Near-Surface Cusp Confinement of Micro-Scale Plasma
AFOSR YIP Grant FA9550-11-1-0029
Prof. Richard E. Wirz
Introduction
Magnetic cusp confinement of plasma at
conducting surfaces involves interactions between
a divergent magnetic field, multiple plasma
species, and the sheath conditions near the
surface.1 The behavior of plasma near the cusp is
especially important for small discharges since the
cusp field represents a large percentage of the
overall discharge volume. The motivation for this
study is to improve the understanding of cusp
confinement for permanent magnet discharge, to
enable
the
development
of
efficient
microdischarges on the order of 1 cm in diameter.
A micro-scale thruster of this size is attractive for
large delta-V missions using small spacecraft and
for precision control and formation flying for
larger spacecraft.2,3,4
In this investigation, we use a combined
experimental and computational effort to examine
the behavior and structure of the plasma very near
the anode wall for a single magnetic cusp.
Measurements were first taken for electron plasma
confined by a single cusp. The results from this
experiment and the computational model were in
good agreement, thus validating critical
components of the computational model. The
configuration was then modified to attain
sufficient ionization to examine the interaction
between primary electrons (primaries), ions and
secondary electrons (secondaries) in the cusp
region. This report provides a detailed description
of the results from this “Cusp Confinement
Discharge Experiment” (referred to as “Discharge
Experiment” below) and related modeling efforts.
In addition, this report gives a brief summary of
preliminary analyses of the macroscopic behavior
for small scale discharge confinement.
Experiment Research Effort
The Discharge Experiment was designed to
provide sufficient ionization to examine multispecies behavior very near a single magnetic cusp.
A ring-cusp configuration was adopted upstream
of the subject cylindrical magnet to improve the
primary confinement while the e-gun spacing
allows the chamber to operate at higher densities
without exceeding the gun’s maximum operating
pressure. Therefore, a magnetic ring placed just
upstream on the gun exit to provide focusing of
the electrons through a relatively small aperture in
the “Aperture Plate”. A ring shape plenum
provides a xenon neutral pressure in the discharge
of approximately 10-3 Torr, which is sufficient to
generate a measureable amount of ionization
during electron gun operation.
Figure 1. Diagram of the “Single Cusp Discharge
Experiment” with magnetic streamlines.
Figure 1 shows a moveable “Wall Probe”
located immediately upstream of the cylindrical
magnet. This probe is pressed against low-friction
surfaces on the downstream end of the discharge
cylinder to minimize neutral leakage from the
chamber as the probe is moved. The effective
probe area is a tapered 400 μm diameter orifice
that is located at the center of the wall to expose
the plasma to the “collector plate”. The right
image of Figure 2 shows that the collector plate
when biased to 30 V causes minimal disturbance
of the plasma. Thus, the probe functions as a
single aperture retarding potential analyzer to take
high resolution measurements across the cusp
surface. This design is specifically well-suited for
large Debye lengths of the low plasma densities
created by this discharge.
Figure 2. The Wall Probe’s 400um diameter orifice (left)
and its cross-section potential contour (right)
Near-Surface Cusp Confinement of Micro-Scale Plasma, Wirz
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The figures below are planar current density
measurements to the downstream anode wall using
the Wall Probe. Operating conditions for the
results below are: 50 μA of 25 eV electrons,
1×10-3 Torr chamber pressure, and a 1200 G cusp
field strength at the anode from the cylindrical
samarium cobalt magnet.
2
0.025
Y (mm)
1
0.02
0.015
0
0.01
-1
0.005
-2
-2
-1
0
X (mm)
1
2
-3
x 10
6
4
8
Y (mm)
2
data are analyzed alongside the computational
results in the following section.
Computational Research Effort
The computational model employs a particlein-cell Monte Carlo collision (PIC-MCC) method.
The PIC model is required to resolve species
interactions and dynamics in the cusp region. In
the model, particles are moved in electric and
magnetic fields and the densities are computed and
used to calculate the electric potential and field on
the grid as illustrated in Figure 4. The key
components of the model are:
 Magnetic field calculation using analytical equations
for permanent magnets5,6 (provides fast and accurate
particle tracking very near the cusp).
 Adaptive Cartesian mesh generation (provides high
resolution near the magnet surface).
 Second-order electric potential7 and field solver.
