Tutorial: Physics and modeling of Hall thrusters [PDF]

Tutorial: Physics and modeling of Hall thrusters Jean-Pierre Boeuf

Citation: Journal of Applied Physics 121, 011101 (2017); doi: 10.1063/1.4972269 View online: http://dx.doi.org/10.1063/1.4972269 View Table of Contents: http://aip.scitation.org/toc/jap/121/1 Published by the American Institute of Physics

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JOURNAL OF APPLIED PHYSICS 121, 011101 (2017)

Tutorial: Physics and modeling of Hall thrusters Jean-Pierre Boeufa) LAPLACE, Universit e de Toulouse, CNRS, INPT, UPS, 118 Route de Narbonne, 31062 Toulouse, France

(Received 17 August 2016; accepted 6 November 2016; published online 3 January 2017) Hall thrusters are very efficient and competitive electric propulsion devices for satellites and are currently in use in a number of telecommunications and government spacecraft. Their power spans from 100 W to 20 kW, with thrust between a few mN and 1 N and specific impulse values between 1000 and 3000 s. The basic idea of Hall thrusters consists in generating a large local electric field in a plasma by using a transverse magnetic field to reduce the electron conductivity. This electric field can extract positive ions from the plasma and accelerate them to high velocity without extracting grids, providing the thrust. These principles are simple in appearance but the physics of Hall thrusters is very intricate and non-linear because of the complex electron transport across the magnetic field and its coupling with the electric field and the neutral atom density. This paper describes the basic physics of Hall thrusters and gives a (non-exhaustive) summary of the research efforts that have been devoted to the modelling and understanding of these devices in the last 20 years. Although the predictive capabilities of the models are still not sufficient for a full computer aided design of Hall thrusters, significant progress has been made in the qualitative and quantitative understanding of these devices. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4972269]

I. INTRODUCTION

Hall thrusters (also called Stationary Plasma Thrusters, SPT) were invented in the 1960s, and the first satellite (METEOR-18) powered by a SPT was launched on December 29, 1971 in Russia.1 The basic properties and fundamentals of Hall thrusters are described in several review papers and textbooks (see, e.g., Refs. 2 and 3). In this paper, we will concentrate on the physics and modelling of Hall thrusters and will not discuss the questions of performance, optimization, and technical aspects of these thrusters. The physics and modelling of the cathode neutralizer and plume plasmas are out of the scope of this paper. Useful information or review on these topics can be found in Refs. 4–6 and 7, respectively.

thrusters, the thrust is transferred by the electrostatic force between the ions and the two grids while in Hall thrusters the thrust is transferred to the magnetic structure by the Lorentz force acting on the Hall electron current and equal to the electrostatic force acting on the ions (see, e.g., Ref. 3). The Hall current, as defined in detail in the following, is the electron current flowing in the direction perpendicular to the discharge electric field and external magnetic field. A plasma is a relatively good conductor and the electric field in a plasma far from walls is generally small. One way to increase locally the electric field is to decrease the electron conductivity. This can be done in principle by applying a magnetic field perpendicular to the electron current from the cathode to the anode, i.e., to the applied electric field.

A. Gridless vs gridded ion sources

The main competitors of Hall thrusters are the gridded ion sources, which can achieve comparable thrust and spe_ where m_ is cific impulse.3 The thrust is defined as: T ¼ mv, the mass flow rate of the propellant and v the propellant velocity, and the specific impulse is the ratio of the thrust to the flow rate of the weight ejected (calculated on the earth ground): Isp ¼ v=g, where g is the acceleration at earth surface. In gridded ion sources, positive ions are extracted from the plasma by applying a voltage between an electrode grid in contact with the plasma and an extracting grid called the accelerator grid. In Hall thrusters, positive ions are extracted from a plasma but without grids, by generating an electric field locally in the plasma (see Fig. 1). In both gridded ion sources and Hall thrusters, ions are electrostatically accelerated out of the plasma, but in gridded a)

FIG. 1. Gridded vs gridless ion source.

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121, 011101-1

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In these E  B devices, electron trajectories are trapped along the magnetic field lines, and electron transport perpendicular to the magnetic field is possible only in the presence of mechanisms leading to abrupt changes in the electron momentum (e.g., collisions). Attempts at generating a large electric field in the bulk of magnetized plasmas based on this principle have been made in various applications, generally with limited success because the formation of different types of instabilities limits the efficiency of electron confinement by the magnetic field (see, e.g., the work on homopolar discharges in the 1970s, where the maximum electric field in the plasma was shown to be related to the critical ionization velocity concept introduced by Alfven8,9). Hall thrusters are very interesting and remarkable examples of cross-field devices where a large electric field can indeed be generated and maintained in the bulk plasma due the lowering of the electron conductivity by a magnetic barrier combined with the depletion of neutral atoms by ionization. Although instabilities are present in Hall thrusters and enhance the electron conductivity through the magnetic barrier, they do not lead to a total collapse of the electric field in the bulk plasma. In gridded ion sources, the extracted ion current density is limited by the space charge between the grids. The maximum current density that can be extracted is a function of the voltage and distance between the grids and is given by the Child Langmuir law (see, e.g., Ref. 3) rffiffiffiffiffi 4e0 2e V 3=2 ; Jmax ¼ 9 M d2 where M is the ion mass, V is the extracting voltage, and d is the effective accelerating distance (i.e., distance between the grids plus height of the plasma meniscus3). The maximum ion current that can be extracted is therefore Imax ¼ Ag Jmax , where Ag is the active grid area. The maximum thrust Tmax is proportional to the maximum ion current times the ion velocity ð2eV=MÞ1=2 so that Tmax / Ag ðV=dÞ2 . The space charge limitation of the current is an important difference between gridded ion sources and Hall thrusters. This limitation does not exist in principle in Hall thrusters because ions are extracted by an electric field present in the quasineutral plasma and not by a voltage drop between two grids. There is however an optimum value of the current density in Hall thrusters, which is about ten times larger than the maximum current density in gridded ion sources.3 B. E 3 B and closed-drift devices

An important aspect of E  B devices (see, e.g., the reviews on E  B devices in Refs. 10 and 11) is that when an external magnetic field B is placed perpendicularly to the applied electric field in a discharge, an electron drift is generated in the E  B direction. Fig. 2 shows the E  B electron flux Ce;EB and the cross-field electron flux Ce;E . The E  B drift velocity is on the order of E/B if the electron collision frequency  is small with respect to the electron angular cyclotron frequency Xce ¼ eB=m. It is essential that the E  B electron current associated with Ce;EB , also called

J. Appl. Phys. 121, 011101 (2017)

FIG. 2. E  B configuration. The electron E  B flux Ce;EB (Hall flux) is much larger than the cross-field flux Ce;E in closed-drift E  B devices. The ratio Ce;EB =Ce;E is the Hall parameter, equal to the ratio of the electron cyclotron frequency to the electron collision frequency (see Section II C).

Hall current, does not flow to a wall. This would induce charge polarization leading to the generation of an electric field in the E  B direction (Hall effect), the Hall electric field EH (parallel to E  B), that would oppose the current. This Hall electric field would in turn generate an EH  B flux antiparallel to the applied electric field (in the same direction as the cross-field flux Ce;EB shown in Fig. 2) and therefore destroy the electron confinement by the magnetic field and enhance cross-field transport. There should not be any Hall effect in a Hall thruster to ensure a good confinement of electrons and a lowering of the electron conductivity in the axial direction. Attempts at designing “linear Hall thrusters”12,13 (i.e., with walls in the E  B directions) have been made, but with little success because of the poor electron confinement (see Ref. 14 for a discussion of “non-closed-drift” in the context of the negative ion source for the ITER neutral beam injection system). Note that Hall thrusters are often improperly called Hall effect thrusters (Hall thruster or SPT should be preferred). One way to ensure that the E  B current does not reach a wall is to use a cylindrical geometry with E  B in the azimuthal direction. E  B devices using such geometry are called closed-drift plasma sources. Hall thrusters are one particular type of closed-drift ion source together with End-Hall ion sources or anode layer thrusters.15 C. From magnetron discharges to Hall thrusters

A very well-known closed-drift plasma source is the magnetron discharge. One particular geometry of magnetron discharge is the planar magnetron. This discharge device consists of two planar electrodes with magnets placed behind the cathode in such a way that the magnetic field has a cylindrical symmetry with a large radial component (Fig. 3). This device operates as a glow discharge except that electrons generated

FIG. 3. Schematic of a planar magnetron discharge.

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at the cathode surface by ion impact are trapped along magnetic field lines until they undergo collisions. The successive collisions allow them to drift across the magnetic field and reach the anode but their residence time in the cathode region is considerably increased by the presence of the magnetic field. This allows the operation of the discharge at a much lower pressure than a regular glow discharge because electrons can undergo collisions and ionize the gas even though the electron mean free path may be much larger than the cathode-anode gap. Good electron confinement is possible because of the closed E  B drift in this geometry and because of the mirror effects at the magnetic field cusps. Electrons are oscillating between the cusps along the magnetic field lines while rotating around the magnetic field and drifting in the azimuthal direction. An important aspect of these types of discharges is that the magnetic field is generally designed such that electrons are strongly magnetized (i.e., the electron Larmor radius is much smaller than the discharge dimensions) while ions are not. The electrons are therefore collisional because they are trapped by the magnetic field, while ions are collisionless. A Hall thruster is in some ways similar to a magnetron discharge. The geometry is cylindrical, with an axially applied electric field and a radial magnetic field. However, the goal in a Hall thruster is to extract ions from the plasma and accelerate them outside the discharge and not to accelerate them toward a cathode. Starting from a planar magnetron, one can keep a similar radial magnetic barrier and remove the planar cathode, as shown in Fig. 4. A way to generate a large electric field in the quasineutral plasma and not in a cathode sheath is to replace the planar cathode by an electron source (Fig. 4). This source provides the electrons that will move across the magnetic field to the anode and generate the plasma by ionization, and also the electrons that are needed to neutralize the space charge of the ion beam ejected out of the discharge. The large electric field must be generated in the plasma at the location of large magnetic field (where the axial electron conductivity should be minimum). The fundamental difference between a planar magnetron discharge and a Hall thruster is the fact that the large electric field is in the cathode sheath in a magnetron while it is distributed in the quasineutral plasma in a Hall thruster. Another difference between planar magnetron discharges and Hall thrusters is the fact, as we will see below, that electrons are more likely to interact with walls in a Hall thruster.

