Preface to Special Topic: Modern issues and applications of E × B plasmas

Plasma discharge devices such as ion sources, Hall thrusters,^1,2 and high power pulsed magnetrons^3,4 operate at low pressure and use an external magnetic field to confine electrons, increasing the electron residence time and allowing ionization and plasma sustainment. The external magnetic field is generally placed perpendicular to the discharge current from the cathode to anode, i.e., perpendicular to the applied electric field. The resulting cross-field or E × B configuration generates an electron drift in the E × B direction (the Hall current), and efficient confinement is obtained only if this current does not intercept a wall and is closed on itself (“closed drift” devices^5,6). This can be achieved if the E × B direction is the azimuthal direction of a cylindrically symmetric device.⁷ In planar magnetron and Hall thrusters, the electric field is axial, and the magnetic field is radial. In cylindrical magnetrons and Penning discharges, the magnetic field is axial, and the electric field is radial. A cylindrical configuration with axial or diverging magnetic field is also often used in other types of magnetized discharges such as RF, helicon, or electron beam sustained plasmas. In many cases, pressure gradients provide additional contribution to azimuthal E × B current. Such configurations are used in many devices and may also occur as a result of the ambipolar electric field, e.g., due to the formation of double layers in expanding the magnetic field.

In most E × B plasmas used in applications, the magnetic field and device parameters are such that the electron gyroradius is much smaller than the device dimensions while the ion gyroradius is not. Because electrons are strongly magnetized while ions are not or are only weakly magnetized, the physics of E × B plasmas is very different from that of many other plasmas, e.g., for fusion applications. This distinct area of basic plasma physics has been relatively poorly explored and is not well understood, but is within the reach of current analytical and numerical capabilities, as well as powerful modern diagnostics tools. Improving our qualitative and quantitative understanding of these plasmas is critical for important technologies such as electric propulsion (Hall thruster, cusp-field thrusters, and gridded ion sources) and material processing [e.g., High Power Impulse Magnetron Sputtering (HiPIMS) and other magnetized plasma sources used in surface processing].

The interest for the physics and applications of cross-field devices and low temperature E × B plasmas is not new. The possibility to generate a plasma at a very low pressure in a magnetically confined, cold-cathode discharge was demonstrated at the beginning of the 20th century and led to the development, in the 1930s, of Penning, magnetron, and inverted magnetron ionization gauges.⁸ Today, ion pumps and magnetron gauges are commonly used in several forms.

Not surprisingly, turbulence and instabilities are present in E × B devices and constitute a major obstacle to the quantitative description of these devices and to the development of predictive codes. Fluctuations are likely to develop in the azimuthal direction and can be triggered by various physical mechanisms including the large E × B electron drift, density gradients, ionization, neutral depletion, etc. Unstable fluctuations across different time and length scales as well as self-organized coherent structures are ubiquitous in E × B devices. The origin and interactions of fluctuations, transport, and structures are not clear but have a dramatic effect on the regimes and performance of these devices.

In the 1960s, the search for new space propulsion technologies allowing weight and cost reduction by operating at a higher specific impulse, i.e., with a higher velocity propellant, led to the development of plasma thrusters. Among them, the closed drift thrusters, which are gridless ion sources, attracted considerable interest, especially in the former USSR where more than 2000–3000 engineers and scientists were involved in the development of Hall thrusters.⁵ At this time, the modern telecommunication satellites rely in large part on the Hall thrusters electric propulsion. It has also become an enabling technology for future space missions such as ongoing projects of satellite constellations to provide high-speed internet services for the globe, as well as long term space missions to Mars and asteroids. Despite this success, the physics of Hall thruster operation is not well understood and new applications and new operational parameters have renewed the need for a less empirical design of magnetized plasmas devices used for space propulsion.

