From collisional to turbulent EXB electron transport
in closed-drift plasma sources
J.P. Boeuf
1
LAPLACE (Laboratoire Plasma et Conversion d'Energie),
Université de Toulouse, UPS, INPT Toulouse
118, route de Narbonne, F-31062 Toulouse cedex 9, France
2
CNRS; LAPLACE; F-31062 Toulouse, France
Since the early works on Hall thrusters, the question of electron transport across the magnetic
field in the acceleration region of these thrusters has been the subject of an impressive number
of papers. The gas density in this region is low because of intense ionization, and it is easy to
check that classical, collisional electron transport cannot explain the measured values of the
electron current across the magnetic field. In the presence of a magnetic field B perpendicular
to the electric field E and to the discharge current, as in a Hall thruster, electron trajectories
are trapped along the magnetic field and in the EXB (closed-drift, azimuthal) direction. Each
collision (momentum exchange) with a neutral atom allows the electron to jump on a new
trapped trajectory, leading to a global (classical, collisional) transport parallel to the E field.
The electron mobility and current density in that case are proportional to the electron-neutral
collisional frequency and inversely proportional to B2. The electron-neutral collision
frequency in the acceleration region of a Hall thruster is typically more than one order of
magnitude too small to explain the measured current.
Two mechanisms are invoked to explain the observations: 1) the possible role of electron
collisions with the walls (secondary electron emission, momentum and energy losses), and 2)
the “anomalous” or turbulent transport associated with fluctuations of the electric field (these
fluctuations may lead to abrupt changes in the electron momentum and therefore their effects
on electron transport can be similar to collisions).
In spite of the considerable literature on the possible role of collisions with walls and
secondary emission, or of turbulence on electron transport across the B field, there is still no
clear conclusion on the relative contribution of each of these two mechanisms to the electron
current in the acceleration region. The consequence is that we still do not have predictive
models of Hall thrusters. Nevertheless we can say that recent works on this important issue
have been very constructive and useful and we have now a much better qualitative and
quantitative understanding of the way electron-wall collisions (see, e.g., Kaganovich et al.1)
and turbulence (see, e.g., Adam et al.2) can affect electron transport across B. There seems to
be a consensus on the fact that both mechanisms are present, the relative importance of each
depending on the operating point (e.g. discharge voltage). For example there is experimental
evidence that the electron current depends on the wall material (this is more clear at high
operating voltages), and recent models describing the role of electron-wall collisions must
also include turbulent transport to obtain quantitative agreement with experiments.
In the present paper we use a simple, one-dimensional Particle-In-Cell model (in the
azimuthal, EXB direction) to explore the role of turbulence on electron transport. The problem
is periodic in the simulated direction x (only a fraction of the closed drift path is actually
described). The magnetic field B and electric field E are both given and are perpendicular to
each other and to the simulated azimuthal direction. Electron and ion transport is described in
the 6-dimensional phase space but Poisson’s equation is solved only in the azimuthal
direction. This allows the description of waves and instabilities with wave vectors in the
azimuthal direction (this approach is similar to that of Ducroq et al.3 except that collisions are
fully included and that the axial electric field is directly taken into account) . The averaged
plasma density is fixed (ionization is treated as an excitation process and because of the
periodic boundary conditions the number of particles in the simulation stays constant). The
gas density (xenon) is also given. Cases without or with electron-wall collisions included have
been studied and will be presented. Simulations are performed for given E and B, and for
different, decreasing values of the gas density, starting from very collisional regimes (gas
density as large as 1022 m-3, i.e. much larger than in a Hall thruster), and going to gas densities
closer to those of a Hall thruster.
The goal is to study the transition from classical, collisional electron transport across B, to
turbulent transport induced by azimuthal instabilities, as the gas density is decreased, and to
assess the influence of electron-wall collisions.
The results show that for E and B values typical of Hall thrusters (respectively 104 -2x104
V/m and 0.01-0.02 T), turbulent transport appears very quickly when the gas density is
decreased. For example, for E=104 V/m, B= 10-2 T, ne=1017m-3, turbulent transport appears
when the Hall parameter / becomes larger than ~ 2 (this corresponds to a xenon gas density
of about 3x1021 m-3 for B=100 Gauss). The mobility perpendicular to B becomes significantly
larger than the pure classical, collisional mobility when the gas density decreases further. For
example, for a gas density of about 1020 m-3 (/ on the order of ~ 60), the calculated electron
mobility is about 10 times larger than the classical collisional mobility, because of the
azimuthal instability. The azimuthal instability predicted in this simple model is very similar
to the instability described in Adam et al.2 and Ducroq et al.3 Above a gas density around 1020
m-3 the average electron properties reach steady states values and the instability is saturated.
For gas densities below this value, the electron energy losses are no longer sufficient and
steady state is not reached (e.g. the electron mean energy continuously increases with time).
Since in a Hall thruster, the gas density in the acceleration region can be significantly lower
than 1020 m-3 the electron mean energy can be limited only if other electron energy losses, e.g.
due to electron-wall collisions are included.
In this paper we will discuss the evolution from collisional to turbulent electron transport
when the gas density is progressively decreased and we will show that a combination of
turbulent transport and electron energy losses to the walls is necessary to explain the
experimental results.
Refeences
1. I. D. Kaganovich, Y. Raitses, D. Sydorenko and A. Smolyakov, Kinetic effects in a Hall
thruster discharge, Physics of Plasmas 14, 057104 (2007).
2. J. C. Adam, A. Heron and G. Laval, Study of stationary plasma thrusters using twodimensional fully kinetic simulations, Phys. Plasmas 11, 295 (2004).
3. A. Ducrocq, J. C. Adam, A. heron and G. Laval, High-frequency electron drift instability in
the cross-field configuration of Hall thrusters Physics of Plasmas 13, 102111 (2006).