 Particle tracking with a modified Boris method.8,9
 Particle weighting using generalized weighting
scheme for axisymmetric domain.10
 Anisotropic elastic scattering for electron-atom
collisions.11
6
0
4
-2
-4
2
-6
-5
0
X (mm)
5
Figure 3. Current density (A/m2) contour plots for the
Wall Probe biased to electron saturation (top) and ion
saturator potential (bottom). The dashed line represents
the location of the cylindrical magnet.
The electron saturation contour represents the
combined electron current density within the loss
region. The peak structure in the top contour is
slightly more diffused compared to the base
pressure case (not shown here) due to the presence
of secondary electrons. The asymmetry is caused
by a slight radial misalignment of the electron gun
with the discharge chamber. This misalignment
causes the electrons to be confined to an azimuthal
region instead of being evenly distributed. The
periodic pattern corresponds with the number
discrete magnets within the ring cusp just
upstream of the cylindrical magnet. The ion
current density is represented with the ion
saturation contour and is seen to coincide within
the same region as the electron loss area. These
Figure 4. Simplified flowchart for the computational
model
In the simulation of Discharge Experiment, the
25 eV primary electrons are injected into the
discharge domain filled with 300 K xenon neutrals
of uniform density 5×1019 m-3, thus experiencing
elastic and inelastic collisions with neutrals and
producing ions and secondaries. The magnetic
field configuration used in the Discharge
Experiment and simulation is shown in Figure 5.
The operating conditions for this experiment result
in sparse plasma conditions with very low
ionization levels.
Near-Surface Cusp Confinement of Micro-Scale Plasma, Wirz
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Figure 5. Magnetic field configuration used in Discharge
Experiment
Figure 6 shows the electric potential computed
by the model. The highest potential value is seen
close to the center of the domain while the
potential drops rapidly toward the grounded
boundaries. This potential structure pushes ions
toward the walls. Figure 7 shows the contour plot
of primary electrons. The density is relatively
higher at the single magnetic cusp. By design, the
stronger magnetic field strength at the ring-cusps
reflect the electrons, while a larger number of
electrons reach the downstream plate because of
the relatively weaker field at that surface. The
contours for ion and secondary generation rate
density are similar to the primary density contour;
thus, the vast majority of ions and secondaries are
created very near the cusp for plasma condition
created by the electron gun.
r (m)
0.015
2 4 6 8 10 12 14
0.01
0.005
0
0
0.01
0.02
0.03
0.04
z (m)
Figure 6. Electric potential (V) calculated by the
computational model.
profiles, leak radii (summarized in Table 1) for
individual species are obtained. In estimating leak
radii from the experiment, the average FWHM
values across the density peaks for radial lines
from the center of the magnet are used. The
computed primary loss radius is on the order of its
Larmor radius, which is much smaller than the
value obtained from the experiment. This
disagreement is somewhat expected since, in the
model, the electron gun is perfectly aligned with
the cylindrical magnet, which results in primary
electron velocities nominally directed largely
along the axis. The primary electron tracking
model has shown that the misalignment of the
electron gun in the experiment reduces the
electron population near the centerline field,
resulting in relatively lower current density peaks
shown in the data. As described above, ions and
secondaries are mostly created close to the cusp.
Because of the axial electrostatic force in this
region, the ions are pushed toward the downstream
wall as shown in Figure 9, resulting in high ion
current density near the cusp. This behavior results
in ion loss radii much smaller than the ion
gyroradius and comparable to the hybrid
gyroradius evaluated for secondary electrons and
ions. The secondary loss radius on the order of
hybrid loss radius is explained by the magnetic
field structure near the permanent magnet cusp.
Unlike a spindle cusp, the permanent magnet
creates a weakly convergent cusp field as shown in
Figure 5, and cross-section of secondary electron
volume remains large even very near the cusp. By
these observations, we see that the sparse plasma
conditions used in this experiment do not exhibit
the same mechanisms that are used to explain the
hybrid gyro behavior for more highly ionized
plasma.
0.01
0.005
0
0
0.01
0.02
0.03
0.04
z (m)
Figure 7. Primary electron density
Primary Electron
Ion
Secondary Electron
0.8
0.008
0.006
0.6
r(m)
r (m)
5E+12 3E+13 6E+13 9E+13
Normalized Current Density
1
0.015
0.4
0.004
0.2
0.002
(m-3).
