Permanent magnet helicon source for ion propulsion
Francis F. Chen
Electrical Engineering Department, University of California, Los Angeles, California 90095-
1594
Helicon sources have been proposed by at least two groups for generating
ions for space propulsion: the HDLT concept at the Australian National
University (ANU), and the VASIMR concept at the Johnson Space Center in
Houston. These sources normally require a large electromagnet and power supply
to produce the magnetic field. At this stage of research, emphasis has been on the
plasma density and ion current that can be produced, but not much on the weight,
size, impulse, and gas efficiency of the thruster. This paper concerns the source
itself and shows that great savings in size and weight can be obtained by using
specially designed permanent magnets (PMs). This PM helicon design, originally
developed for plasma processing of large substrates, is extended here for ion
thrusters of both the HDLT and VASIMR types. Measured downstream densities
are of order 10
12
cm
-3
, which should yield much higher ion currents than reported
so far. The design principles have been checked experimentally, showing that the
predictions of the theory and computations are reliable. The details of two new
designs are given here to serve as examples to stimulate further research on the
use of such sources as thrusters.
I. Introduction
Helicon sources fall into the category of Inductively Coupled Plasmas (ICPs), which use
radiofrequency (RF) antennas to create plasma without internal electrodes. Since they are related
to whistler waves, helicons exist only in a steady (DC) magnetic field (≡ B
0
). Sources based on
helicon waves have been found to produce 3-10 times higher plasma density (≡ n) than field-free
ICPs in manufacturing applications. In 2003, supersonic ions from helicon sources were found
by S. A. Cohen et al.
1
and by C. Charles and R.W. Boswell, who discovered that a current-free
double layer (DL) occurs downstream of a helicon source expanding into a diverging magnetic
field
2,3
, and that the ion beam is accelerated in the thin, collisionless layer
4
. This concept was
named the Helicon Double Layer Thruster (HDLT). In addition to the potential jump in the DL,
Ref. 4 contains detailed measurements of the ion velocity distribution using a retarding-field
energy analyzer at typically 250 W of RF power and 0.35 mTorr of Ar. These ion energy peaks
at the 29 eV plasma potential and at 47 eV behind the DL. The latter is supersonic at 2.1 c
s
,
where c
s
is the ion sound velocity. The DL has been shown pictorially by Charles
5,6
. These
results were confirmed by Keesee et al.
7
using laser-induced fluorescence and by Sun et al.
8
on
the helicon machine at West Virginia University. Further confirmation was given by the PIC-
MCC simulations of Meige et al.
9,10
A machine
11
constructed at the Ecole Polytechnique in
France to reproduce the HDLT experiment fully confirms and extends the ANU results. All
experiments on the HDLT were recently reviewed by Charles
12
. The specific impulse and thrust
of the HDLT were estimated to be low compared with conventional thrusters, but the authors
anticipated that these could be increased with improvements in source efficiency. This paper is
intended to provide such improvements.
2
That a single layer should form in an expanding B-field was shown by Chen
13
to be a
result of the Bohm sheath criterion, which is normally applied to the condition at a wall or
floating probe that will maintain quasineutrality. In this instance, the ion acceleration to the
Bohm velocity c
s
that normally occurs in a pre-sheath occurs in an expanding B-field, since
perpendicular ion energy is converted into parallel energy. In the absence of a wall, the single
sheath has to turn into a double sheath so that the potential will flatten out, else the ions will be
accelerated indefinitely without an additional source of energy. Chen
13
predicted that the DL
should occur where the B-field has decreased by e
−½
= 0.61. A measurement of the DL position
by Sutherland et al.
14
verified this prediction to within 3%. In varying the B-field, Charles
15
found that the DL depended on the field near the back plate. This effect is probably related to
wave reflection at the back plate, an effect used in source optimization in this paper. Finally,
Gesto et al.
16
have calculated the surface where ion detachment from the B-field occurs. This is
deemed important in hydrogen plasmas (next paragraph) where high beta is necessary for
detachment, but it is probably not important for the argon plasmas used so far in DL
experiments, since the ion Larmor radii are larger than the chamber radii.
The second large group using a helicon source for ion propulsion is that of F.R. Chang-
Diaz in the Variable Specific Impulse Magnetohydrodynamic Rocket (VASIMR) project
17
. In
this concept, hydrogen or deuterium ions are ejected from a helicon source immersed in B
0
. The
charge-neutralized ion beam is then compressed when it enters a stronger magnetic field and
through a small aperture used for differential pumping. In the low-pressure, high B-field region,
the ions are heated by ion cyclotron resonance (ICRH); and the subsequent expansion into a
weak B-field converts the ions’ perpendicular energy into parallel energy. Finally, magnetic
nozzles
18
are used to shape the exiting beam for maximum thrust. In the helicon section, the
requisite B-field cannot be created with small solenoids because the field lines curve back,
preventing the plasma from propagating downstream unless its beta is large enough to break
through the field lines.
