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Electric potential in a magnetically confined virtual cathode fusion device
Article in Physics of Plasmas · April 2021
DOI: 10.1063/5.0040792
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Electric potential in a magnetically confined
virtual cathode fusion device
Cite as: Phys. Plasmas 28, 042702 (2021); https://doi.org/10.1063/5.0040792
Submitted: 16 December 2020 . Accepted: 11 March 2021 . Published Online: 02 April 2021
Richard Bowden-Reid, and Joe Khachan
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Phys. Plasmas 28, 042702 (2021); https://doi.org/10.1063/5.0040792
© 2021 Author(s).
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Electric potential in a magnetically confined virtual
cathode fusion device
Cite as: Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Submitted: 16 December 2020 . Accepted: 11 March 2021 .
Published Online: 2 April 2021
Richard Bowden-Reida)
and Joe Khachanb)
AFFILIATIONS
Department of Plasma Physics, School of Physics, University of Sydney, Sydney 2006, Australia
a)
Author to whom correspondence should be addressed: [email protected]
[email protected]
b)
ABSTRACT
The magnetically confined virtual cathode (MCVC) is an approach to nuclear fusion in which multipole magnetic traps are used to confine a
dense cloud of electrons and thereby establish a deep electrostatic potential well for the heating and trapping of ions. We describe
preliminary studies conducted in MCVC-0, a two-coil, biconic cusp trap, in which high impedance, floating Langmuir probe measurements
were used to characterize the electrostatic potential. Contrary to previous studies in six-coil “polywell” devices, no potential well formation
was observed and this is attributed to the particular configuration of magnetic fields within the new device. A computational model was
developed, based on the anisotropic electrical conductivity of discharge plasmas within magnetic fields, and shown to accurately describe the
obtained experimental results. Electrostatic boundaries that were intersected by magnetic field lines were found to strongly dominate the
form of the electric potential within the device, with strong implications for the design of future MCVC/polywell machines.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0040792
I. INTRODUCTION
Inertial electrostatic confinement (IEC) is a method of producing
nuclear fusion wherein a spherically convergent electric field is used to
heat and trap ions. First proposed by Oleg Lavrent’ev in 1950,1–4 the
approach uses two or more spherically concentric electrodes to mimic
the radial gravitational well present in stars. Although Lavrent’ev’s
machine was never built in the Soviet Union, American inventor Philo
Farnsworth independently devised and tested a similar device.5 Robert
Hirsch and Gene Meeks, working under Farnsworth, also successfully
demonstrated a small scale electric fusion device based on Lavrent’ev’s
concept.6–8
Obtaining nuclear fusion in an IEC device is almost trivial when
compared to standard, thermonuclear style devices due to the ease
with which high energy ions (25–100 kV) may be obtained simply by
increasing the applied system voltage. Small scale, deuterium fueled
machines routinely achieve neutron production rates of 106 –108 s1
in steady state operation and hence a great many IEC systems have
been operated in the United States,9–14 Japan,15–20 Australia,21–24
Germany,25 Denmark,26 the Netherlands,27 Iran,28–30 and Turkey.31
While the devices find use as small scale neutron generators for the
production of medical isotopes, neutron imaging, and security applications,13,32–34 theoretical studies by Rider35 and Nevins36 indicate little
opportunity for net energy production. It has been noted by other
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
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authors, however,37,38 that the assumptions made by Rider and Nevins
in their analyses may be overly pessimistic and in some cases are selfcontradictory, and that a net energy gain in IEC is, in principle,
possible.
The polyhedral well machine, or polywell, is a hybrid electromagnetic device that was devised with the view to circumvent the losses of
gridded IEC machines.39,40 Invented by Robert Bussard,41,42 the device
makes use of magnetic cusp trapping to confine a dense population of
electrons, thereby generating a virtual cathode. The requirement that
only electrons need be magnetically confined relaxes the design
requirements on such a trap, in terms of both size and magnetic field
strength, when compared with conventional thermonuclear machines.