0
0
Figure 8 shows current density profiles at the
downstream plate; these data are normalized to the
maximum values for each species to show the
proportionality of the current density. By taking
full width at half maximum (FWHM) of these
0.034
1
2
3
Radial Positon (mm)
4
5
0.036
0.038
0.04
0.042
0.044
z(m)
Figure 8. (Left) Normalized current density profile along
the downstream plate radius. The data are normalized to
the maximum values for each species.
Figure 9. (Right) Examples of ion trajectories very near
the cusp
Near-Surface Cusp Confinement of Micro-Scale Plasma, Wirz
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Table 1. Characteristic radii (mm) for the Cusp Confinement Discharge Experiment
Primary Electrons
Ions
Leak Radius (Experiment) a
0.62
0.77
Leak Radius (Model) a
0.08
0.35
Gyroradius b
0.14
2.50
Hybrid Gyroradius
0.59 c
0.59 c / 0.37 d
Secondary Electrons
0.59
0.30
0.055
0.37 d
a
Leak radii determined for full width at half maximum (FWHM)
Gyroradius calculated for each species using local conditions
c
Hybrid gyroradius estimated using primary electrons and ions is 0.59 mm (i.e., 𝜌ℎ ≅ √𝜌𝑝 𝜌𝑖 )
d
Hybrid gyroradius estimated using secondary electrons and ions is 0.37 mm (i.e., 𝜌ℎ ≅ √𝜌𝑠 𝜌𝑖 )
b
Microdischarge Design
In parallel to the near-cusp analysis effort
described in the previous section, we are also
examining the macroscopic confinement behavior
of smaller discharges. To this end, analysis of the
plasma structure of a miniature discharge provides
insights into confinement physics that is crucial to
the development of a microdischarge. Plots in
Figure 10 are plasma density contour for a 3 cm
discharge measured at 1.5 A hollow cathode
discharge current. Results from this study show
that the plasma properties are dominated by the
magnetic field and invariant to the discharge
current.
Additionally,
stronger
magnet
configurations demonstrate reduced discharge
efficiency due to the reduction in usable plasma
volume.
improved even with weaker magnets and larger
loss area by expanding the confinement volume.
Using this design philosophy, Figure 11 shows a
design where magnets are sized and configured to
maximize cusp strength while countering each
other in the bulk region. Results show desirable
primary electrons ionization levels of nearly 25%.
Figure 11. Plot of primary electron trajectories for a 1.5
cm microdischarge. Magnets shown not to scale
Conclusion
A combined experimental and computational
effort has been undertaken to improve the
understanding of near-surface cusp confinement
for microdischarges. These efforts have thus far
examined electron plasma and multi-species
interactions in sparse plasma conditions. The next
step is to extend this analysis to the partiallyionized plasma conditions expected for a
microdischarge by replacing the electron gun with
a hollow cathode and using a hybrid PIC model
with a two-fluid approximation for the ions and
secondaries.
References
1
Martinez-Sánchez M., Ahedo E., Phys. Plasmas, 18 (2011)
Conversano R., Wirz R., J Spacecraft and Rockets, submitted
June 2012, accepted with revisions Aug 2012
3
Wirz R. et. al., AIAA-2004-4115
4
Martin, S. et al., Aerospace Conference,2008 IEEE, 1-8 March 2008
5
Engel-Herbert R. and Hesjedal T, J. Appl. Phys., 97, 074504 (2005)
6
Babic S. I., Akyel C., Prog. Electromagnetics Res. C, 5 (2008)71-82
7
Fox J. M., Ph.D. Dissertation, MIT, 2007
8
Wirz R., Ph.D. Dissertation, Caltech, 2005
9
Mao H.-S., Wirz R., AIAA 2011-3739
10
Verboncoeur J., J. Comput. Phys., 174 (2001) 421–427
11
Okhrimovskyy A. et al., Phys. Rev. E, 65, 037402 (2002)
2
Figure 10. Langmuir traces for a 3 cm discharge with
weaker single-stack (top) and stronger double-stack
(bottom) ring cusp magnets. Contours are normalized to
its maximum density
The design of a microdischarge will require
careful placement of magnets to maximize the
effective volume to surface area ratio. Preliminary
particle tracking results have shown that effective
primary confinement may be significantly
Near-Surface Cusp Confinement of Micro-Scale Plasma, Wirz
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