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The principles of Hall thrusters as described in this section seem extremely simple and straightforward. We will see in the rest of the paper that the physics of a Hall thruster is actually very complex and non-linear, mainly for the following reasons: (1) Imposing a large electric field in a quasineutral plasma by applying a magnetic barrier perpendicular to the cathode-anode current is a challenge since various types of instabilities can develop, some of them being completely detrimental to the electron confinement. (2) Electron interactions with the walls containing the plasma have a strong influence on cross-field electron transport. The characteristics of the thruster depend on some properties of the wall materials which are often not very well known (secondary electron emission, surface roughness, etc…). (3) The gas flow must be practically completely ionized. This is an essential condition not only for a high thruster efficiency but also for good ion extraction from the plasma since the profile of the extracting electric field depends on the gas density through the electron conductivity. The coupling between transport instabilities driven by the E  B configuration and non-uniformity of the gas density due to strong ionization makes the physics of Hall thrusters extremely non-linear. II. PHYSICS OF HALL THRUSTERS A. Principles of Hall thrusters

The typical geometry of a Hall thruster is shown in Fig. 5. The plasma is generated in a channel between two concentric dielectric cylinders of radii R1 and R2 (channel width H ¼ R2-R1, mean diameter D ¼ R1 þ R2). The dielectric material must have a low ion sputtering yield and good properties with respect to secondary electron emission due to electron impact3 (see below). Ceramic materials such as BN or BN-SiO2 are often used for the channel walls. The anode is at the end of the channel and an electron emissive cathode is placed outside the channel. Electrons flowing from the cathode to the anode ionize the gas (usually xenon), which is injected from the anode side. The magnetic field distribution is created by a magnetic circuit schematically shown in Fig. 5, with magnetic poles at the channel exit. The magnetic field is mainly radial in the region of the poles and is maximum near the channel exhaust plane. Because of the large magnetic field and low neutral atom density in the exhaust region, the axial electron conductivity is strongly reduced, leading to a local increase of the electric field. This electric field extracts the un-magnetized positive ions from the plasma. B. Typical operating conditions and orders of magnitude

FIG. 4. Scheme illustrating the general principle of a closed-drift thruster. The left part is an axial-radial (x,r) view of the device while the right part is a radialazimuthal (r,h) view, showing the radial magnetic field and the Hall current.

We show below the typical nominal operating conditions for a Hall thruster in the kW range. We take the example of the SPT100M of Fakel.16

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FIG. 5. Schematic of a Hall thruster.

1. Electrical characteristics and performance

Discharge current: ID ¼ 4:5 A, discharge voltage: UD ¼ 300 V, power: PD ¼ 1350 W, xenon mass flow rate: m_ ¼ 5:3 mg=s, thrust: T ¼ 90:2 mN, specific Impulse: Isp ¼ 1734 s, i.e., the mean velocity of the beam formed by the ejected ions is Vi;b ¼ 1734  g ¼ 17 km=s. 2. Channel dimensions

For a SPT100 Hall thruster, the dimensions of the channel are: Channel length: L ¼ 2:5 cm, inner and outer channel radii: R1 ¼ 3:5 cm; R2 ¼ 5 cm, i.e., a channel width H ¼ 1:5 cm. The channel cross-sectional area is therefore: Ac ¼ pðR22  R21 Þ ¼ pHD, where D ¼ R1 þ R2 is the mean channel diameter. For the SPT100 Ac  40 cm2 . 3. Mass flow rate, ion current, and gas density

Assuming that all xenon atom flux is ionized and neglecting losses and multiply charged xenon ions, the _ (M is the ion extracted ion current would be Ii ¼ em=M mass). For a mass flow rate of xenon m_ ¼ 5:3 mg=s, the ejected ion current is therefore on the order of Ii  3:9 A for a total discharge current ID ¼ 4:5 A (the electron current entering the channel would be 0:6 A in this example). For a neutral velocity va;0  200  300 m=s at the anode, the neutral density in this region should be na ¼

m_ 1  2  3  1019 m3 : M Ac va;0

configuration are essential for electron confinement (even though instabilities and electron-wall collisions tend to limit the confinement efficiency). 5. Hall thruster current density

A useful order of magnitude is the discharge current density JD ¼ ID =Ac , which is on the order of JD  1000 A=m2 ¼ 0:1 A=cm2 in a Hall thruster. It is interesting to note that as the thruster size and power change, the current density does not change and stays on the order of 1000 A=m2 in most Hall thrusters.3 6. Mean velocity and density of the ejected ions

If all the ions were accelerated to the maximum energy for an applied voltage UD ¼ 300 V, the ion velocity at exhaust would be Vi;max ¼ ð2eUD =MÞ1=2 ¼ 20 km=s. The measured ion velocity (17 km=s in a SPT10016) is slightly below this value because not all the ions “see” the whole potential drop (i.e., the ionization and acceleration regions overlap). The smaller mean axial ion velocity can also be attributed to the beam divergence and to the fact that the acceleration voltage is lower than the applied voltage because the potential of the cathode plasma is above the ground potential. Since the current density of the ion beam Ji is close to the total current density, i.e., on the order of 1000 A/m2, the density of the ion beam in the exhaust region, ni ¼ Ji =ðeVi Þ, is about 3  1017 m3 . The ion and plasma density in the channel before ion acceleration is much larger, typically between 1018 and 1019 m3.

4. Electron collisions

7. Magnetic field

For an electron energy of 10 eV, the electron momentum cross-section17 is r  2  1019 m2 . Assuming a gas density na  2  1019 m3 in the anode region, the electron neutral collision-frequency in the anode region is  ¼ na < rve > 5  106 s1 . This corresponds to an electron mean free path of about 30 cm for 10 eV electrons. In the exhaust region, the gas density is more than ten times smaller because of ionization and the electron mean free path is therefore more than ten times larger. This clearly shows that the presence of the magnetic field and the closed-drift

The maximum radial magnetic field in the exhaust region in a SPT100 is in the range Br;max ¼ 15– 20 mT ð150–200 GÞ and its profile is shown in Fig. 6. For such values of the magnetic field, the electron cyclotron frequency is on the order of Xce ¼ eB=m  2  3  109 rd=s, which corresponds, for an electron mean energy of 20 eV, to a Larmor radius smaller than qe;L ¼ ve =Xce  1 mm. The electron Larmor radius is therefore much smaller than all the dimensions defined above. The ion cyclotron angular frequency in the exhaust region is Xci ¼ eB=M  104 rd=s, and

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FIG. 7. Schematic representation of the acceleration and ionization regions and of the radial magnetic field profile.

FIG. 6. Axial profile of the normalized radial magnetic field in the middle of the channel (top) and magnetic field lines in the channel (bottom) in a SPT100M (after Ref. 16).

the ion Larmor radius qi;L ¼ vi =Xci is in the range [0.2–2] m for ion velocities between 2 and 20 km/s, i.e., much larger than all dimensions above. Electrons are therefore strongly magnetized but ions are practically not sensitive to the presence of the magnetic field in a Hall thruster. The characteristic length of the decrease of the magnetic field inside the channel, from the exhaust plane to the anode, LB , is an important parameter and is on the order of 1 cm in a SPT100 (Fig. 6). This length roughly coincides with the length of the ionization region where most of the gas flow is ionized. In more recent designs, the magnetic field distribution is more symmetric with respect to the channel centreline than in the SPT100 design of Fig. 6 (see, e.g., Figs. 7–3 of Ref. 3). Some variations around the standard design have been proposed by different authors. For example, in the Cylindrical Hall Thruster (CHT) of Raitses et al.,18–20 well adapted to low power applications, the central magnetic pole is shifted toward the channel end, leading to a larger volume to surface ratio of the channel, with a cusp-like magnetic field distribution in the exhaust region. In the recent magnetically shielded design of Refs. 21 and 22 described in Section IV A, the magnetic poles are outside the channel and the magnetic field lines do not intercept the walls inside the channel, which considerably reduces the wall erosion. In the wall-less configuration of Refs. 23 and 24, the magnetic poles are also located outside the channel and a ring anode is place near the exhaust plane. 8. Electric field distribution

The axial electric field profile is schematically shown in Figure 7 together with the ionization rate and radial magnetic

field profiles. As said above, the large magnetic field associated with neutral depletion, e.g., smaller electron collision frequency in the exhaust region, lowers the axial electron conductivity, leading to an increase of the axial electric field. This electric field provides the energy necessary for the electrons to ionize the gas flow and at the same time accelerates freely the positive ions outside of the channel. The magnetic field intensity and profile control the relative positions of the ionization and acceleration regions and must be optimized to ensure efficient ion extraction. The value of the maximum axial electric field is on the order of 4  104 V/m in the nominal conditions of a SPT100. Since most of the gas flow is ionized in the ionization region, the neutral gas density in the exhaust region is very small. This raises the question of cross-field electron transport in the exhaust region and outside the channel. Because of the very low electron collision frequency, electrons should be efficiently trapped along the magnetic field lines. The mechanisms that allow cross-field electron transport in these regions are still not fully understood. C. Classical cross-field electron transport

The physics of classical collisional electron transport in E  B configurations can be easily understood by writing the steady state electron momentum transfer equation for a uniform plasma.  me ½E þ v  B  v ¼ 0, where v is the mean electron velocity and  the electron collision frequency. With E in the x direction and B in the y direction (EXB in the z direction) of a rectangular coordinate system, this equation gives: vx ¼ 

eE 1 ; m 1 þ h2

vz ¼

eE h ; m 1 þ h2

where h ¼ Xce and Xce ¼ eB m are, respectively, the Hall parameter and the electron cyclotron angular frequency. For Xce  , (which is true in the acceleration region of Hall thrusters where the Hall parameter h is larger than 103), the components of the electron velocity parallel to the electric field and in the E  B direction can be written as: vx ¼ vE  

1E ; hB

vz ¼ vEB 

E B

Note that the ratio of the Hall current density JH ¼ JEB ¼ envEB to the axial electron current density JE ¼ envE is equal to the Hall parameter: j JEB JE j ¼ h. The equations above show that in the presence of a magnetic field perpendicular to the electric field, the electron

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D. Plasma properties from Hall thruster fluid and hybrid models

FIG. 8. Examples of electron trajectories in a Hall thruster, in the radialazimuthal (r,h) plane (left), and in the axial-radial (r,x) plane (right). These trajectories were calculated by integration of the electron equations of motion with a given magnetic field distribution and an electric field distribution obtained from a hybrid model (a Monte Carlo module was used for collisions assuming a constant neutral atom density).

mobility le;E ¼ 1=ðBhÞ parallel to the electric field is strongly reduced (divided by the factor 1 þ h2  h2 ) with respect to the case without magnetic field (where e le;E ¼ j vEx j ¼ m ). The mobility and diffusion parallel to the magnetic field are not affected. The classical, collisional electron mobility in the E direction therefore varies as 1=B2 and is proportional to the collision frequency . In principle, the electrons are perfectly trapped in the absence of collisions (the mobility parallel to the electric field is zero) but in practice, perfect trapping is never realized and instabilities and turbulence or wall scattering enhance the transport across B. Fig. 8 shows examples of electron trajectories in a Hall thruster. The electrons rotate along the magnetic field lines (which are radial in the exhaust region) and bounce back and forth between the wall sheaths while drifting in the azimuthal direction (they are scattered by the walls and may generate secondary electrons if their energy in the direction perpendicular to the walls is larger than the sheath potential drop). Azimuthal instabilities as well as electron interactions with the wall can contribute to cross-field electron transport and enhance the axial electron conductivity. Since these mechanisms are not perfectly understood nor quantified, their contribution to crossfield transport is often termed as “anomalous” (anomalous transport, mobility, or conductivity).