Further, the absence of grid electrodes enables a higher degree of ion
recirculation within the core and hence the probability of fusion for
any given ion is greatly increased.
Bussard described a device in which magnetic field coils are positioned on the faces of a regular polyhedron, most commonly a cube.
The coils are oriented such that their magnetic moments point
inwards, resulting in a high order multipole cusp with a central magnetic null. High energy electrons are injected into the core from electron guns placed outside of the magnet coils and the resulting buildup of negative space charge produces the desired virtual cathode or
“potential well.” Bussard reasoned that if the kinetic plasma pressure
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within the device could be made to approach the magnetic pressure,
that is if
b¼
Pplasma
nkB T
¼
! 1;
Pmagnetic B2 =2l0
(1)
then the diamagnetic circulation of electrons about magnetic field lines
would expel the applied field from the core. This “high beta” mode,
sometimes referred to as the “Wiffle ball” after the popular American
children’s toy, was expected to exhibit the favorable particle trapping
properties first described by Grad and Berkowitz43,44 leading to a sharp
increase in device performance once the high beta mode was obtained.
A number of low beta polywell devices have also been operated
at The University of Sydney (USYD).45–51 The work has been principally concerned with the phenomenon of potential well formation and
characterization of the electron trapping properties of the polywell
field configuration. The first USYD polywell, constructed by Carr,45
consisted of teflon reels fastened together by aluminum brackets.
Electrons were injected using a hollow cathode style electron gun52
and floating Langmuir probe measurements showed 100–200 V
potential wells over a broad range of background gas pressures. Carr
subsequently upgraded his machine using toroidal aluminum coil
cases that were conformal to the magnetic field. The resulting machine
operated at lower voltages than its predecessor (100–200 V) but produced potential wells of comparable relative depth (10–20 V). Carr
also used biased Langmuir probe measurements to characterize the
energies of the trapped electrons and proposed a novel energy distribution function.47 Cornish49,50 conducted a survey of scaling laws
within the Polywell, with particular consideration given to the dependence of virtual cathode formation on the size and spacing of the magnetic field coils. Finally, studies conducted by Ren and Poznic51
examined electron density and energy distributions in a larger device
using both biased Langmuir probe and radio frequency plasma wave
techniques.
In this paper, we detail the operation of magnetically confined
virtual cathode (MCVC-0), the University of Sydney’s most recent
polywell-style machine. MCVC-0 represents both the largest and highest power virtual cathode fusion experiment at the University of
Sydney and was designed to operate in a parameter regime (in terms
of applied voltages and plasma currents) where measurable deuterium–deuterium fusion was possible. Due to size limitations, this
machine consists of two magnetic field coils, rather than the traditional
six, arranged such that they form a biconic cusp. As such, the device
may no longer be referred to as a polywell, and hence we adopt the
more generalized nomenclature magnetically confined virtual cathode.
In Sec. II, we describe the MCVC-0 system and the associated diagnostics. In Sec. III, we present results from single-ended floating probe
measurements of the MCVC-0 plasma and illustrate significant discrepancies between the measured electrostatic profiles and the
expected potential wells. In Sec. IV, we outline a theoretical model
based on the solution of the Poisson equation in the presence of a conductive medium and demonstrate how anisotropic plasma conductivity sufficiently accounts for the unexpected behavior of the MCVC-0
machine. It is found that the electric potential within the device is
dominated by those electrostatic boundaries that are intersected by
magnetic field lines, which has significant implications for the design
of future MCVC/polywell style fusion machines.