The size and complexity of the helicon source in either HDLT or VASIMR can be greatly
reduced by employing permanent magnets to create the B-field. Previous attempts to do this by
placing the plasma tube inside ring magnets suffered from the same deficiency as small
solenoids: the field lines curved backward upon leaving the tube, preventing ejection of the
plasma. However, annular magnets have a stagnation point beyond which the field lines extend
toward infinity. By placing the plasma tube in the weaker field beyond the stagnation point,
plasma can be ejected from the source even at low beta. Furthermore, the permanent magnets
can be relatively small and require no power supply. This concept has been proven
experimentally
19,20
. The original design in Ref. 19 was for large-area plasma processing but can
be adopted without change as an upgrade of the source in the HDLT. This is done in Sec. III.
Experiments on the VASIMR engine have been carried out at the Johnson Space Center
in Houston using light gases in a 9-cm diam helicon source
21
. The use of gases lighter than Li
was dictated by the ICRH section, where heavier gases would require overly high B-fields to
keep the ion Larmor radii small. The helicon section itself has been studied at the Oak Ridge
National Laboratory
22
with deuterium in a 5-cm diam tube. Supporting experiments on a high-
power helicon discharge in argon, also with an inner diameter of order 5 cm, were done at the
University of Washington
18,23
.
To correlate with these existing experiments, the present
calculations were done for both 9- and 5-cm diameter plasmas in Secs. IV and V, respectively.
Although enhancement of ICPs with PMs has been investigated by a number of authors,
very few papers report on the use of PMs for helicon discharges
24,25
. In particular, there are no
papers other than Ref. 19 on the use of the external field of annular magnets.
3
II. Methodology
This design of an optimized PM helicon source relies on two innovations: 1) the low-field
peak and 2) the HELIC code. In its simplest form, the dispersion relation for a helicon wave of
frequency
ω
/ 2π in a long circular cylinder can be written
0
r
z
ne
k
kB
μ
ω
= , so that
1 n
akB
ω
∝ (1)
where k
r
is an effective radial wavenumber (inversely proportional to the plasma radius a), k
z
(or
k) is the axial wavenumber, and n and B the plasma density and RF B-field. In a uniform
plasma, k
r
is given by a Bessel function root, and n is a constant; but radial non-uniformity
requires computation. Nonetheless, the proportionalities in Eq. (1) allow us to predict the
direction to go during optimization. Equation (1) shows that n should increase linearly with B,
but it was found that n has a small peak at low B-fields. This low-field peak was subsequently
explained
26,27
by constructive interference of the helicon wave reflected from the endplate near
the antenna. It occurs with bi-directional antennas, such as the Nagoya Type III and m = 0
(azimuthally symmetric) antennas, but not with helical antennas, which excite m = +1 helicon
waves in only one direction. Diverging (cusp) magnetic fields upstream of the antenna can also
create a low-field peak by bringing the field lines against the sidewalls, which then act as an
endplate
26
. This effect was observed long ago
28
and has been rediscovered in VASIMR
experiments
21
. By using the low-field peak, a given density can be produced with a smaller B-
field; and by using m = 0 antennas, plasma wall losses under the antenna can be greatly reduced.
The HELIC code of D. Arnush
29
, based on analytic theory, made possible rapid scans of
parameter space to find optimum absorption of RF power. A user-friendly version of this code
for personal computers can be obtained from the author
30
. The program assumes a plasma of
radius a, a thin antenna of radius b, and a ground plane at radius c, as shown in Fig. 1. The
confining cylinder can be infinitely long or bounded by insulating or conducting endplates
a
b
c
Antenna
Fig. 1. Geometry of the HELIC program
separated by the cylinder length L
c
. The antenna can be any of the common types, or a new one
can be specified by its Fourier transform. The midplane of the antenna can be set at an
adjustable distance h from one endplate. The main constraint which allows for rapid
computation is that the equilibrium density n and magnetic field
B
0
must be uniform in the axial
direction z. To simulate ejection from a small discharge tube, L
c
in this paper is set at a large
value of 2 meters, while h is of the order of centimeters. The radial profiles of density, electron
temperature T
e
, and neutral pressure p
0
can also be specified. We use the convenient
parametrization of Eq. (2) to vary n(r). Here three arbitrary numbers s, t, and w (or f
a
) are used
4
to match an arbitrary, smooth profile between r = 0 (where n = n
0
) and r = a (where n = n
a
).
Also, f
a
is the fractional density at r = a, and w is the “width” of the profile and is directly related
to f
a
. Electrons are assumed to be Maxwellian. Consequently, T
e
affects mainly the plasma
potential, which does not enter the design, though it affects the ejected ion energy. The collision
frequency does depend on T
e
, but in high-power helicon discharges T
e
usually lies between 3 and
4.5 eV. The collision frequencies of a few gases are built into HELIC, but an adjustable collision
factor can accommodate other gases.