To complement the orders of magnitude given above, Fig. 9 presents the space distribution of the main plasma parameters as predicted by a hybrid model. In this hybrid model (see Section III A), electrons are treated as a fluid while ions and atoms are described kinetically. Although the models provide a good qualitative and a reasonable quantitative description of the plasma properties they are not fully self-consistent because, as explained below the cross-field electron transport is described empirically, by adjustable coefficients, in the absence of a complete theory of anomalous electron transport in Hall thrusters. We see in Fig. 9 that the neutral atom density drops by two orders of magnitude from the anode to the exhaust region and more than 90% of the xenon atom flux is ionized. The maximum plasma density, in the ionization region, is on the order of 3  1018 m3 and decreases as the ions are accelerated in the acceleration region. The time averaged axial electric field is distributed over about 1.5 cm and is maximum close to the exhaust plane. The ionization rate maximum is shifted upstream by about 7 mm with respect to the peak electric field and about 75% of the total ionization takes place inside the channel over a length of 1.5 cm from the exhaust plane (ionization region). The results of Fig. 9 are only indicative since, for example, the exact locations of the acceleration and ionization regions and their respective positions depend on the adjustable parameters that are used to characterize anomalous electron transport. This is illustrated in Fig. 10 from Ref. 25, where the plasma potential and ionization rate are shown for different values of the adjustable parameters. Anomalous electron transport was taken into account by adding “anomalous” collision frequencies representing the effect of wall scattering or turbulence (w and B , respectively, inside and outside the channel), to the electron-neutral collision frequency en . The total effective collision frequency,  ¼ en þ w þ B . en was deduced from the local gas density and electron temperature. The effective electron-wall scattering frequency, w , was taken as a constant of the form w ¼ aref with  ref ¼ 107 s1 , giving a contribution a Xrefce B1 to the mobility. B , which can be defined as a modified Bohm collision frequency, and was of the form B ¼ ðK=16ÞXce , giving a

FIG. 9. Axial distributions of time averaged (a) plasma potential and ionization rate, (b) external radial magnetic field and axial electric field, (c) plasma density and xenon atom density, from a 2D hybrid model of a Hall thruster (SPT100 type, xenon mass flow rate 5 mg/s, applied voltage 300 V). After Adam et al.26

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contribution K=ð16BÞ to the mobility (K ¼ 1 corresponds to the usual Bohm value). a and K were adjustable coefficients. Their values are indicated on each of the 4 cases represented in Fig. 10. Another empirical parameter was included in the model to describe the energy loss to the walls.25 Note that similar approaches have been taken in many hybrid or fluid models of Hall thrusters, some using a less empirical way of representing the wall effects. For example, the effect of walls on electron transport is described by empirical and adjustable coefficients in Refs. 25, 27–33, while the models of Refs. 34–36 use an analytical model of the wall sheath to estimate the effect of the walls on the electron momentum and energy losses (see the discussion of these models in Section III B). Attempts at including self-consistently in hybrid models the effect of anomalous electron transport induced by instabilities have been made recently.37 We see in Fig. 10 that the adjustable coefficients completely control the position and length of the acceleration and ionization regions. When the a coefficient is decreased (CASE 1 to CASE 3) for a constant K coefficient (i.e., less anomalous transport inside the channel), the electric field is larger and distributed over a smaller length (with less potential drop outside the channel), and the ionization rate is more peaked. When K is increased at constant a (CASE 1 to CASE 4), the electric field is also much smaller outside the channel and more peaked inside. Hall thrusters are subject to various types of instabilities and oscillations (see Section II E 3). An important type of oscillation is due to the periodic depletion and replenishment of neutral atoms in the ionization region. This leads to the well-known ionization instability (called “breathing mode” in Ref. 27), leading to current oscillations in the 10–20 kHz frequency range. Fig. 11 shows that the amplitude of the oscillations in this frequency range is very sensitive to the anomalous transport coefficients used in the model. One can also see in this figure that higher frequency oscillations (100–500 kHz), called “transit time oscillations,” are present and more visible on the ion current.

J. Appl. Phys. 121, 011101 (2017)

FIG. 11. Total current (thick line) and ion current (thin line) leaving the thruster as a function of time (the same conditions as in Fig. 10). After Ref. 25.

The use of adjustable anomalous transport coefficients in 2D (r,x) hybrid and fluid models is necessary and no solution can be obtained from the models if some anomalous cross-field transport is not included. Evidence of anomalous transport can also be obtained from experiments and more sophisticated simulations. This is described below. E. Evidence of anomalous transport

The classical, collisional transport associated with electron-neutral atom collisions is not sufficient to explain the cross-field electron transport, especially in the exhaust region and outside the channel where the neutral atom density is very low due to the strong ionization upstream. Two mechanisms have been invoked: electron scattering by the channel walls associated with secondary electron emission (SEE) by electron impact (the contribution of this mechanism to cross-field transport has been termed as “near wall conductivity,” NWC, by Morozov and his colleagues2) and plasma oscillations and turbulence. Both mechanisms certainly play a role but it is still difficult to decide which one is dominant in particular thruster configurations or operating conditions. We briefly discuss in this section the different experimental evidence of anomalous cross-field electron transport in Hall thrusters. 1. Current-voltage characteristics—Role of the wall materials

FIG. 10. 2D (r,x) time averaged distributions of the electric potential (contour lines) and ionization rate (grey scale) in a SPT100 type thruster for a mass flow rate of 5 mg/s for 4 different cases with different adjustable parameters for anomalous electron transport inside and outside the channel. The applied voltage is 300 V or 600 V. After Ref. 25.

The current-voltage characteristics of a Hall thruster and its dependence on the channel wall material can provide information on the electron conductivity. Fig. 12 displays three current-voltage characteristics measured for the same thruster but with different wall materials operating at a xenon mass flow rate of 5 mg/s. We first note that if the atom flux is completely ionized the maximum ion current extracted from the channel must be, assuming only singly charged ions, Ii  3:7 A. The discharge current ID is the sum of the ion and electron current, so the electron current entering the channel is Ie ¼ ID  Ii . Assuming a quasi-constant ion current along the I–V characteristics, we find that the electron current entering the channel in Fig. 12 depends on the channel wall materials and tends to increase with voltage for Silicon Carbide and

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applied voltage of 300 V), we get a lower limit of the electron mobility le;E ¼ ve;x =E  0:2 m2 =V=s. Assuming collisional cross-field electron transport with a mobility le;E ¼ B1 Xce and for a magnetic field on the order of 1:5  102 T, we get an estimate of the electron collision frequency at the channel ve;x exhaust:  ¼ Xce E=B  107 s1 . This is between one and two orders of magnitude larger than the electron-atom collision frequency in the exhaust region (the electron-atom collision frequency in the anode region is 5  106 s1 and more than 90% of the atom flux is ionized). Since this estimation is made in the region of the characteristics where the wall materials seem to play a less important role, we can infer that electron scattering with the walls may not be the only mechanism responsible for anomalous cross-field transport. FIG. 12. Measured current-voltage characteristics for a SPT100 for 3 different channel wall materials (after Gascon et al.38).

Alumina walls. This clearly indicates that the channel wall material has a significant influence on the electron conductivity in the channel. The strong influence of the channel walls on the plasma properties and electron transport is reported in many papers. For example, Fig. 13 displays measurements of the maximum electron temperature in the channel for two different types of channels walls (note the very large electron temperature at large voltages). Below 200 V, we see in Fig. 12 that the current (i.e., cross-field electron transport) is not very sensitive to the nature of the channel walls. One possible reason is the lower electron mean energy and consequently lower secondary electron emission from the walls (see Section III B). In this region of the I–V characteristics, electron-atom collisions are however not sufficient to explain cross-field electron transport, as shown below. Around 200 V, we have ID  4:5 A and therefore Ie ¼ Ii  ID  0:8 A. Since the plasma is quasineutral, IIei ¼ Vvexi;b and the ion beam velocity Vi;b is on the order of 15 km/s for an applied voltage of 200 V, the axial electron drift velocity vex at the channel exit should be on the order of vex  4  103 m=s. Assuming a maximum value for the electric field in that region E  2  104 V=m (Fig. 9(b) gives a maximum electric field of E  3  104 V=m for an

FIG. 13. Measured maximum electron temperature in the channel as a function of discharge voltage for two different channel wall materials (after Raitses et al.39).

2. Hall parameter measurements

Meezan et al.40 combined a series of diagnostics to measure the variations of the effective axial electron mobility, 1 le;E ¼ B1 Xce ¼ Bh , along the channel axis. Knowing the profile of the radial magnetic B, this gives the inverse Hall parameter, h1 , which is proportional to the effective collision frequency. If the electron-atom collision frequency can be deduced from the experiments, one can then compare the effective electron collision frequency (or effective inverse Hall parameter) with the electron-atom collision frequency (or inverse Hall parameter due to electron-atom collisions). In the experiments of Meezan et al., the ion current density distribution was deduced from probe measurements. The electron current density at a given location in the channel is the difference between the total current density (deduced from the measured discharge current) and the ion current density. The plasma density was obtained from probe measurements or deduced from the ion current density and the ion mean velocity obtained from Laser Induced Fluorescence (LIF) measurements. The axial electric field was calculated from the measured plasma potential. The effective electron mobility and Hall parameter distributions can then be deduced from the electron current density, plasma density, and electric field distributions. The profile of the atom mean velocity along the channel was also deduced from LIF measurements. Continuity imposes that the sum of atom flux and ion flux along the channel is constant, and this gives the axial variations of the gas density. The electron-atom collision frequency (and associated Hall parameter) was obtained by multiplying the atom density by the collision rate (deduced for the momentum electron-xenon cross-section and the measured electron temperature, assuming a Maxwellian distribution). From this (complex) set of measurements, Meezan et al. were able (Fig. 14) to compare the effective inverse Hall parameter with the Hall parameter associated with electronatom collisions, (respectively, indicated as “experimental” and “classical” in Fig. 14) (it is possible to show that the contribution of electron-ion collisions is small in Hall thrusters). We clearly see in this figure that the experimental and classical electron collision frequencies are close together in the anode region where the atom density is relatively large and

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FIG. 14. Inverse Hall parameter h1 ¼ =Xce as a function of axial position along the channel deduced from experiments by Meezan et al.40 (the channel length is 7 cm in this thruster).

that the effective collision frequency is much larger than the electron-atom collision frequency in the exhaust region (more than one order of magnitude). This confirms that mechanisms other than electron-atom collisions must contribute to cross-field electron transport in a large part of the channel (and outside the channel). 3. Role of plasma oscillations, instabilities, and turbulence