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
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II. APPARATUS
A. MCVC-0 device
The MCVC-0 device was constructed within a cylindrical vacuum
chamber 300 mm in diameter and 300 mm in length. A pair of
100 turn magnetic field coils (152 mm diam, 1770 lH and 0:15 X)
were situated within toroidal metal cases, which were the positive electrodes in the MCVC-0 system. The tori were constructed from 316
stainless steel and had major and minor radii of approximately 76 and
30 mm, respectively, and a wall thickness of 4 mm. In order to allow
biasing of the cases relative to the field coils, the coils were supported
free floating within the tori using sets of interlocking insulating collars
made from polyether–ether–ketone (PEEK). The coils were mounted
on a support frame consisting of a pair of stainless steel plates fastened
together by three vertical struts. The magnetic field coils were mounted
to the frame on PEEK bushings, such that the cases could be biased
610 kV relative to the support structure. The PEEK supports were
machined with multiple fins to increase the arc creep distance and
were sheathed in 60 mm O.D. alumina ceramic tube to prevent sputtering damage. In order to improve the high vacuum compatibility of
the apparatus, all steel components were baked at 500 C for 72 h. The
coil support structure is depicted in Fig. 1. Note that the magnetic field
lines within the device are everywhere conformal to the toroidal cases
such that particle collection on the electrodes occurs transverse to the
magnetic field. This so-called “magnetic shielding” reduces the collected current during a shot, thereby improving the efficiency of the
device.
The magnetic field coils were driven by a 108 mF, 900 V capacitor bank resulting in a maximum coil current of 2000 A. The peak
achievable magnetic field strength, when measured in the center of the
coils, was 1.15 T. The coil cases were biased at a maximum voltage of
6 kV, switched to the cases from a 180 lF capacitor using a MOSFET
switch. Electron beam injection was necessary for the formation of a
virtual cathode. A thermionic electron source was therefore constructed in the lower, axial vacuum port such that electrons could be
injected through the face of the lower magnetic field coils. A custom
filament support stage was positioned within the lower port of the vacuum chamber and populated with six 12 V, 50 W tungsten filaments
obtained from halogen light bulbs. The filament stage was supported
on ceramic rods such that it could be biased to 2000 V with respect
to the vacuum chamber. A grounded molybdenum mesh spanned the
port opening to accelerate the thermionic electrons into the chamber.
The entire electron gun assembly was sheathed with a 62 mm inside
diameter borosilicate glass tube in order to provide additional electrical
standoff between the floating filament stage and the wall of the vacuum port. In all experiments the electron gun current was set to its
maximum value of 150 mA, at an injection energy of 1 kV. The total
energy gained by an electron as it enters the core of the device is therefore the applied coil bias plus 1 keV.
Prior to all experiments, the MCVC-0 system was pumped down,
with baking at 165 C, until the ultimate base pressure of
1 107 Torr was reached. Optical spectra obtained during plasma
shots at base pressure were found to be dominated by strong Ha H
Balmer lines, identifying the primary gas component as residual atmospheric water. All experimental runs were therefore conducted without
the addition of hydrogen or deuterium gas and hence all presented
results were obtained using the base system pressure of
1 107 Torr. During a typical shot, the electron gun was activated
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FIG. 1. Magnetically confined virtual cathode machine.
and allowed to heat for 2–3 s such that the thermionic emission stabilized. The high voltage (HV) bias was switched to the coils for a duration of 400 ms and the magnetic coils were fired at the mid-point of
the HV pulse. An example shot trace is given in Fig. 2. A single-ended
floating Langmuir probe was positioned at the core of the device and
the resulting trace is given in red. The floating potential in the core is
seen to mirror the applied bias voltage up to time t ¼ 0 at which point
the magnetic field pulse (given in black) causes a sharp drop. Magnetic
trapping of electrons in the core results in a build of negative space
charge and hence the reduction in floating potential. As the magnetic
field pulse decays, the floating potential is once again seen to rise as
the trapped virtual cathode dissipates. Note that before (t < 200 ms)
and after (t > 200 ms) the HV pulse, both the coil cases and floating
probe are charged negatively by the electron gun beam.