00
1, 1
tt
ss
a
a
n
nr a
f
nw nw
⎡⎤ ⎡⎤
⎛⎞ ⎛⎞
⎢⎥ ⎢⎥
=− =− ≡
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
(2)
Once the geometry, n(r),
B
0
, p
0
,
ω
, antenna type, and gas have been specified, the
program solves a fourth-order differential equation for each of several hundred values of k to
obtain the radial and axial profiles of the RF current and electric and magnetic fields, as well as
the profiles of the energy absorbed. The latter are integrated to obtain the plasma loading
resistance R
p
, our principal result. The absorption of RF energy is dominated by mode
conversion to heavily damped Trivelpiece-Gould (TG) modes
31
at the radial surface, and this
Shamrai
32
effect is fully accounted for in HELIC. Electromagnetic radiation, ion mass effects
(such as lower-hybrid resonance), and Landau damping are included, but none of these effects is
important in the parameter regime of this paper. Helicon discharges with endplates are known to
exhibit axial resonances
33,34
. The low-field peak treated here is a special case in which there is
only one endplate. For reasons which will become clear, our design uses m = 0 antennas. The
behavior of such antennas has also been studied by numerous authors
35-37
.
The HELIC program can also scan ranges of n or B
0
, or both at the same time, with either
linear or logarithmic spacing of the points. Data for R
p
vs. n for various B
0
such as the ones
shown below typically take 2-3 hours to generate on a 400-MHz PC. Programming to take
advantage of faster CPUs was not available when HELIC was written. The code has not been
calibrated in a dedicated experiment, but its predictions have been verified in at least two cases.
Blackwell et al.
38
detected the Trivelpiece-Gould modes directly, in agreement with HELIC
calculations, by measuring the RF current. Chen and Torreblanca
39
found that the absolute value
of R
p
from HELIC agreed within experimental error with the value obtained from the jump into
the helicon mode. The fact that the assumed uniform B-field disagrees with the actual diverging
B-field in ejection experiments is not important because little ionization or wave reflection
occurs outside the source. This was proved in Ref. [19] and subsequent experiments, in which
the optimum conditions agreed with those predicted.
III. Design of an HDLT source
Medusa 2 is an 8-tube array of small helicon sources at UCLA to test production of a
large-area, high-density, uniform plasma with multiple sources. A single source of the same
design can be used for the low-power applications of the HDLT without modification. The field
of a stack of annular permanent magnets is shown in Fig. 2. Note that a plasma created inside
the strong field region cannot be ejected because of the field lines run into the wall. However,
there is a stagnation point at which the B-field reverses sign. A plasma placed in the field
beyond the stagnation point will be in a slowly diverging field. Figure 3 shows a proof-of-
concept experiment in which it was shown that a helicon discharge could be produced in the far-
field. The magnetic field can be varied by changing the magnet height D. Figure 4 shows radial
profiles taken with fully RF-compensated Langmuir probes at an RF power P
rf
= 500W at 13.56
5
MHz and D = 15 cm. The probe positions are Z1 = 7.4 cm and Z2 = 17.6 cm below the tube-
flange junction. Two pressures, 4 and 10 mTorr of argon are shown. These are for plasma
processing and are higher than would be used in a thruster. They are fill pressures, not the
neutral pressure inside the tube during the discharge. The flattened profiles at Z2 are caused
both by the B-field divergence and by diffusion at these pressures. The peak density at Z1 is 0.8
× 10
12
cm
-3
.
34 cm
36 cm
D
Z1
Z2
Pyrex tube,
2-in. diam
Ceramic magnet
stack
Fig. 2. Field lines of annular magnets. Fig. 3. Discharge tube in the far-field.
0
1
2
3
4
5
6
7
8
9
-5 0 5 10 15 20
r (cm)
n (10
11
cm
-3
)
Z1, 10 mTorr
Z1, 4 mTorr
Z2, 10 mTorr
Z2, 4 mTorr
500W, D = 15"
Fig. 4. Radial density profiles in Fig. 3 at two probe positions and two pressures.
The discharge tube and magnet were then optimized in the manner described in the next
section. The details will be omitted for this case, since they are available elsewhere
19,40
. The
tube, magnet, and antenna optimized for 13.56 MHz are shown in Fig. 5. The tube is quartz or
alumina, and the top plate (without gas feed) can be a standard 50-mm diam aluminum cover
with O-ring mount. The NeFeB magnet is made in two pieces, each with 7.6 cm I.D., 12.7 cm
O.D., and 1 cm thickness. They are supported by an aluminum sheet. The B-field at the antenna