Hall thrusters are fascinating E × B plasma devices where large electric fields (several 10⁴ V/m) can be maintained in a quasineutral plasma over lengths much larger than the electron gyroradius and Debye length. This is quite unusual, and, as stated by Professor Morozov,⁹ the Russian father of Hall thrusters, the suggestion “to create a thruster with an electric field distributed along the accelerating channel was met with the negative reaction of experts in gas discharge physics.” This is because turbulence and instabilities in magnetized plasmas are expected to considerably increase electron transport and conductivity across magnetic fields and thus to prevent the formation of large electric fields in quasi-neutral plasmas. The Hall thruster is therefore a perfect counterexample of the application of the CIV (Critical Ionization Velocity, a concept introduced by Alfven¹⁰) phenomenon to E × B plasma discharges. Early (up to the 1970s) experiments and theories of the CIV in relation to the E × B discharges were summarized as follows by Piel:¹¹ “In nearly all cross-field discharges, where at least part of the volume is not fully ionized, the discharge voltage obeys a linear relationship on the magnetic field: $V=V0+v*Bd$⁠, where $d$ is a distance, e.g., the electrode separation, and the constant $v*$ has the dimension of a velocity. $V0$ is the voltage drop at electrode sheaths.” $B$ is the magnetic field and $v*$ was found to be the critical ionization velocity, i.e., the velocity of ions of mass $M$ having a kinetic energy equal to the ionization threshold $Ui$⁠, $v*=2eUi/M$ . This means that the electric $E$ in the quasineutral plasma in these experiments could not be larger than $Bv*.$^10–12 The electric field in a Hall thruster can be hundreds of times larger than $Bv*$ and this indicates that the CIV concept does not apply, at least in the acceleration region of a Hall thruster.

One common characteristic of E × B plasmas is the presence of luminous rotating plasma non-uniformities (“rotating spokes”). Such rotating structures were observed in CIV experiments, Hall thrusters^2,13–16 and more recently in HiPIMS.^17,18 Some of the rotating instabilities observed in HiPIMS and Hall thrusters may or may not be related to the CIV concept, but the complex and non-linear physics of these devices can be responsible for the development of many different types of instabilities^15,16,19,20 and rotating azimuthal nonuniformities. In Hall thrusters, for example, particle-in-cell simulations have evidenced the development of azimuthal waves with frequencies on the order of the ion plasma frequency and wavelengths in the mm range, exhibiting large azimuthal electric field amplitudes^2,21–23 (see Fig. 1). Fluctuations of the same wavelength and frequency have been observed in collective light scattering experiments.²⁴ These fluctuations are thought to be of similar nature as the instabilities studied in the 1970s in the context of space plasmas and collisionless shocks and related to the lower-hybrid,²⁵ “electron cyclotron drift” or “beam cyclotron”^26–30 instabilities (see also the more recent work of Muschietti and Lembège³¹).

The above presentation is intended to illustrate the need for a better understanding of the complex physics of E × B devices. This need has also been well-recognized in the context of space propulsion and plasma processing in view of the device optimization and the development of predictive codes. This need has led to the organization in the last few years of mini-workshops in both communities. It therefore appeared to us in 2017 that organizing a workshop on “E × B plasmas for space and industrial applications” would be very timely. The workshop was organized in Toulouse from June 21–23, 2017. The attendance was limited to 40 researchers and students in order to facilitate interactions and encourage discussions between participants. Each session (composed of two 30-min lectures followed by a 90-min discussion) was organized by two moderators (R. Boswell, E. Choueiri, B. Jorns, I. Kaganovich, T. Lafleur, S. Mazouffre, I. Mikellides, Y. Raitses, and A. Smolyakov) whose efforts led to long and lively discussions. The overall goals of the workshop were to help clarify the current state of research and understanding of E × B plasma devices, to review and discuss the present state of knowledge that has been obtained from diagnostics, theory, and simulations, and to define new research directions. The focus was on charged particle transport and instabilities in various low pressure magnetized plasma devices with large E × B or ∇P × B drift of magnetized electrons and weakly magnetized ions. These devices included Hall thrusters, magnetron discharges such as those used for high power impulse magnetron sputtering, Penning sources, as well as an electron-beam sustained magnetized plasma column used for basic studies of instabilities and turbulence. The related physics of magnetic nozzles in partially magnetized plasmas was also discussed. One presentation was devoted to the study of E × B instabilities in the context of space plasmas. The program of the workshop and list of participants are available on the conference website at https://exb-2017.sciencesconf.org.

The workshop was a success, and the participants left with a much greater awareness of the specificity and richness of the physics of partially magnetized plasmas for space and industrial applications. Discussions during the workshop revealed the need for the community to strengthen its cohesion and visibility and to clearly identify common objectives. A first step in this direction has been to define a project for benchmarking simulation codes related to partially magnetized plasmas: the LANDMARK project.³² The purpose of the project is codes verification and validation. An initial important goal is to provide a reliable basis for a deeper understanding of the physics underlying the behavior of various E × B plasma devices and processes. Several test cases have been discussed and formulated which involve 1D and 2D simulations of the Electron Cyclotron Drift Instability (ECDI) and breathing mode oscillations.³² Experimental validation of the codes, which is the ultimate goal of these efforts, will be pursued, in parallel. The first results of these benchmarking work will be discussed at a second E × B workshop, to be held in Princeton in the fall 2018.

This special issue includes a subset of the invited papers presented at the E × B-2017 workshop.