B. High impedance floating Langmuir probe
A Langmuir probe was assembled from telescoping sections of
alumina ceramic tubing, sealed using TorrSeal adhesive and installed
in a translational stage such that it could be precisely positioned within
the vacuum chamber. The exposed probe tip consisted of a 50 lm
diameter, 5 mm long cylindrical tungsten wire.
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Published under license by AIP Publishing
The spatial distribution of relative electric potentials was obtained
through measurement of the floating potential, Vf, obtained by connecting the Langmuir probe, via a 2 GX series resistor, directly to an
oscilloscope input. The oscilloscope channel is terminated internally
by a 1 MX input resistor in parallel with a 13 pF capacitance. The
resulting RRC circuit acts as a 2001:1 voltage divider between the
probe tip and ground up to a roll-off frequency of 12 kHz. The temporal response of the probe is therefore very short relative to the timescale of a plasma shot. The floating potential is always smaller than the
space potential and may be approximated, for a Maxwellian plasma, as
kB Te
2mi
;
(2)
ln
Vf Vs 2e
pme
where kB is the Boltzmann constant, Te is the electron temperature, e
is the fundamental charge and me and mi are the electron and ion
masses, respectively.53 For hydrogen, deuterium and water the logarithmic factor is about 7.1, 7.7, and 9.9, respectively. The floating
potential is therefore only a valid measure of the space potential in systems where the electron temperature, Te, does not vary significantly
with time and the electron velocity distribution does not contain a
non-thermalized, high energy beam components. In systems where
ðkB Te =eÞ Vs , however, the floating potential serves as a convenient
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FIG. 2. Example shot of the MCVC-0 device. The applied system voltage and field
coil current are given in blue and black, respectively. The floating potential in the
center of the device is given in red and is seen to mirror the applied system voltage
before and after the magnetic field pulse. During the shot, the core potential is
reduced relative to the biased coils.
approximation to the space potential without the need for probe biasing and current sensing. This is particularly important in IEC systems
where the space potential may be many tens of kilovolts and traditional swept probe measurements may not be possible.
In order to demonstrate that validity of the floating probe diagnostic in our system, biased probe measurements of the type described
by Langmuir54 were also conducted. A DC bias was applied to the
probe and the collected current recorded over a single plasma shot.
Varying the probe bias over repeated shots resulted in a set of current
traces, Ip ðtÞ, and probe voltages, Vp ðtÞ. The space potential was then
computed from the resulting IV curves by fitting the current–voltage
relationship derived by Langmuir for a cylindrical collector, that is,
8
sffiffiffiffiffiffiffiffiffiffiffi > exp ðgÞ;
g<0
kB Te < pffiffiffi
Ip ¼ 2prlNe e
2 pffiffiffi
2pme >
g þ exp ðgÞerf g ; g 0;
: pffiffiffi
p
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FIG. 3. Indicative IV curve, averaged over the t ¼ 70–73 ms period of the shot in
shown Fig. 4. The computed plasma parameters are shown and the space potential
of 26.24 V is in good qualitative agreement with the measured floating potential of
approximately 15 V over the same period.
presented in Fig. 4. Prior to the magnetic field pulse, the floating probe
is a poor indicator of the space potential. In the absence of a magnetic
field, the plasma density due to the electron beam is extremely small,
and hence the collection of charge on the unbiased probe is slow. This
results in long charging times and a significant lag in the probe
(3)
where
g¼
eðVp Vs Þ
;
kB Te
Ne is the electron density and r and l are the probe radius and length,
respectively. An example IV curve is given in Fig. 3, averaged over the
70–73 ms period of the shot shown in Fig. 4. The solid line indicates
the best fit of Eq. (3) to the data and the computed plasma parameters
are given. The estimated space potential of 26.24 V is in good qualitative agreement with the measured floating potential of approximately 15 V over the same period. A comparison of the space
potential and floating potential measurements over an entire shot is
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Published under license by AIP Publishing
FIG. 4. Comparison of space potential and floating potential as determined via
biased and high impedance floating Langmuir probe measurements, respectively.