Numerous oscillations have been detected in Hall thrusters, in a large frequency range, from kHz to 10 s of MHz. Choueiri41 gave a comprehensive review of the plasma oscillations and instabilities that had been identified and studied in the 1960s to 1990s. From that period of time to recent years, these oscillations have been thoroughly studied with powerful diagnostic techniques such as Laser Induced Fluorescence, fast imaging probe and current measurements on segmented electrodes. An instability present in the acceleration region, the E  B electron drift instability, has been predicted by particle simulations26,42 relatively recently and detected in collective laser scattering experiments.43 This instability seems to contribute significantly to the anomalous electron transport and will be described in more details below. The main types of oscillations in a Hall thruster are the following: •

• • •

Low frequency (10–20 kHz) ionization oscillations. These oscillations are due to the periodic ionization of the atom flux in the region of large magnetic field. Azimuthal low frequency oscillations, also called “rotating spokes” in the kHz range. Axial transit time oscillations in the 100–500 kHz frequency range. Azimuthal, small wavelength oscillations in the 1–10 MHz range, called E  B Electron Drift Instabilities (EDI) in the rest of this paper.

a. Ionization oscillations. The physical reasons for the ionization oscillations in the 10–20 kHz frequency range are relatively well understood. Due to strong ionization of the

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atom flow in the large magnetic field region, the neutral atom front moves upstream in a lower magnetic field region where ionization becomes less efficient because the electron mobility is larger. This leads to a decrease of the current till the neutral atom front moves back to the exhaust region where intense ionization can take place again. This oscillation mode has been termed as “breathing mode” in Ref. 27 where a 1D model confirmed this qualitative description. According to this interpretation, the period of the oscillation should be related to time necessary for the atom to replenish the ionization region, which gives the correct order of magnitude. A less qualitative description, based on a simplified analytical approach, has also been used to study this oscillation mode. It consists in writing the two continuity equations for ions and neutral atoms subject to transport at a constant velocity in the ionization region and generation (ions) or loss (atoms) due to ionization. This system of equations is of the Lotka-Volterra type (also called predator-prey equations) and leads, after linearization, to the equation 1=2 of an undamped va Þ harmonic oscillator at a frequency fi ¼ ðVi2pL , where Vi , va , and L are, respectively, the ion velocity, the neutral velocity, and the length of the acceleration region.3,41,44,45 b. Azimuthal, low frequency oscillations—Rotating spokes. The first evidence of this type of rotating instability

was given by Janes and Lowder46 who used a Hall thruster with a rather long (10 cm) channel and detected density fluctuations moving azimuthally in the direction of the electron drift with a phase velocity of 0.2E/B. More recent observations of similar rotating spokes include those of Chesta et al.,47 McDonald et al.,48,49 and Ellison et al.50 McDonald et al. deduced from image processing high speed camera videos of a 6 kW Hall thruster plasma, the existence of low frequency oscillations in the kHz range.48,49 These oscillations were characterized by rotating spokes, i.e., regions of higher plasma emission rotating at velocities from several hundred to a few thousand meters per second, with long wavelengths on the order of several to over tens of centimeters (Fig. 15). The oscillations reported by McDonald et al. were consistent with measurements of the currents on a segmented anode. The authors deduced from the current measurements that about 50% of the anode current was carried by the rotating spokes. Ellison et al.50 drew similar conclusions from their measurements of the current on a segmented anode in a cylindrical Hall thruster. The experiments above show evidence of azimuthal oscillations of the electron current on the anode side which seems to indicate that the rotating spokes are present in the anode region. Additionally, from detailed mapping of the time variations of the plasma potential in the near field and cathode region of a Hall thruster, Smith and Cappelli51 deduced that the plasma potential in that region appears to fluctuate in a helical fashion and that this could also play a role in driving cross-field electron transport (as shown by the authors with single particle simulations in Refs. 52). The presence of a non-zero azimuthal electric field Eh (note that the integral of the azimuthal field along the azimuthal direction should be zero) can contribute to electron transport because it can create an Eh  B drift in the axial

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FIG. 15. Processed images (top) of rotating spokes in the H6 thruster, and currents waveforms measured on 3 segments of a segmented anode (bottom). After McDonald et al.49

direction. The average axial electron current density due to the azimuthal field can be written as (see, e.g., Ref. 46) ð Eh Jex ¼ e n dA=A; B where the average is made over the azimuthal direction. The net current depends on the correlation between the azimuthal electric field and plasma density fluctuations. For example, for a given phase shift between field and density fluctuations defined by:

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FIG. 16. Spoke velocity as a function of radial position in the Hall thruster channel for three different gases in the experiments of Janes and Lowder.46 The value of the Critical Ionization Velocity (CIV) for each gas is also indicated.

Recent Particle-In-Cell simulations have predicted the presence of rotating spokes in a cylindrical magnetron and provided a clear possible interpretation of the physical mechanisms involved.54,55 Fig. 17 shows an example of potential distribution obtained in these simulations, where the rotating spoke is characterized by an abrupt potential drop at the front of the rotation structure. One can deduce from these simulations that the CIV spokes are likely to be present under conditions where the plasma source (cathode region in the case of a magnetron discharge) is sufficiently far from the anode and if the magnetic

Eh ¼ E0 sin h n ¼ n0 þ n1 sin ðh þ uÞ the average axial electron current density would be Jex ¼ en1 EB0r cos u. The nature of the rotating spokes observed in recent experiments is not completely clear. In their early work, Janes and Lowder46 observed the presence of rotating spokes in a long channel (10 cm) Hall thruster by using probes. To explain the presence of rotating spokes in their experiment, they invoked the Critical Ionization Velocity (CIV) concept. The CIV concept implies that the rotation of the spoke is associated with a rotating double layer (charge separation) with an azimuthal potential drop on the order of the ionization energy of the gas, E i . The rotation is related to the azimuthal acceleration of the ion at the critical ionization velocity, VCIV ¼ ð2eE i =MÞ1=2 . The spoke rotation velocity measured by Janes and Lowder for different gases showed good agreement of this scaling with ion mass and ionization energy (Fig. 16). Spoke rotation associated with the CIV concept has been observed and thoroughly analysed in the 1970–1990s in the context of homopolar discharge devices,8,9 and more recently in magnetron discharges.53

FIG. 17. Potential distribution in a rotating spoke in a cylindrical magnetron discharge as predicted by Particle-In-Cell simulations (uniform magnetic field perpendicular to the simulation domain. After Ref. 55.

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field is still large in the region between the plasma source and the anode. The CIV mechanism in this case provides the source of ionization necessary to maintain a quasineutral plasma in that region while allowing electron transport to the anode. This seems to be the case in the experiment of Janes and Lowder, but the magnetic field distribution in modern optimized Hall thrusters and the characteristic length of the magnetic field decay should be sufficiently low in the anode region to avoid this regime. We finally note that, in recent experiments,56,57 the presence of spokes does not seem to affect the performance of the thruster. This tends to suggest that the spokes are present in a region of the channel where the power dissipated by electrons is small (i.e., not in the acceleration region, where the power dissipated by electrons is large and where a significant change in electron conductivity would affect the overall power balance and performance). This is consistent with the measurements of Sekerak et al.,56,57 which show that the spoke velocity matches the ion acoustic velocity for 5 eV electrons that are present in the near-anode and near-plume regions of the discharge. c. Axial transit time oscillations. The frequency of these oscillations is in the 100–500 kHz range and is associated with the transit time of ions in the acceleration region (the transit time of 10 km/s ions over 1 cm is 1 ls). These axial oscillations were first observed by Esipchuk et al.58 and a linear analysis of their mechanism was proposed by the same authors59 (see also the more recent paper of Barral et al.60). The dispersion relation for these waves can be obtained by writing and linearizing a system of 1D fluid equations based on electron and ion continuity equations, collisionless momentum equation for cold ions, and a drift-diffusion momentum equation for electrons, with the constraint of constant current (neglecting displacement current). The system can be further simplified by removing the effect of ionization and electron pressure leading to the resistive mode, as described by Fernandez et al.61 The authors note that “the resistive mode results from the interaction of the ion beam with the resistivity mediated electron flow under the constraint of current continuity and quasineutrality.” The growth rate of this mode was shown to vary with the electron mobility as le;E 1=2 . The excitation of the transit time oscillations and their non-linear growth were illustrated in a 2D hybrid model of a Hall thruster by Bareilles et al.29 This paper showed that the amplitude of the oscillations was strongly dependent on the electron resistivity (controlled by an adjustable effective mobility) and increased with decreasing electron mobility, in total agreement with the resistive mode described by Fernandez et al. The results of the 2D hybrid model of Bareilles et al. also showed that the instability is associated with important kinetic effects coupled with oscillations of the acceleration zone.29 During the oscillations of the acceleration zone, some ions may “see” a total potential drop larger than the applied voltage while a group of low energy ions is formed. Although this cannot be predicted by the linear analysis based on fluid equations, this instability is due to

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a too large electron resistivity, as predicted by the fluid dispersion relation of Fernandez et al.61 Transit time instabilities have been predicted and observed in analytical and numerical models. Experimental studies of these instabilities are rather scarce and it is only recently that time-resolved Laser Induced Fluorescence measurements of the local ion velocity distribution function have confirmed the presence of this instability in a Hall thruster.62,63 d. E  B electron drift instability. 2D Particle-In-Cell Monte Carlo Collisions simulations have shown the presence of large amplitude, small wavelength azimuthal oscillations of the azimuthal electric field.26,42,64,65 These instabilities result from the coupling between electron Bernstein waves and ion acoustic waves and had been studied in the 1970s in other contexts66–70 (e.g., shockwaves propagating perpendicular to a magnetic field). Experimental evidence of these instabilities has been obtained using collective light scattering experiments.26,43,71,72 We will describe these instabilities in more detail in Section III D). F. Basic concepts and scaling 1. Importance of the magnetic field design

The axial distribution of the radial magnetic field barrier in a Hall thruster controls the electron conductivity and therefore the axial electric field. Electron transport, electric field distribution, and ionization are directly related to the distribution of the radial magnetic field. Moreover, the magnetic field distribution controls the ion beam divergence through the potential distribution and controls the wall erosion by ions. An important aspect concerning the magnetic field distribution and magnetic field lines is that magnetic field lines tend to be equipotential (this would be exactly true for a zero electron temperature). In the case of a Hall thruster, the electric field force along a magnetic field line is balanced by the electron pressure along B. Therefore Ek  Te rk lnðne Þ   kTeo ne ln ) /o ¼ /  ¼ constant: e no This shows that, in the case of small density gradients, the magnetic field lines are essentially equipotential and so determine the ion acceleration trajectories and the beam divergence. In the presence of large plasma density gradients or large electron temperatures, the equipotential lines may be significantly different from the magnetic field lines and the design of the magnetic field distribution to optimize divergence and wall erosion is not straight forward. 2. Length of ionization region

The length of the ionization region is an essential parameter in Hall thrusters. For a given neutral atom density in the anode region, and electron temperature, it can be estimated from the 1D atom continuity equation:

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@t na þ @x Ca ¼ ne na ki ¼ Ca ne ki =va na is the atom density, Ca ¼ na va is the atom flux, va the atom mean velocity, ne the electron density (plasma density), and ki the ionization rate (which depends on electron temperature). The right hand side of the equation corresponds to the atom losses due to ionization. Assuming steady state, this equation gives an estimate of the decay of the neutral atom flux Ca ¼ na va due to ionization (obtained assuming a constant product of the plasma density and ionization rate): Ca ðxÞ ¼ Ca;0 exp ðx=ki Þ; where na;0 is the atom density at the anode, and ki is defined as ki  nveaki : For typical operating conditions and in the ionization region of a Hall thruster, va  200 m=s, ri  5  1020 m2 (values of the xenon ionization cross section ri can be found in Ref. 17) ki ¼ hri ve i  1013 m3 =s, and ne  1018 m3 , we obtain ki  2 mm (this value is only a rough estimation). Ionization of a large fraction g of the atom flux will therefore take place on a length (the ionization length), deduced from:3 1  g ¼ exp ðLi =ki Þ. The ionization length is therefore Li ¼ ki lnð1  gÞ. For g ¼ 0:95 (95% of the xenon flux is ionized over the ionization length Li ) this gives an ionization length of Li  6 mm. The ionization length must be of course sufficiently smaller than the channel length. Another way to look at the ionization length is to start from the steady state electron continuity equation written for the electron flux: @x jCe j ¼ jCe jna ki =vex , where vex is the electron drift velocity (absolute value) parallel to the applied electric field. Integration of this equation for the electron flux, assuming that the other parameters are constant, gives another estimate of ki (continuity implies that Ca  jCe j should be constant, i.e., the decay of the atom flux from the anode to the cathode takes place on the same length as the increase of the electron flux from the cathode to the anode): ki  vex =ðna ki Þ. One can see in this expression that for a given atom density and ionization rate, the electron mean axial velocity controls ki and therefore the length of the ionization region Li . Since the electron mean axial velocity is strongly related to the magnetic field (and varies as 1/B2 or 1/B for collisional or turbulent transport), it is clear that the value of the radial magnetic field controls the length of the ionization region. The length of the ionization region therefore decreases with increasing magnetic field strength.

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scaling of Hall thrusters, depending on the parameters to optimize (efficiency, lifetime) and constraints (magnetic circuit, thermal load,…). In this section, we just give a few basic ideas on scaling. Detailed discussions on the subject can be found, e.g., in Refs. 73–79. The size of Hall thrusters increases with power. On analysing a large number of existing optimized thrusters,77,78 it appears that the channel mean diameter D and width H stay proportional when the discharge power PD is changed and that the thrust and power are proportional to DH (or D2), PD / HD, and to the square root of the applied voltage, since _ i;b and the the thrust T is given approximately by T ¼ mV beam ion velocity varies as Vi;b  ð2eUD =MÞ1=2 . This is illustrated in Fig. 18. This means that the mass flow rate should vary as DH. Therefore, the injected flux of neutral atoms and the ion current density (and discharge current density) should stay constant when the power is changed. The discharge current density in optimized Hall thrusters does not vary much with power and is around 1000 A/m2 as estimated in Section II B for a SPT100 (a range of 1000 to 1500 A/m2 is indicated in Ref. 3). Since the injected flux of xenon stays constant for different powers, the gas density should also stay constant,77,78 on the order of 1–3  1019 m3. When changing the power PD of a Hall thruster, the simplest scaling is therefore to increase the radial dimensions 1=2 homothetically as PD and change the mass flow rate proportionally to PD . For the axial dimension, the simplest scaling is to keep the same axial magnetic field distribution (for the same operating voltage) and the same channel length. The questions of magnetic field distribution and channel length are actually more complex because other parameters, such as the electron interaction with walls (which is more important for smaller thrusters) or the thermal load, must be considered (see the references cited above for a discussion of these questions).

3. Thruster scaling

Designing Hall thrusters of different dimensions and powers is still rather empirical because of the incomplete understanding of cross-field electron transport. However, a considerable experience has been accumulated in (empirically) optimizing thrusters with different powers and thrusts (from 10 s of watts to 100 kW and thrust from a few mN to several N). There are different strategies for the design and

FIG. 18. Analysis of existing Hall thrusters of different powers showing that the channel width H stays proportional to the mean channel diameter D (top) and that the thrust (and discharge power) increases as D2 (bottom). After Ref. 77.

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III. MODELING AND SIMULATION A. Fluid and hybrid models

Fluid models of Hall thrusters are based on fluid equations for electrons, ions, and neutrals assuming quasineutrality. In quasineutral models, the electron and ion densities are supposed to be equal and the electric field is not obtained from Poisson’s equation but rather from a continuity equation for the total current. For electrons, the fluid transport equations consist of continuity, momentum in the drift-diffusion form, and energy equations. The electron velocity distribution function is highly anisotropic and some models include different energy equations for the electron temperature parallel to the applied electric field and parallel to the magnetic field (because electron-wall interaction depends on the latter). The positive ion fluid equations are generally the continuity and momentum equations (for cold ions or for ions with a given ion temperature). Because of quasineutrality, the electron and ion continuity equations are identical and are only one single equation. In hybrid models, electrons are treated as a fluid while ion transport is described with a particle model, i.e., ions are represented by super-particles (one ion of the simulation represents a large number of real ions). In a Hall thruster, the gas density is quite low so neutral atoms collide with the walls much more often than among themselves so a DSMC (Direct Simulation Monte Carlo) approach is not needed. Neutral atoms can be described as a fluid with more or less simple fluid equations or as particles in a Particle-In-Cell model. One parameter of the neutral atom transport model is the nature of the reflexion of atoms by the channel walls, i.e., specular or diffuse. The principles of a quasineutral fluid or hybrid model are illustrated below with the description of one integration time step. We assume that the model is one-dimensional (1D) in the axial direction x or two-dimensional (2D) in the radial-axial, (r,x) plane. The azimuthal direction is generally not described in fluid or hybrid models and it is assumed that the plasma is axisymmetric and uniform in the azimuthal direction. 1. Ion transport

In a fluid model, knowing the electric field E and ionization rate na ki ðTe Þ at the beginning of the time step, the ion continuity and momentum equations are solved for one time step. In a hybrid model, ion super-particles are moved for one time step according to the electric field force and taking into account particle generation by ionization and losses to the walls. The plasma density n and ion flux Ci ¼ nvi are known at the end of the time step. 2. Electron temperature

Knowing the plasma density and electric field, and assuming a drift-diffusion electron flux, the new electron temperature is obtained by integrating the electron energy equation over one time step assuming a Maxwellian electron distribution,

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3 @nTe 5 5 þ r:ðCe Te Þ  r:ðle nTe rTe Þ 2 @t 2 2 ¼ E:Ce  na nj  nW; where j is the energy loss rate due to collisions with atoms (deduced from the electron-atom cross-sections and electron temperature) and W corresponds to the energy losses to the channel walls. (The electron mobility le is the mobility parallel to the applied field and perpendicular to B in a 1D model and is a tensor in a 2D model.) 3. Electric field

The new electric field at the end of the time step is obtained from the combination of the current continuity equation r:ðCi  Ce Þ ¼ 0 and the electron momentum equation in the drift-diffusion form (inertia is neglected) Ce ¼ nle E  le rðnTe Þ. This gives: r:ðnle EÞ ¼ r:½Ci  le rðnTe Þ, which can be solved for E knowing the plasma density, ion flux, electron temperature, and electron mobility (including the contribution of anomalous transport). Note that the electron mobility is actually not isotropic (mobility is much smaller in the direction perpendicular to the magnetic field than in the direction parallel to B), so that the temperature and electric field equations are actually tensor equations. 4. Neutral atom transport

Atoms are moved for one time step taking into account wall collisions and ionization. If atoms are represented by super-particles, ionization can be taken into account by decreasing the weights of the super-particles according to the local ionization rate. The procedure above is straightforward in 1D models where the electric field can be obtained simply from: 9 ID = Ci  Ce ¼ e ; Ce ¼ nle E  le rðnTe Þ 1 )E¼ ½eCi  ID þ ele rðnTe Þ; enle where ID is the discharge currentÐ calculated for a given d applied voltage UD . By writing that 0 Edx ¼ UD (d is the cathode-anode distance), this gives: UD  ID ¼

ðd

Ci þ lrðnTe Þ dx nl 0 : ðd 1 dx 0 enl

The problem is actually much more complicated in a 2D model because of the strong anisotropy of the electron mobility. Instead of trying to solve the electric field (or potential) equation in the form given above, r:ðnle EÞ ¼ r:½Ci le rðnTe Þ, the direction perpendicular and parallel to the magnetic field lines can be treated separately.28,80 In the direction parallel to the magnetic field lines, the electric force

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in the electron momentum equation is supposed to balance the pressure gradient (we note l? and lk the mobility perpendicular and parallel to B, respectively): Ce;k ¼ nle;k Ek  le;k rk ðnTe Þ  0; which gives; for the potential : Uðx; rÞ ¼ U ðkÞ þ Te ðkÞlnðn=n0 Þ; where k is constant along a magnetic field line, i.e., is a stream function deduced from the magnetic field distribution: @x k ¼ rBr ; @r k ¼ rBx . The problem then becomes one-dimensional in the direction perpendicular to the magnetic field where the perpendicular electron flux can be written as: Ce;? ¼ rBl? n@k U  rBl? @k ðnTe Þ Integrating this equation along the magnetic field lines and using current continuity (with the known ion flux), one gets a first order differential equation for the electric potential V. More details on 1D and 2D fluid and hybrid models can be found in Refs. 27, 28, 32, 33, 35, 36, 80, and 81. We note that the recent 2D model of Mikellides and Katz36 does not treat separately the directions parallel and perpendicular to the magnetic field, as shown above, but solves the equations on a computational mesh that is aligned with the applied magnetic field (which is still a numerical challenge because the electron mobility parallel to the magnetic field can be orders of magnitudes larger than in the direction perpendicular). The 1D model of Hara et al.33 is different from most hybrid models since it is based on a hybrid-direct kinetic simulation, where the electrons are treated as a fluid while ion and neutral atom transport is described by directly solving the Boltzmann equation with a high accuracy scheme instead of using a particle simulation (this would be probably more challenging in 2D). Finally, we note the recently published paper by Lam et al.82 on a 2D hybrid Hall thruster model that resolves the E  B direction (axial-azimuthal model). This is certainly an interesting approach since the most important instabilities contributing to cross-field transport develop in the E  B directions. The model indeed predicts the development of such instabilities and it would be interesting to compare these predictions with those of the Particle-In-Cell simulations described in Section III D. The computational aspects of the 2D axial-azimuthal fluid equations are however very challenging and extensive validation tests should be performed.

