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The E × B drift instability is studied in Hall thruster using one-dimensional particle in cell (PIC) simulation method. By using the dispersion relation, it is found that unstable modes occur only in discrete bands in k space at cyclotron harmonics. The results indicate that the number of unstable modes increases by increasing the external electric field and decreases by increasing the radial magnetic field. The ion mass does not affect the instability wavelength. Furthermore, the results confirm that there is an instability with short wavelength and high frequency. Finally, it is shown that the electron and ion distribution functions deviate from the initial state and eventually the instability is saturated by ion trapping in the azimuthal direction. Also for light mass ion, the frequency and phase velocity are very high that could lead to high electron mobility in the axial direction.
Ion thruster is one of the electrical thruster types used in spacecraft propulsion and satellites. Electrical thrusters use different acceleration methods for producing the thrust. These methods could be separated in to three classes: electro-thermal, electrostatic, and electromagnetic. Hall thrusters and gridded ion thrusters are in the electrostatic class. In both ion thrusters, ions are electrostatically accelerated out of the plasma. In the gridded ion thrusters, thrust is transferred by the electrostatic force between ions and two grids, but in the Hall thrusters, thrust is created by an external axial electric field established perpendicular to the radial magnetic field.[1,2] Hall thrusters are well known as the stationary plasma thrusters,[3] which use noble gases such as xenon, argon, and krypton for propellant. Xenon is usually used because it has high molecular weight and low ionization potential.[4] The applied potential between the anode and external hollow cathode produces the axial (z-direction) electric field and the outer and inner magnetic polls create the radial (x-direction) magnetic field. Electrons coming from the external hollow cathode are accelerated to the anode by the electric field. However, they will be confined by the radial magnetic field and thus they are trapped in an azimuthal direction (y-direction) around the annular channel. The trapped electrons collide with neutrals and generate ions. The ions with respect to the electrons have larger Larmor radius and therefore they are not affected by the magnetic field. The extracted ions from the system without neutralizing can cause dangerous spacecraft charging. So, it must be neutralized by part of electrons that come from the external cathode. Despite many studies have been done on different aspects of Hall thrusters,[5–9] there are some problems in physical interpretations of Hall thrusters such as electron anomalous transport across the magnetic field.[10,11] Among different mechanisms proposed in Refs. [12–14], in order to explain the anomalous electron transport, most physically explanations are relevant to the formation of instabilities in azimuthal, E × B direction. The azimuthal velocity of the ions is almost zero because they are not affected by the magnetic field. However, the electrons are magnetized and they have the azimuthal drift velocity (Ved = E/B) and the amount is as high as 106 m/s. This difference between the ion and the electron drift velocities could lead to the azimuthal instability with short wavelength and high frequency.[15,16]
Some researchers have studied the instabilities in the azimuthal direction of the Hall thruster theoretically and experimentally. In 1970, Gary and Sanderson investigated the transfer of magnetized Maxwellian electrons and unmagnetized ions in the plasma. They have shown an instability excited in the azimuthal direction because of coupling between Bernstein and ion acoustic modes.[17,18] In 2004, Litvak et al.[19] experimentally identified and characterized the high-frequency plasma azimuthal waves in Hall thrusters. In 2012, McDonald[20] experimentally showed that the azimuthal drift instability could increase the electron mobility in the axial direction of the thruster. Lafleur et al.[21] have used one-dimensional (1D) azimuthal particle in cell (PIC) simulation to study the effect of the plasma density on the electron transport perpendicular to the magnetic field in Hall thrusters. They found that the electron transport can be explained by the classical electron–neutral collision theory in low-density plasma. However, in high-density plasma, a strong instability can be observed which causes considerable increase in electron transport mobility, even without considering the electron–neutral collisions with respect to the classical theory.
Although simulation in two or three dimensions is more accurate comparing to that in one dimension but it requires long time. Therefore, it is useful to study the simulation in one dimension in the azimuthal direction to investigate the azimuthal instability.
In this work, the azimuthal E × B drift instability is studied in the Hall thruster using one dimensional PIC and Monte Carlo collision (MCC) simulation method. In Section
Many oscillation modes in the range of a few
In order to understand the origin of the E × B drift instability, we need to study the dispersion relation. In this paper, the dispersion relation is studied using the Gordeev function because in this method, the solution of the dispersion relation converges very quickly.[23] Considering constant electric and magnetic fields, where the axial electric field is perpendicular to the radial magnetic field, the dispersion relation is obtained by the electrostatic growth of the wave in uniform plasma in the Hall thruster. The dispersion equation in the Hall thruster is[16,23]
![]() | Fig. 1. The effect of kx on (a) angular frequency (real part of ω) and (b) growth rate (imaginary part of ω) at kz = 0. The used parameters are presented in Table |
![]() | Table 1. The wavelength of the first harmonic of the electron cyclotron frequency for different Bx. . |
It is obvious from Fig.
Furthermore, the effect of the electric and magnetic fields on the number of unstable modes is studied. In Fig.
![]() | Fig. 2. The effect of magnetic field on the unstable modes with constant electric field E0 = 2000 V/m. |
![]() | Fig. 3. The number of unstable modes as a function of the electric field with constant magnetic field B0 = 0.02 T. |
In Table
![]() | Table 2. The wavelength of the first harmonic of the electron cyclotron frequency for different Ez. . |
The results could be physically interpreted as follows. By increasing the magnetic field, more electrons are confined and thus the number of unstable modes and consequently the wavelength decreases, while by increasing the electric field, the electron drift velocity increases to distort the electron confinement, which leads to more unstable modes with longer wavelength. Although a small electric field has less unstable modes and smaller amplitude of unstable modes but it is not good for Hall thruster because of the low ion beam velocity.
In Fig.