For the duration of the magnetic field shot (0–100 ms), the two measurements are
shown to agree within a few tens of volts, indicating that the floating probe is a valid
measure of the space potential.
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FIG. 5. Probe positions within MCVC-0 at
which the floating potential was measured.
Circular, triangular, and square markers
correspond to the floating potential measurements presented in Figs. 8–10,
respectively, as well as simulated potentials given in Figs. 12–14, respectively.
response relative to changes in the space potential. Note, however, that
for the duration of the magnetic field pulse the two measurements are
in good agreement. The high impedance floating probe therefore offers
a far more convenient means of measuring the space potential during
the shot, without the need for large numbers of repeat pulses. It is
expected that the space potential and floating voltage should agree to
within tens of volts.
III. FLOATING POTENTIAL MEASUREMENTS IN MCVC-0
The MCVC-0 system was pumped down, as described in Sec.
II A, until the base pressure of 1 107 Torr was reached. The high
impedance Langmuir probe was positioned in the center of the
MCVC-0 device and moved horizontally through the radial plane of
the device in increments of 10 mm. The floating potential was
recorded at each position for varying magnetic field strengths of 0 to
0.2 T. Magnetic field strengths are quoted according to the peak
strength along the device axis. Biases of 2–6 kV were applied to the
coil cases, and the collected plasma currents recorded as a voltage drop
across a 700X series load resistor. The above procedure was repeated,
moving the probe in 10 mm increments vertically along the central
axis of the device, as well as along a vertical line 40 mm off axis. In the
second case, the off axis probe was positioned such that it approached
within 5 mm of the biased coil casing. The positions at which the floating potential was sampled are indicated in Fig. 5.
Presented in Fig. 6 is the floating potential in the core of the
MCVC-0 device for an applied coil bias of 6 kV and varying magnetic
field strengths. The traces are labeled according to the maximum magnetic field attained during the shot. The core potential is seen to drop
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Published under license by AIP Publishing
FIG. 6. Floating potential as measured in the core of the MCVC device for varying
magnetic field strength. Each trace is labeled according to the peak field strength
during the shot. The lower panel illustrates a representative magnetic field coil current pulse in normalized units.
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significantly with the application of even modest magnetic field
strengths, eventually saturating at a floating potential of approximately
200 V at 0.2 T. Increasing the magnetic field strength beyond 0.2 T
resulted in asymptotic approach of the floating potential toward
ground. The observed well depth of 5.8 kV represents approximately
82% of the injected electron energy of 7 kV, in keeping with previous
measurements.55 The floating potential was averaged over the 15 ms
window at the peak of the magnetic field shot and plotted against the
average field strength over the same period. The results are given in
Fig. 7. Note that there is very little dependence on the applied bias voltage as in all cases the core potential reaches saturation at around
60 mT. In an electron trapping model, we expect the electron trapping
time, and hence the resulting virtual cathode potentials, to be dependent on the electron injection energy in this cusped magnetic field
geometry. Gummersall56 estimated the electron confinement time in a
zero-beta, six-coil polywell machine to be given by
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The above expressions were formulated for a six-coil cubic device and
so are not exact for our two coil system. They do, however, adequately
demonstrate the expected behavior of the potential dip as a function of
magnetic field strength for varying electron energy. Computing the
derivative of Eq. (6) results in
djDVf j 1 R2coil 1
3=4 1=2
dBcusp
2 E Bcusp
(7)
(6)
and we note the dependence of the slope on the kinetic energy of the
injected electrons. This is found to be inconsistent with the curves in
Fig. 7, which exhibit almost identical gradients in both the pre- and
post-saturation regions. This apparent independence between the
injected electron energy, E, and the slope of the Vf ðBcusp Þ curve suggests that electron trapping is not the primary cause of the drop in
core potential during a shot. The floating potential is given as a function of radial and axial position in Figs. 8 and 9, respectively, corresponding to the circular and triangular markers in Fig. 5. In the
absence of a magnetic field, the measured floating potential is found to
approximate the calculated vacuum potential, given as a solid red line.