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CSEE ¼ cCe;w is the flux of secondary electrons emitted by the walls under electron impact. The secondary emission coefficient c is a function of electron temperature and can become larger than one for sufficiently large electron energy (Fig. 19). Writing the usual expressions for the charged particle fluxes to the walls, one gets the classical sheath potential modified by secondary emission (see, e.g., Ref. 34): " rffiffiffiffiffiffiffiffiffi# M Us ¼ Te;?w ln ð1  cÞ 2pm Note that the electron temperature Te;?w in this expression is the temperature in the direction parallel to the magnetic field (i.e., perpendicular to the wall), which can be significantly smaller than the temperature in the axial direction in a Hall thruster. This expression is not valid when the secondary electron yield becomes near or larger than 1, which is common in Hall thrusters operating at sufficiently large voltages. When c approaches 1, the sheath becomes space charge saturated and can no longer extract the secondary electrons (SCS, space-charge-saturation regime). In that case, the theory of Hobbs and Wesson83 indicates that: cSCS  0:98; DUs;SCS  1:02Te;?w Note that for xenon, the Debye sheath potential without qffiffiffiffiffiffi M secondary emission, Us ¼ Te;?w ln 2pm  5:28Te;?w is about five times larger than the Debye sheath potential calculated in the SCS regime. The ion flux to the walls is the Bohm flux, which can be estimated knowing the plasma density at the sheath edge and the electron temperature. The electron flux to the wall can then be deduced from the ion Ci;w . This flux can be very large flux to the wall by Ce;w ¼ 1c when the sheath approaches the saturation regime, leading to very large electron collision frequencies with the walls and large energy losses. Considering a volume element Ac dx of the channel, where Ac ¼ pðR22  R21 Þ, the electron-wall collision frequency can be defined as the number of electrons hitting the walls per unit time divided by the number of electrons in the

B. Accounting for anomalous electron transport in fluid and hybrid models

In this section, we briefly summarize the research efforts to include the effects of electron collisions with the channel walls in the electron momentum and energy equations of fluid models. We start by writing that no net current can flow to the walls, Ci;w ¼ Ce;w  CSEE ¼ ð1  cÞCe;w , where Ci;w and Ce;w are the electron and ion fluxes to the channel walls and

FIG. 19. Secondary electron yield for several wall materials used in Hall thrusters. After Goebel and Katz3.

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ðR1 þR2 ÞdxCew ew volume element, i.e., w ¼ 2p ¼ 2C hniH , where H ¼ hnipðR22 R21 Þdx R2  R1 is the channel width and hni is the averaged plasma density in the volume element. This gives, as in Barral et al.34

2Cew 2Ciw 1 1 ¼  2v uB ; hniH hniH ð1  cÞ H 1c  1=2 eT where uB ¼ Me;?w is the ion Bohm velocity calculated for the electron temperature perpendicular to the wall and v is the ratio of the ion density at the sheath edge to the density at the middle of the channel. In the SCS regime, this gives w  60 uHB  107 s1 , where we assumed v  e1=2  0:6 (ratio of the density at the sheath edge to the plasma density). The energy loss power flux density to  the wall can then be 2Te;?w þ DUw Ci;w , where DUw is estimated84 as qe  e 1c the total (sheath plus presheath) potential drop. This corresponds to very large energy losses per unit time, close to W  2Te;?w w . Kaganovich et al.84–86 showed that this fluid approach actually leads to a large overestimation of the energy losses to the walls. Using kinetic (Particle-In-Cell) simulations in the radial direction of a thruster channel they showed that the electron distribution function is not only strongly anisotropic in the channel (because electrons gain energy in the axial direction and the small electron-atom collision frequency leads to a small redistribution of the energy in the radial direction) but is also depleted at high energy due to wall collisions and secondary electron emission. On the basis of these kinetic simulations, they provided analytical estimates of the wall potential, electron-wall collision frequencies, and energy loss frequency. Fig. 20 displays an example of electron velocity distribution function perpendicular to the walls obtained in the kinetic simulations of Ref. 85 and shows the depletion of electrons in the tail of the distribution above the sheath potential, and the presence of secondary beam electrons. w ¼

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The effective collision frequency (to include in the effective electron mobility) obtained in the theory of Kaganovich et al.85 writes w 

cp 1 uB H 1  cb

In this expression, the authors distinguish the secondary emission coefficient cb due to the “beam electrons” emitted by the walls and accelerated by the sheath toward the opposite wall, from the secondary coefficient of bulk plasma electrons cp . Quantitatively, this expression gives values of the effective collision frequency that are not very different from those of the theory above of Barral et al.34 (especially at large applied voltages) but the physics underlying the two expressions is completely different. In the theory of Kaganovich et al., the secondary electrons emitted by one wall are accelerated toward the other wall, lose energy while crossing the other sheath and reach the wall. While crossing the channel, these electrons can drift (about one Larmor radius) in the axial field, toward the anode. The flux of electrons to the walls is much smaller in this theory and is given by: rffiffiffiffiffiffiffiffiffiffiffiffiffi   H eTe;?w U ; exp  Cew ¼ ne 2pm 2ken Te;?w where ken is the electron mean free path for collisions with neutrals and ne the plasma density in the center. We see in this expression that there is no electron flux to the walls in the limit of no electron collisions with neutrals (infinite ken ). Physically this is due to the fact that collisions are needed to redistribute the electron energy and generate electrons with sufficient energy in the direction perpendicular to the walls to overcome the Debye potential. This also results in much smaller electron energy losses than in the above theory. The electron energy loss frequency obtained from this theory is proportional to the electron-neutral collision freTe;x , the sheath quency and is given by:85 W  en Te;x Te;x þU s  q ffiffiffiffiffiffiffiffiqffiffiffiffiffiffi Te;x M potential Us being given by: Us ¼ Te;x ln kHen Te;?w 2pm . Kaganovich et al. also give a relation between the axial electron temperature and the temperature perpendicular to the wall Te;?w ¼

FIG. 20. Electron Velocity distribution function (EVDF) normal to the walls as a function of energy perpendicular to the wall. Solid black line, total EVDF; dotted blue line, bulk electrons; dashed green line, SEE beam electrons; dot-dashed magenta line, Maxwellian EVDF at 10.1 eV. After Kaganovich et al.85

Us Te;x : Us þ Te;x

Numerically, the equations above give, for example, for xenon and for ken =H  300 (which is consistent with a neutral density between 1018 and 1019 m3) Us  Te;x , Te;x  2Te;?w , and W  12 en Te;x . Note that turbulence can also contribute to redistribute the electron energy in all directions so the expression above of W tends to underestimate the energy lost per unit time to the walls (the electron-neutral collision frequency en should be replaced by an effective collision frequency taking into account the effect of turbulence or instabilities).

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To our knowledge, the effective collision frequency and energy loss frequency obtained from this theory have not been used in a 1D or 2D fluid or hybrid model of a Hall thruster. It would be, however, very interesting and useful to compare the results of fluid and hybrid models using anomalous electron transport from the kinetic theory above of electron-wall interaction in the presence of SEE with experimental results (and with results obtained with models using the fluid theory above of electron-wall collisions). C. Particle simulations of azimuthal instabilities

Particle-In-Cell simulations have been used in Hall thruster models to study the electron-wall interaction (see above) or in 2D radial-axial simulations (see, e.g., Refs. 87–90). 3D simulations of electron trajectories in a non-self consistent electric field (obtained from plasma potential measurements) have been performed by Smith and Cappelli52,91 in order to better understand the near-field electron transport from the cathode to the channel entrance. They show that the three-dimensionality of the E and B fields (e.g., nonaxisymmetric E or B fields), together with electron-wall collisions, appears to be important drivers of cross-field transport in this region of the discharge, and could lead to sufficient levels of electron transport to the channel without invoking plasma turbulence. Since fully self-consistent 3D simulations of Hall thrusters are not available, we must consider that the question of the role of instabilities and turbulence in electron transport in Hall thruster is still an open question. In this section, we concentrate on the use of selfconsistent (2D and 1D) particle simulations to understand and quantify anomalous electron transport due to instabilities and turbulence. The advantage of fully kinetic Particle-InCell simulations is that no assumptions are made on the charged particle distributions functions. Moreover, some of the instabilities and turbulence phenomena are associated with kinetic effects and cannot be described accurately with a fluid approach. We briefly recall here the principles of Particle-In-Cell Monte Carlo Collisions (PIC MCC) simulations, with emphasis on the specific features associated with the simulations of Hall thrusters. 1. PIC MCC method: Principles and constraints

The basic principles of the PIC MCC method can be found in Refs. 92–94. Fig. 21 summarizes the principles of

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the method by showing the successive operations performed during one time step of an explicit, electrostatic PIC MCC simulation. The trajectories of a large representative number of particles (super-particles) are followed in phase space under the influence of the electric forces (here, due to the self-consistent electric field and external magnetic field) and collisions. The electric field is calculated (step 4 of Fig. 21) at each time step on a grid from the charged particle densities on that grid, deduced from the positions of the superparticles (step 3 of Fig. 21). The electric forces are then calculated on the grid. The equations of motion of the superparticles are then integrated in time assuming that the electric field and forces stay constant during the time step (explicit method (step 1 of Fig. 21). Then, a Monte Carlo module (step 2 of Fig. 21) is used to decide which particles are undergoing collisions during the time step, the nature of the collisions, the change in velocity, etc…These choices are made on the basis of random numbers following probability densities associated with the collision cross-sections. The new charged particle densities can then be calculated from the new super-particle positions, and so on. There are important constraints associated with PIC MCC methods. In explicit simulations, because the electric field is supposed to be constant during one time step, the integration time step must be less than the inverse of the plasma frequency. The grid spacing must also be limited to make sure that one particle does not move over more than one grid interval during the time step. Another constraint is that the number of particles per spatial cell of the simulation must be large enough to avoid errors and numerical heating. The constraints usually imposed in explicit PIC MCC simulations are as follows:93 Dt <

0:2 ; Dx < kDe ; Ncell > 50: xpe

Here, Dt is the integration time step, Dx is the grid spacing, Ncell is the number of particles per spatial cell, xpe is the electron plasma angular frequency, and kDe is the electron Debye length. The constraints on the time step and grid spacing are quite severe for high plasma densities. In the conditions of a Hall thrusters, with plasma densities and electron temperatures on the order or larger than 3  1018 m3 and 20 eV, respectively, we have xpe  1011 rd=s and kDe  20lm, which gives Dt < 2  1012 s; Dx < 20lm. Steady state is reached after more than 100 ls in Hall thrusters (the breathing oscillations are in the 10–20 kHz

FIG. 21. Diagram showing one integration time step of a PIC MCC simulation

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range), i.e., in about 5  107 time steps. The number of grid points over a length Lx ¼ 4 cm would be Nx ¼ Lx =kDe ¼ 2000. The total number of grid points for a 2D (r,x) simulation on a 2 cm  4 cm would be Nx  Nr ¼ 2000  1000, and, assuming 50 super-particles per cell, the total number of particles of each type would be Nparticles ¼ 108 . An estimation of the computation time assuming parallelization over 8 processors with 5 cores each gives several months for the conditions above (this corresponds to a fraction of CPU second per time step). It is therefore clear that 3D simulations in these conditions would be extremely difficult. One way to reduce the computation time is to use implicit PIC methods. These methods use a prediction of the field variations during one time step and allow use larger time steps and grid spacing than in explicit simulations (see, e.g., Adam et al.42 in the context of Hall thrusters). 2. Scaling to decrease computation time