In order to obtain the velocity distribution of electrons, it is useful to study the plasma system using the kinetic theory. In other words, in the kinetic approximation, the plasma system is characterized by the electron distribution function. The analytical solution of the kinetic equation is complicated and therefore numerical solutions such as Vlasov[25] and PIC[26–28] simulation methods should be used. In PIC simulation, the system is described by the electromagnetic fields, distribution functions, and Newton’s equations.
In this paper, the numerical method is organized by 1D azimuthal-PIC-MCC simulation algorithm.[29] Radial magnetic field B0 and axial electric field E0 are uniform and fixed during the simulation. At the beginning of the simulation, the charged particles (ions and electrons) are separated uniformly in the axial (acceleration length) and azimuthal domain of the simulation. Both initial velocity distribution functions of electrons and ions are Maxwellian. All physical parameters used in this work are summarized in Table
![]() | Table 3. Physical parameters used in the PIC simulation. . |
The self-consistent electric field is solved along the E × B direction using Thomas tridiagonal algorithm with periodic boundary conditions.[30] The reference potential at the boundaries is zero. Furthermore, in this simulation, the axial boundary conditions are considered. The electrons and ions exited from the axial region are replaced by new electrons with temperature 10 eV and ions with temperature 0.2 eV (see Fig.
For checking the code, the average kinetic energy of electrons and ions is plotted in Fig.
![]() | Fig. 6. Mean energy of electron and ion with number of particles per cell Npcc = 500, number of cells Nc = 500, and azimuthal length Ly = 0.026 m. The parameters are presented in Table |
![]() | Fig. 7. Electron and ion density fluctuations at t = 12 μs. The used parameters are presented in Table |
As it is known, the azimuthal instabilities could lead to non-uniformity and polarization of the plasma in the azimuthal direction. These instabilities are excited by the formation of an azimuthal electric field. However, the electron confinement can be destroyed by the azimuthal electric field. The oscillating electric field in the azimuthal direction as a function of position (y) at t = 3 μs is depicted in Fig.
The wavelength, frequency, and velocity of the E × B drift instability can be characterized by the contour plot (time and space evolution) of the azimuthal electric field which is given in Fig.
Since particles motion depends on the electric and magnetic fields, it is useful to investigate the effect of the magnetic and electric fields on the propagation velocity of the instability. In Table
![]() | Table 4. The characteristics of instability with different magnetic fields. . |
![]() | Table 5. The characteristics of instability with different electric fields. . |
It is shown in Table
Finally, one can conclude that the dependence of the wavelength, frequency, and velocity of the instability on the magnetic field is more than their dependence on the electric field.
The velocity of particles plays a key role in the interaction between travelling waves and charged particles. Therefore, it is necessary to distinguish between the velocity distribution functions of charged particles. The kinetic equations give us a suitable explanation of particle–wave interaction. Using these equations, the electron and ion densities and velocity distribution functions fi,e(r v t) can be obtained. It is important to note that the distribution functions of electrons and ions play a key role in predicting and controlling the plasma parameters.
The correlation between the oscillating electric field and distribution function has a strong effect on the distribution function shape at different time. In Fig.
![]() | Fig. 9. (a) Electron velocity distribution function and (b) electrons temperature in azimuthal length at different time. |
In Fig.
![]() | Fig. 10. (a) Ion velocity distribution function, (b) fitted curve at t = 6 μs, (c) ion temperature in azimuthal length at different time. |
The ion temperature in azimuthal direction for three different time is plotted in Fig.
It is important to note that the main reason for the deviation of the distribution function from the initial state is the nonlinear process. However these lead to the saturation of the instability. Usually, the saturation process occurs after increasing the instability growth rate up to several times. The saturation happens because of ion–wave trapping. The study of ions and electrons phase space at two different azimuthal and axial directions shows that in the axial direction, there is no evidence of trapping for both electrons and ions. Furthermore, the phase space diagram in the azimuthal direction for electrons shows that there is no electron trapping, but for ions, phase space vortexes are created as illustrated in Fig.
The ion mass is one of the most important parameters in Hall thruster operation. Considering different cross section of excitation and ionization energy for each gas (Table
![]() | Table 6. Ionization and excitation energies for noble gases. . |
By using the contour plot of the azimuthal electric field for different ion masses (Figs.
![]() | Fig. 12. Time–space evolution of the azimuthal electric field up to t = 12 μs for different ions: (a) He, (b) Ne, (c) Ar, (d) Kr, and (e) Xe. |
![]() | Table 7. Characteristics of instability for noble gases. . |
The dispersion relation was investigated using Gordeev function to understand the drift instability theory in Hall thruster. The results indicated that the unstable modes occur only in the separated lines corresponding to the electron cyclotron frequency harmonics. Furthermore, the effect of the electric field, magnetic fields, and ion mass on the number of unstable modes was studied. It was found that the increasing axial electric field leads to the increase of the number of unstable modes. While by increasing the radial magnetic field, the number of unstable modes decreases and ion mass does not affect the number of unstable modes. Furthermore, by using 1D Azimuthal PIC-MCC simulation, the drift instability was developed in the Hall thruster. The results confirmed that there is an instability with short wavelength in the range of millimeter and high frequency in the range of megahertz. Moreover, the results showed that the wavelength, frequency, and velocity of the instability depend more on the magnetic field than the electric one. Also, the result showed that for light mass ions, the frequency and phase velocity are very high that lead to higher electron mobility in the axial direction.
In addition, the effect of the drift instability on the electron and ion distribution functions was investigated using 1D PIC simulation. The results indicated that in the presence of the azimuthal instability, the electron and ion distribution functions deviate from the initial state (Maxwellian) in the azimuthal direction. Finally, ion trapping in the azimuthal direction confirmed the instability saturation process.
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