With the application of increasing magnetic fields, the potential profile
flattens until the floating potential is uniform from the core of the
device to the chamber wall. Vertical dashed lines denote the radius
and central plane of the magnetic field coils and we see there is no distinction between the interior and exterior regions of the device. The
off axis potentials, indicated as squares in Fig. 5, are shown in Fig. 10
and exhibit considerably more structure, while the potentials measured
in the z ¼ 0 mm plane are consistent with the corresponding radial
measurements in Fig. 8; out of the central plane of the cusp, we see the
formation of a magnetic sheath about the coil casing with increasing
field strength. The maximum potential is observed in the central plane
of the coil, at the point of closest approach between the probe tip and
FIG. 7. Floating potential in the core of the MCVC-0 device for varying field
strengths and electron energies. Electron energy is given as the sum of the electron
gun voltage (1 kV) and the applied coil bias (2–6 kV). The pre- and post-saturation
regions are denoted by a dashed vertical line at 60 mT.
FIG. 8. Floating potential, for varying magnetic field strength, as measured radially
in the central plane of the MCVC-0 device, positions denoted by circular markers in
Fig. 5. The solid red line represents the computed vacuum potential while the
dashed vertical line indicates the major radius of the field coil.
1=2 3=4
Icoil Rcoil
;
(4)
E3=4
where Icoil and Rcoil are the coil current and radii, respectively, and E is
the electron energy, given by the sum of the electron gun injection
energy and the positive coil bias. If we take the magnitude of the dip in
the floating potential, jDVf j, to be proportional to the electron density,
itself proportional to the confinement time, and assume that the cusp
field varies as
s/
Bcusp /
Icoil
;
Rcoil
(5)
then we find that
1=2
jDVf j /
Bcusp R2coil
:
E3=4
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
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FIG. 9. Floating potential, for varying magnetic field strength, as measured in the
vertical direction along the central axis of the MCVC-0 device, positions denoted by
triangular markers in Fig. 5. The solid red line represents the computed vacuum
potential while the dashed vertical line indicates the central plane of the field coil.
the high voltage surface. The size of the sheath region is found to be
25–30 mm.
From the observed floating potentials within MCVC-0, it is
apparent that the device is not operating according to the magnetically
confined virtual cathode principle. Magnetic trapping of electrons
within the core of the device would be expected to form an
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electrostatic potential well in space, evident by a large differential
potential between the center of the machine and the regions of strongest magnetic field. Instead, the observed potential profiles are flat,
extending from the core to the chamber wall with very little variation.
Only where the floating probe is translated across magnetic field lines
is the potential seen to vary significantly; however, this effect is attributed to magnetic sheath formation rather than any more desirable
electron-trapping effect.
A valid criticism of the presented work is that the injected electron currents (150 mA) are very small relative to those in comparable
Polywell machines.57 While insufficient electron current is a possible
explanation for the observed behavior, the injected currents are nonetheless an order of magnitude larger than those used in smaller devices
in which potential well formation has been observed.48,50,51 The
biconic magnetic field configuration within MCVC-0 is expected to
exhibit larger particle losses than a comparable six-coil configuration;
however, this geometric difference is unlikely to account for such a
large departure from the expected behavior. In Sec. IV, we attempt to
address the observed characteristics of MCVC-0 through application
of a computational model based on anisotropic plasma conductivity in
a magnetic field.