The paper by Krueger et al. presents a method to construct the analytical mathematical model for the complex magnetic fields (as in a planar magnetron) based on the input provided by the measurements in the experimentally accessible limited region. The obtained analytical expressions can then be embedded in simulations of magnetron and other discharges.

Ramos, Merlino, and Ahedo formulate the hybrid fluid-kinetic model for stationary flow of collisionless plasma in diverging-converging magnetic field. The model enables the study of non-Maxwellian and anisotropic electron population and both magnetized and non-magnetized (or weakly magnetized ion) cases. The challenges of the case with weakly magnetized ion flow are discussed and some solutions are proposed.

Tichy et al. present the measurements of the electron temperature, plasma density, and the electron energy distribution function in a permanent magnetic field Hall thruster. The experimental measurements are complemented and compared against PIC simulations results. This study shows the existence and different roles of two distinct groups of electrons: higher energy electrons emitted by the cathode and the slower electrons produced by the ionization inside the channel.

The paper by Claire et al., reports the experimental study of the m = 1 rotating instability in the linear magnetized plasma device MISTRAL. The laser-induced fluorescence diagnostic is used to infer the radial and azimuthal velocities, ion fluxes, and electric fields. These measurements reveal a complex 3D structure of the m = 1 mode which are not easily interpreted with existing theories.

Several papers in the special issue deal with various instabilities of E × B plasmas.

The paper by Koshkatov et al. show that the nonlinear stage of the axial lower-hybrid instability induced by the ion beam across the magnetic field in a finite length system results in large amplitude wave-breaking type nonlinear structures. It is suggested that such structures may be relevant to the breathing-mode oscillations as well as to the experimentally observed non-monotonous profiles of the electric field.

Lafleur et al. perform a comparison between quasi-linear kinetic theory and PIC simulations emphasising the importance of self-consistent non-Maxwellian electron distribution function for accurate predictions of the Electron Cyclotron Drift Instability (ECDI) growth rate and resulting anomalous electron-ion friction force. By using the electron distribution functions obtained from the PIC simulations, they find that the predictions of quasi-linear kinetic theory are in good agreement with the simulation results. By contrast, the use of Maxwellian distributions leads to a factor of 2–4 larger values of the growth rate and electron-ion friction force that significantly overestimates the electron transport. A possible method for self-consistent modelling the distribution functions without requiring PIC simulations is proposed.

Carlsson et al. reports 2D (azimuthal-radial) PIC simulations of the Penning discharge. A persistent nonlinear spoke-like structure rotating in the direction of E × B and electron diamagnetic drifts has been observed providing enhanced (anomalous) radial electron transport with most of the radial current within the spoke region. The spoke rotation velocity is within about a factor of two of the ion acoustic speed and the spoke frequency is found to follow the experimentally observed scaling with ion mass.

In a follow up paper, Powis et al., via two-dimensional full-sized kinetic simulations of a Penning discharge, also find the rotating spoke and enhanced radial electron transport. The parameter scans with discharge current, magnetic field strength, and ion mass show that the spoke frequency scaling is consistent with that of the collisionless Simon-Hoh instability therefore suggesting that the spoke development is related to a nonlinear stage of the Simon-Hoh instability.

In highly resolved 2D azimuthal-radial PIC simulations of ECDI, Janhunen et al. analyze two-dimensional effects involving the parallel dynamics along the magnetic field. It is found that the instability develops as a sequence of the cyclotron modes demonstrating non-quasineutral large amplitude ion density fluctuations at short wavelengths. The intense long wavelength mode with distinct variations along the magnetic field was identified as the modified two-stream instability leading to strong parallel heating of electrons.

Boeuf and Garrigues have studied Electron Cyclotron Drift Instability in 2D azimuthal-axial geometry with a simplified assumption of the fixed ionization profile but self-consistent profiles of the density and electric field. They find that the wavelength and frequency of the azimuthal wave scale well with predictions from the modified ion acoustic instability theory. The amplitude of the fluctuations is however significantly smaller than that corresponding to saturation of the instability by ion trapping.

Taccogna and Minelli investigate the physics of the instability and electron transport in Hall thruster using the 3D PIC simulations. The azimuthal E × B instability is confirmed. The radial direction terminated by the walls results in interesting radial standing wave structures with overall oblique (radial-azimuthal) pattern. The effect of the electron secondary emission is shown to contribute to the mode saturation leading to smaller amplitude oscillations.

The E × B-2017 workshop has been organized with partial support from the RTRA STAE Foundation in the framework of the IMPULSE project and of the French Space Agency, CNES.