4626

[43] Choueiri E 1999 Phys. Plasmas 5 2290[44] Lampe M, Manheimer W M, McBride J B, Orens J H,

Papadopoulos K, Shanny R and Sudan R N 1972 Phys.Fluids 15 662

[45] Lampe M, Manheimer W M, McBride J B, Orens J H,Shanny R and Sudan R N 1971 Phys. Rev. Lett. 26 1221

[46] Baalrud S D 2010 Kinetic theory of instability-enhancedcollective interactions in plasma PhD Thesis University ofWisconsin-Madison

[47] Baalrud S D, Callen J D and Hegna C C 2008 Phys. Plasmas15 092111

[48] Wei L, Han K, Wang Ch, Zhang Ch and Yu D 2012 Contrib.Plasma Phys. 52 761

[49] Stix T H 1992 Waves in Plasmas (New York: Springer)

[50] Baalrud S D, Hegna C C and Callen J D 2009 Phys. Rev. Lett.103 205002

[51] Coche P 2013 Modélisation cinétique d’un propulseur à effetHall PhD Thesis Universitéde Toulouse

[52] Lenard A 1960 Ann. Phys. 3 390[53] Balescu R 1960 Phys. Fluids 3 52[54] Baalrud S D and Hegna C C 2011 Phys. Plasmas 18 023505[55] Katz I, Mikellides I G, Jorns B A and Ortega A L 2015 Joint

Conf. 30th Int. Symp. on Space Technology and Science34th Int. Electric Propulsion Conf. and 6th Nano-satelliteSymp. (Hyogo-Kobe, Japan: The Electric Rocket PropulsionSociety) IEPC-2015-402

[56] Cappelli M A, Young C V, Cha E and Fernandez E 2015 Phys.Plasmas 22 114505

[57] Mazouffre S 2013 Plasma Sources Sci. Technol. 22 013001[58] Katz I, Ortega A L, Jorns B A and Mikellides I G 2016 52nd

AIAA/ASME/SAE/ASEE Joint Propulsion Conf. (Salt LakeCity, UT: American Institute of Aeronautics andAstronautics) AIAA-2016-4534

[59] Huang W, Drenkow B and Gallimore A D 2009 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conf. & Exhibit(Denver, Colorado: American Institute of Aeronautics andAstronautics) AIAA-2009-5355

Characteristics and transport effects of the electron drift...

Documents

Transcript of Characteristics and transport effects of the electron drift...

ProtoDUNE-SP Electron Lifetime & Drift Time Analysis · 3 ProtoDUNE-SP Electron Lifetime & Drift Time Justin Hugon Louisiana State University Measuring Electron Lifetime Bruce Baller

Resonance cone probe for measuring electron density ... · Resonance cone probe for measuring electron density, temperature, drift speed and beam components A. Piel Institut f¨ur

Lesson 5 Current and Resistance Batteries Current Density Electron Drift Velocity Conductivity and Resistivity Resistance and Ohms’ Law Temperature.

PHYS1511: College Physics 1 Mechanics, Heat and Soundhomepage.physics.uiowa.edu/~sbaalrud/1511s16_materials/lecture_1.… · PHYS1511: College Physics 1 Mechanics, Heat and Sound

Numerical calculations of relativistic electron drift loss effect

Prof. Jøran Moen - EISCAT kick-off seminar . Future perspectives ... Patches (GDI - Gradient Drift Instability) ICI-2 GDI -electron clouds ICI-3 KHI ...

Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending.

The Theory of Continental Drift Continental Drift Discovery.

Confinement of pure electron plasmas in the CNT stellaratorsites.apam.columbia.edu/NNP08/slides/Pedersen.pdf · improvement for CNT •This is essentially because the ExB drift dominates

Development of the Silicon Drift Detector for Electron ......2020 September • 47 Silicon Drift Detector The transfer of signal charges in a CCD was performed by rap-idly changing

Effects of finite electron temperature on gradient drift ...physics.usask.ca/~smolyak/as/Papers/lakhin_phys_plasma_2018.pdf · In particular, they are important for magnetron plasma

Edinburgh Research Explorer Electron capture dissociation and … · 2016. 1. 14. · 1 1 Electron capture dissociation and drift tube ion mobility-mass spectrometry coupled 2 with

Collisionless kinetic theory of oblique tearing instabilitieshomepage.physics.uiowa.edu/~sbaalrud/papers/BaalrudPOP2018.pdfCollisionless kinetic theory of oblique tearing instabilities

Effect of E × B electron drift and plasma discharge in dc magnetron ...

Lecture 31 - UMD · PDF filedecrease with distance ... (like relating T, p f gas to energy of molecules): ... velocity for single electron (drift speed, )

Electron capture dissociation and drift tube ion mobility ... · Electron capture dissociation and drift tube ion mobility-mass spectrometry coupled with site directed mutations provide

IEPC-95-58 - 2 - OF THE CLOSED ELECTRON DRIFT …

STARE velocities: 2. Evening westward electron ﬂoM. Uspensky et al.: STARE velocities: 2. Evening westward electron ﬂow 1079 component of the plasma VE×B drift (Greenwald et al.,

Electron and hole drift velocity in chemical vapor

Cosmic Ray Test of Mini-drift Thick Gas Electron ... · Comparing with the gas electron multiplier (GEM) chamber rst intro-duced in 1996 at CERN [7], the THGEM chamber is one of the