IV. ANISOTROPIC POISSON EQUATION
A common feature of hybrid electro–magnetic fusion devices,
such as the Polywell, is the use of transverse magnetic fields to limit
plasma conduction to electrode surfaces, so-called “magnetic
shielding,” with the view to reduce particle losses and therefore
improve the energy efficiency of the machine. In this section, we will
apply an anisotropic conductivity model to demonstrate the importance of boundary conditions within magnetic virtual cathode devices
and how complete magnetic shielding of electrode surfaces may prove
to be deleterious in the formation of electrostatic potential wells.
A standard approach in plasma modeling is to first consider the
vacuum fields, resulting from applied currents and electrostatic
boundaries, and subsequently evaluate perturbations to those fields
due to plasma currents and charges. Vacuum electric potentials may
be determined using the discrete form of the Poisson equation, the
solved electric potential being valid in all cases where the solution volume is filled with a medium of uniform electrical conductivity, r.
However, if the solution domain is instead filled with a medium whose
conductivity varies with position and direction, the Poisson equation
requires modification. Consider a current flowing in a bulk conductor,
J c Þ terms, and taking
written as a sum of source ð~
J s Þ and conduction ð~
the conduction term to be governed by Ohm’s law:
~
Jc ¼~
Js ~
S rU;
J ¼~
J s þ~
(8)
where U is the electric potential and ~
S is the conductivity tensor
within the medium. From the principle of charge conservation, we
determine that in the steady state the divergence of the current must
be zero and so taking the divergence of Eq. (8) and equating to zero
yields
r ðS rUÞ ¼ r ~
J s:
FIG. 10. Floating potential, for varying magnetic field strength, as measured in the
vertical direction at a radial distance of 40 mm from the central axis of the MCVC-0
device, positions denoted by square markers in Fig. 5. The solid red line represents
the computed vacuum potential while the dashed vertical line indicates the central
plane of the field coil.
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
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(9)
Equation (9) is a generalized Poisson equation which describes the
electrostatic potential in the presence of an arbitrary conductive
medium. We will consider the case in which the volume source current is zero and the conductivity is both inhomogeneous and
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anisotropic. The conductivity tensor therefore consists of nine unique
elements, each of which is given as a function of position
rl ¼ rl ðx; y; zÞ
0
1
rxx rxy rxz
~
S ¼ @ ryx ryy ryz A:
(10)
rzx rzy rzz
In a magnetized plasma discharge, such as that in MCVC-0, the inhomogeneities in the electrical conductivity are determined locally by the
strength and direction of the magnetic field. As MCVC-0 and polywell
devices may not be reduced to simpler one- or two-dimensional geometries, we require a method for computing the conductivity tensor in
the (x, y, z) coordinate set for any arbitrary magnetic field direction.
We therefore define a local coordinate system, (i, j, k), such that the ^i
unit vector lies in the (x, y) plane, and ^k is aligned to the local magnetic
field. The local (i, j, k) unit vectors are thus given by
^ ^
^i ¼ k z ;
jj^k ^z jj
^j ¼ ^k ^i;
where rk and r? are the conductivities along and across magnetic field
lines, respectively, and rH is the Hall conductivity, resulting from particle drift in crossed electric and magnetic fields. The elements of ~
S ijk
may be written in terms of the charges, qa , plasma frequencies, xa ,
gyro frequencies, Xa and collision frequencies, a , of each species a
within the plasma according to58
x2a a
;
a
2
ð a þ X2a Þ
P qa x2a Xa
rH ¼ 0 a
;
jqa j ð 2a þ X2a Þ
P x2
r k ¼ 0 a a ;
a
where the gyro and plasma frequencies are given, in units of Hertz, as
P
1 qa B
Xa ¼
;
2p ma
sffiffiffiffiffiffiffiffiffiffi
1
na q2a
xa ¼
:
2p 0 ma
The collision frequencies may be computed by summing over all possible particle–particle interactions within the plasma. A coordinate
transform is subsequently used to compute the equivalent conductivity
tensor in the global Cartesian coordinates. Given two sets of unit vectors ð^i; ^j; ^kÞ and ð^x ; ^y ; ^z Þ, the tensor transformation from the local
magnetic field frame to the global Cartesian frame is given by
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Published under license by AIP Publishing
(13)
~ is
where the transformation tensor Q
0
1
^x ^i ^x ^j ^x ^k
B
C
~ ¼ B ^y ^i ^y ^j ^y ^k C:
Q
@
A
^z ^i ^z ^j ^z ^k
(14)
Assuming a uniform plasma density and temperature throughout the
MCVC-0 device, and an applied voltage of 6 kV, we subsequently compute solutions to Eq. (9). Biased Langmuir probe measurements gave
estimates for the electron density and temperature during the plasma
shot as 5 1014 m3 and 1 eV, respectively, and hence we take
these parameters to be fixed. Using the expressions for electron–
electron and electron–ion collision frequencies given by Miyamoto,59
assuming singly charged ion species (Z ¼ 1) and taking ln K ¼ 12,60
gives
(11)
The conductivity tensor about the field line is then computed in the
local frame as
0
1
r? rH 0
~
0 A;
(12)
S ijk ¼ @ rH r?