PIC MCC simulations are very time consuming because of the small time step and grid spacing that are necessary to resolve the plasma frequency and Debye length. One way to reduce the computation time is to perform the simulations at lower plasma density (the ionization rate must be scaled to describe correctly the neutral depletion and low frequency oscillations). The obvious problem when scaling the plasma density is that the ordering of the Debye length with the other characteristic lengths (Larmor radius, mean free path, device dimensions, etc…) is not conserved, which may or may not have important consequences depending on the plasma that is simulated. The scaling may have important consequences when physical phenomena associated with charge separation take place. This can be the case when instabilities or turbulence are present so that simulations using a scaled plasma density are not able to reproduce the physics in these conditions (see Ref. 95 in the context of Hall thrusters). Reducing the plasma density by a factor s is actually equivalent to multiplying the vacuum permittivity by the same factor (see, e.g., Ref. 96). Other authors have been using a scaling based on reducing the dimensions of the device while keeping the real plasma density. In that case, the magnetic field and collision cross-sections are increased by the same scale factor. In all these scaling, the ratios kD =L, qL =L, kn =L are the same (kD ; qL ; kn ; and L are, respectively, the Debye length, Larmor radius, mean free path for collisions with neutrals, and device dimensions). qL =L and kn =L are the same as in the simulated problem but kD =L is not (the plasma density is reduced and the dimensions are the real ones in the density scaling, or the plasma density is the real one and the dimensions are reduced in the dimensions scaling). The questions of scaling in PIC MCC simulations are discussed for example, in Refs. 87 and 88. A third type of scaling, based on changing the electron to ion mass ratio, is used in Refs. 89 and 90. The authors claim that this scaling allows a better description of the effect of plasma turbulence but (1) plasma turbulence may depend on the ion mass and (2) axial-radial models, which have been used in these

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papers, cannot describe the most important instabilities in a Hall thruster, which are azimuthal. To summarize this section, scaling can be an efficient way to reduce the computation time in PIC simulations but cannot be used to describe instabilities and turbulence and must be used with care or adapted when sheath phenomena play an important role. 3. Radial-axial (r,x) vs axial-azimuthal (x,h) PIC simulations of Hall thrusters

The E  B configuration in Hall thrusters plays an important role in the triggering of instabilities and turbulence. The large Hall current in the acceleration region of Hall thrusters is clearly an important potential source of instabilities. Such azimuthal instabilities are generally associated with the formation of a non-zero fluctuating azimuthal electric field that can contribute to electron cross-field transport. Obviously, such instabilities cannot be described in a 2D (r,x) simulation and thus potentially important aspects of plasma turbulence in Hall thrusters are not present in 2D (r,x) simulations. A description of the azimuthal instabilities requires that the azimuthal direction be included in the simulations; the simulation plane in a 2D model must be (x,h) or (r,h). The (x,h) plane allows study of the development of azimuthal instabilities and their impact on axial electron transport26,42 but cannot give detailed information on the electron-wall interactions, while the (r,h) plane can be used to study both azimuthal instabilities and electron-wall interaction (and their coupling)65 but gives limited information about axial transport. Note also that 1D radial PIC simulations have also been used to derive useful information about electron-wall interaction and their effect on axial transport85,86 (see Section III B above) and that 1D azimuthal PIC simulations have been performed to study in a simpler way and with more detail the azimuthal instabilities present in a Hall thruster.55,64,97,98 We summarize in Sec. III D the development of azimuthal instabilities as predicted by axial-azimuthal PIC MCC simulations. D. Anomalous transport from axial azimuthal particle simulations and theory

The total length of the azimuthal direction in Hall thrusters is quite long so axial-azimuthal simulations are generally restricted to a small part of the azimuthal direction with periodic boundary conditions. Note that this limits the wavelength or the number of modes of instabilities that could be potentially present in that direction. In the simulations of Adam et al.,26,42 the length of the simulation domain was 5 mm while the length in the axial direction was slightly longer than the length of the thruster channel, i.e., from 3 to 4 cm. The problem can be further simplified by neglecting the curvature of the channel cylindrical walls (which is a reasonable approximation if the channel width is much smaller than the mid-channel radius, i.e., H D=2). In that case, the simulation domain (see Fig. 22) is a rectangular domain in the plane (x,z) where z represents the azimuthal direction. In

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FIG. 22. Simulation domain in an axial-azimuthal PIC MCC simulation. The geometry is rectangular, x being the axial direction of the thruster and z the azimuthal direction. Cec is the electron flux generated in the simulation on the cathode side and is taken equal to the electron flux minus the ion flux reaching the anode, Cea  Cia . The net electron flux entering the channel is Cec1 while an electron flux Cec2 neutralizes the ion beam flux Cic (these three fluxes are results of the simulation).

the simulation domain, electrons are injected from the cathode side plane boundary at a rate ensuring current continuity (Fig. 22). Neutral atoms are injected uniformly in the anode plane and their transport can be described by a simplified 1D fluid equation. The axial-azimuthal PIC MCC simulations of Adam et al. were able to predict most of the known features of the Hall thruster physics: periodic depletion of the neutral density, current oscillations in the 10–20 kHz range, distribution of the axial electric field, etc.,… The important point of these simulations is that they did not make any assumptions on cross-field electron transport and that the observed anomalous electron transport was entirely self-consistent and driven by an azimuthal instability that the authors characterized analytically. The azimuthal instability predicted by the axialazimuthal PIC MCC simulation can be seen in Fig. 23. We see large amplitude fluctuations of the azimuthal electric field (as large as 300 V/cm while the maximum axial electric field, also shown in the figure is on the same order or slightly larger) with a small spatial period, on the order of one mm. This instability is responsible for a substantial enhancement of the electron cross-field mobility as seen in Fig. 24. The electron mobility le;E in the anode region is close to the collisional mobility while the mobility in the exhaust region is one order of magnitude larger. The Hall parameter at the exhaust that can be deduced from this mobility, h ¼ ðBle;E Þ1 , is on the order of 250 instead of several thousand if one assumes that 90% of the atom flux is ionized in the exhaust region, indicating that the effective collision frequency is at least ten times larger than that corresponding to electron-atom collisions (see also the experimental measurements of the inverse Hall parameter of Fig. 14). Even more intriguing is the very large mobility that can be seen in Fig. 24 outside the channel. The azimuthal instability predicted by the axialazimuthal PIC MCC simulations has been analysed by Adam et al. in several papers26,42,64,65 and shown to be of the same nature as the E  B Electron Drift Instabilities (EDI) studied in the 1970s in other contexts66–70 such as shockwave propagation across a magnetic field. This instability has also been

FIG. 23. (a) Axial-azimuthal distribution of the azimuthal component of the electric field at a given time of a PIC MCC simulation of a Hall thruster (channel length 2.5 cm). (b) Radial magnetic field and axial electric field (averaged over time and azimuthal direction) profiles. The exhaust plane is indicated by a dashed line. The applied voltage is 300 V and the xenon mass flow rate 5 mg/s. The dashed green line indicates the position of the exhaust plane. After Adam et al. 26

called electron cyclotron drift instability or beam cyclotron instability. For Hall thrusters, the most appropriate name is E  B electron drift instability and we will use this name in the rest of this paper. The E  B EDI results from the coupling between electron Bernstein modes and ion acoustic modes. The instability is kinetic and its dispersion relation can be obtained by linearizing the electron Vlasov equation coupled with ion cold fluid equations (continuity and momentum) and Poisson’s equation. The dispersion relation can be written as64,99

FIG. 24. Time integrated effective cross-field electron mobility deduced from the PIC MCC simulations of FIG. 23. The dashed green line indicates the position of the exhaust plane.

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k2 k2De x2pi x  kz Vd 2 1 þ k2 k2De þ g ; kx þ kz2 ; ky2 q2  Xce ðx  kx Vi;b Þ2 ¼0 where g is the Gordeev function,64,99 kx ; ky ; kz are the components of the wave vector k parallel to E; B; E  B, respectively, Vd is the electron azimuthal drift velocity, xpi is the ion angular plasma frequency, kDe the electron Debye length, q the electron Larmor radius at the thermal velocity, and Vi;b the mean velocity of the ion beam. Ducroq64 et al. considered the 2D dispersion relation only in the (x,z), axial-azimuthal plane and found that transitions from stability to instability occur when x  kz Vd ¼ nXce . The wave angular frequency was found to be much smaller than the electron cyclotron angular frequency so that the instabilities were in small intervals around kz Vd  nXce (resonance points), where n in an integer (wave vector multiple of the inverse electron Larmor radius at the electron azimuthal drift velocity). The mode frequencies are on the order or smaller than the ion plasma frequency. Their wavelength as seen in the PIC simulations is in the mm range in the conditions of a Hall thruster. The mechanism of the instability can be described as follows:68 Berntsein waves are Doppler shifted towards low frequencies by the large electron drift velocity Vd and reach the ion acoustic wave range. The instability occurs when the two modes merge. In Ref. 99, Cavalier et al. present a very clear parametric study of the three-dimensional dispersion relation of the

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EDI. Some results of this study are shown in Fig. 25. They show that the amplitude of the oscillations of the dispersion relation at the resonant points kz Vd  nXce smooths out as long as the wave-vector along the magnetic field is non-zero (typically ky kDe > 0:03). The dispersion in these conditions tends to an asymptotic curve corresponding to what they call the “modified ion acoustic instability.” The instability is triggered by the electron drift velocity Vd , hence the name E  B electron drift instability. The angular frequency of the wave is not strongly affected by the value of the electron drift velocity as long as it stays on the order or below the electron thermal velocity and is on the order or smaller than the ion angular plasma frequency. The instability can also be analysed in 1D PIC MCC simulations in the azimuthal direction assuming a given axial electric field and external magnetic field.55,97,100 As predicted by the dispersion relation, the instability develops on very short time scales (for sufficiently large plasma density). An example of the results is shown in Fig. 26. In Fig. 26(a), the charged particle densities and electric field distributions are plotted along the azimuthal direction (a finite length of 5 mm of this direction is simulated, with periodic boundary conditions. The given axial electric field and radial magnetic field are fixed at E0 ¼ 2  104 V/m and B0 ¼ 2  102 T. The average plasma density, n0, is 5  1016 m3. Collisions are included in the simulation, with a xenon density na ¼ 1.6  1020 m3. The length of the acceleration region is 1 cm. Although the axial direction is not considered in the simulation, particles are followed along this direction and each

FIG. 25. Solutions of the 3D E  B EDI dispersion relation—Adapted from Cavalier et al.99 kz and ky are the wave vector components in the azimuthal and radial (parallel to B) directions, respectively. x and c are the real part (angular frequency) and the imaginary part (growth rate) of the instability; xpi is the ion angular plasma frequency, vthe the electron thermal velocity, kDe the electron Debye length. The calculations are made for kDe ¼ 8.3  105 m, xpi ¼ 5.1  107 rd/s, (n ¼ 2  1017 m3, Te ¼ 25 eV, Vi,b ¼ 16 km/s).

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FIG. 26. (a) Example of E  B electron drift instability obtained with a 1D PIC MCC simulation in the azimuthal direction (see text); (b) phase diagram of the ions in the same simulation. After Ref. 55.

electron (resp. ion) leaving the simulation domain on the anode (resp. cathode) side is replaced by a new electron (resp. ion) at the cathode (resp. anode) side. In this particular simulation, the instability is saturated by electron-neutral collisions and by the finite acceleration length, and steady state is reached in about 1 ls. Fig. 26(b) shows a phase diagram (vz, z) of the ions along the azimuthal direction illustrating the ion acoustic instability and showing ion trapping in the wave potential. The phase velocity of the azimuthal wave shown in Fig. 26(b) is on the order of the ion acoustic velocity. An important result of Ref. 55, which has been confirmed in Ref. 97 and analysed in details in Ref. 98, is that the amplitudes of the density and field fluctuations increase with the plasma density and that the resulting effective mobility is an increasing function of the plasma density. Lafleur et al.,97 citing the work on E  B electron drift instability of Lampe et al.,69,70 note that after a few growth times, nonlinear turbulent effects set in, which act to smear out the discrete nature of the EDI, causing a transition to an acoustic ion instability, which is consistent with the experimental observations of Tsikata et al.43 based on collective light scattering measurements (see also Refs. 26, 43, 71, 72, 99, and 101–103 for more details on the method and analyses of the results). The ion-acoustic wave then saturates due to ion-wave trapping. In Ref. 98, the authors present a study of the dispersion relation of the E  B EDI that is consistent with the work of Cavalier at al. Assuming that the instability

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is predominantly in the azimuthal direction, they show that pffiffiffi the maximum growth rate is obtained for pffiffikffi z;max kDe  1= 2 and for angular frequency xmax  xpi = 3 (consistent with Fig. 25) giving a phase velocity in the azimuthal direction on the order of the ion acoustic velocity. The maximum growth rate can be written as cmax  axpi Vd =vthe (with a  0:2). These results are consistent with the solutions of the dispersion relation of Cavalier et al.99 shown in Fig. 25. Lafleur et al. also deduce from these parameters the fluctuations of the azimuthal electric field associated with the wave (assum~ ¼ p1 ffiffi Te , ing a sinusoidal waveform). They show that jdEj 3 2 kDe i.e., the amplitude of the wave electric field increases with the plasma density, which is consistent with the PIC simulations.55,97 In their derivation of the dispersion relation, Lafleur et al. define and calculate an electron-ion friction force that can be used to obtain an estimate of the “effective collision frequency” characterizing the effect of the turbulence on electron cross-field transport. This approach is interesting although the expression obtained for the effective mobility depends on parameters that depend themselves on anomalous transport. This mobility can therefore not be used in a simple way for example, in fluid models of electron transport. Attempts at including in a Hall thruster fluid model the growth of the E  B EDI and its saturation and convection in the plume region had also been done in a recent paper by Katz and Mikellides.37 The studies of Lafleur et al. and Katz and Mikkellides are certainly very good steps toward including anomalous transport effects due to instabilities and turbulence in a fluid model, on a more physical basis than just using a 1/B mobility. We conclude that the PIC simulations indicate that the E  B EDI is a significant instability in Hall thrusters and certainly contributes to anomalous transport in the acceleration and near plume regions. Studies of the saturation and convection of the instability and of its consequence on anomalous transport have been undertaken very recently but more work is still needed to derive expressions of the effective cross-field mobility that could be used in fluid models. More experiments are also needed to study the presence of this instability in Hall thrusters. As said above, collective laser scattering experiments have been performed by Tsikata et al.43 to study the E  B EDI. Although these experiments showed the presence of density fluctuations of wavelength and frequency in agreement with the PIC simulations and theory, the measured amplitude of these fluctuations was much smaller than the model predictions. Other types of instabilities may be present and play a role in cross-field electron transport, especially in the region between the acceleration zone and the anode where the E  B EDI is not important. Research efforts to study these instabilities and their effect on electron transport are also necessary.104–106 IV. MAGNETICALLY SHIELDED AND DOUBLE STAGE HALL THRUSTERS

In this section, we describe the principles of a new and promising magnetic field design, the magnetically shielded Hall thruster, which considerably limits the wall erosion, and

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the concept of double-stage Hall thruster which is aimed at allowing separate control of thrust and specific impulse. A. Magnetically shielded configuration

A recent Hall thruster configuration, the magnetically shielded Hall thruster,21,22,107,108 has the interesting property to limit the plasma interaction with the channels walls and therefore reduce wall erosion. This is achieved by a special magnetic field design where the magnetic field lines do not intercept the channel walls (Fig. 27). The magnetic poles are outside the channel, which tends to shift the maximum value of the radial magnetic field and acceleration region to outside the channel. In this configuration, the magnetic field lines close to the channel walls are parallel to the walls. Moreover, because the magnetic field lines are approximately equipotential and because the magnetic field lines close to the wall are also the one penetrating deeper inside the channel, their potential is close to the anode potential and therefore the electron temperature along these lines is relatively small (the electrons moving along these lines have crossed the acceleration region and have lost their energy through collisions with xenon). A consequence of the reduction of plasma-wall interaction in this configuration is that near-wall-conductivity, i.e., cross-field transport due to electron-wall interaction should not play an important role. Mikellides et al.21 conclude from their numerical and experimental study of a magnetically shielded 6 kW thruster that the ion flux to the walls is reduced, with respect to an unshielded thruster, by factors of 8 and 50 at the outer and inner walls, respectively. Combined with the lower ion energy to the walls predicted by the simulations, this leads to a very large reduction of the sputtering rate (estimations of several orders of magnitudes reduction of the sputtering rate are given by the authors). A consequence of this new magnetic field topology is that the ionization region is shifted downstream. Also, erosion of the magnetic inner pole has been observed in this configuration109 (the magnetic poles are outside the channel in the magnetically shielded thruster, as can be seen in Fig. 27).

FIG. 27. Top schematic of the upper half of the annular channel in a standard, unshielded thruster and in a magnetically shielded configuration, bottom: axial profile of the electric potential and electron temperature close to the wall. After Ref. 108.

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B. Double stage Hall thruster concept

In Hall thrusters, the same electric field extracts the ions from the plasma and provides the energy necessary for the electrons to ionize the gas flow. Acceleration and ionization are therefore linked and it is difficult to adjust separately thrust and specific impulse. For some specific tasks, it may be useful to control these parameters separately. The concept of double stage Hall thruster has been proposed for this purpose. Double stage thrusters could also be an interesting alternative to single stage Hall thrusters when using gases (such as argon) which are less easy to ionize (and less expensive) than xenon. The basic idea of a double stage Hall thruster is to add an ionization stage, i.e., an independent plasma source upstream of the acceleration stage (see Fig. 28). The ionization stage can be in a relative large volume chamber behind the acceleration channel, as in the NASA173GT or Galatea concepts shown in Fig. 28. Since the area of ion extraction through the thruster channel is relatively small with respect to the total wall area of the ionization stage, it is essential, in such a configuration, to have an excellent magnetic confinement in the ionization stage. This

FIG. 28. Examples of double stage Hall thrusters; (a) schematic of the NASA-173GT thruster developed at Michigan State Universitry,110 (b) Galatea concept of Morozov and Savelyev.111–113

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is done with magnetic cusps in the NASA-173GT developed at Michigan State University,110 and with an interesting magnetic configuration based on the Galatea magnetic trap concept of Morozov and Savelyev in the Galatea double stage thruster. The plasma source in both designs is an electron emitter at a potential about 50 V below the anode potential (the emitter is at a voltage between the external cathode voltage and the anode voltage). These designs are very interesting but their magnetic configuration can be complex. Note that the ionization stage of the NASA-173GT is very similar to the plasma source of a gridded ion source. More work is needed to prove that such double stage designs can operate efficiently in a large range of thrust and specific impulse. The design of the magnetic configuration at the interface between the ionization stage and the acceleration stage in double stage thrusters is certainly crucial and must be optimized to (1) collect most of the ions created in the source and (2) minimize the neutralizing electron current entering the channel from the external cathode side. The double stage concept also implies that the distance between the ionization region and the acceleration region will be different from that in a single stage thruster. This could have important consequences on anomalous electron transport between the acceleration and ionization zones. Simpler double stage thrusters where both ionization and acceleration stages are inside the channel have also been designed and tested.114–117 In these configurations, ion production is closer to acceleration but the possibility of efficient operation in a large range of thrust and specific impulse has yet to be demonstrated. Other types of plasma sources for the ionization stage have been studied: microwave,118 electron cyclotron resonance microwave,119 and helicons.120,121 We will not discuss here the physics or performance of these thrusters. We conclude this section by saying (1) that the double stage concept is very appealing but that efficient operations of double stage thrusters in a large range of thrust and specific impulse have not been demonstrated yet and (2) that the crucial question of the magnetic connection between the ionization stage and the acceleration stage should be carefully considered before designing a new double stage concept. V. CONCLUSION

The Hall thruster is a unique and remarkable plasma device where a large electric field can be maintained in the bulk plasma by lowering the electron conductivity in an E  B configuration to extract positive ions from the plasma. Electron interactions with walls as well as instabilities and turbulence contribute to the enhancement of electron transport across the magnetic field, but at the optimized operating point of Hall thrusters, the electron conductivity is sufficiently small to allow efficient ion extraction by a large electric field localized close to the maximum magnetic field. The physics of anomalous cross-field transport due to electronwall collisions or turbulence is complex and this complexity has prevented the development of fully self-consistent and predictive Hall thruster models.

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Important progress has been made in the last 20 years in the diagnostics, modelling, and theory of Hall thrusters. The influence of electron-wall interaction of cross-field transport can now be calculated more accurately, and the origin of the most important oscillations and plasma instabilities and their contribution to anomalous transport are much better understood. The recent advances in the understanding of crossfield electron transport in these devices should help the development of more predictive modelling tools in the near future. ACKNOWLEDGMENTS

The author and his group at the LAPLACE laboratory would like to acknowledge continuous support from the French Space Agency CNES (Centre National d’Etudes Spatiales), and especially from A. Cadiou, D. Arrat, N. Arcis, and C. Boniface. Numerous and enlightening discussions with members of the French Group on Space Propulsion are also acknowledged, especially with JC Adam, S Barral, G Bonhomme, A Bouchoule, M Dudeck, L Garrigues, D Gresillon, G Hagelaar, A Heron, N Lemoine, S Mazouffre, D Pagnon, and S Tsikata. 1

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