0
0
rk
r? ¼ 0
~~
~T ;
~
S ijk Q
S xyz ¼ Q
ne e4
ln K ¼ 28:9 kHz;
p
ffiffiffiffiffiffiffiffi
6p20 3me ðkB Te Þ3=2
(15)
ne Z 2 e4
ln K ¼ 10:9 kHz:
p
ffiffiffiffiffiffiffiffi
51:620 pme ðkB Te Þ3=2
(16)
ee ~
^k ¼ B :
jj~
Bjj
scitation.org/journal/php
ei As the measured plasma density is found to be almost an order of
magnitude smaller than the neutral density at base pressure
(3:2 1015 m3 at 1 107 Torr), we also deduce electron–neutral
collisions to be a significant collision process within the system.
Taking the primary gas component to be residual atmospheric water
and using scattering cross sections for elastic and inelastic collisions
between electrons and water molecules compiled by Itikawa61 and
Mu~
noz,62 we estimate the e H2 O collision frequency for a
Maxwellian electron distribution as
e H2 O nhrvi
¼
P
kB Tg
ð1
v3 rðvÞ exp me v2 =2kB Te dv
0ð
1
v2 exp me v2 =2kB Te dv
¼ 6:7 kHz;
0
(17)
where P and Tg are the neutral gas pressure and temperature, respectively. The absolute electron collision frequency is therefore taken as
the sum of the above processes
e ¼ ee þ ei þ e H2 O 46:5 kHz:
(18)
The solutions to Eq. (9), for four values of magnetic field strength, are
given in Fig. 11 as slices in the (R, Z) plane. The positions of the biased
coil cases, which form the system anode, are indicated by dark circles.
The left-hand edge of each panel represents the central axis of the
device, while the top, bottom, and right-hand edges form the grounded
wall of the vacuum chamber (cathode). Each panel is labeled according
to the maximum magnetic field strength along the central axis of the
device. For the smallest values of the applied magnetic field, the solutions to Eq. (9) are consistent with the expected vacuum potential.
However, as the magnetic field strength is increased the contours of
electric potential are found to increasingly conform to the magnetic
28, 042702-8
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FIG. 11. Solutions to the electric potential within MCVC-0 as defined by Eq. (9). The solutions are given as radial slices in the (R, Z) plane and are computed for point cusp
magnetic field strengths of 1–12 lT. Increasing the magnetic field strength results in the electric potential flattening in directions parallel to magnetic field lines, such that the
wall potential (0 V) is projected along magnetic field lines and into the core of the device.
field lines, such that the potential approaches uniformity along a given
line. As the ratio of parallel to perpendicular conductivity
rk =r? ! 1, the magnetic field lines increasingly represent discrete
conducting paths, and where those field lines intersect Dirichlet
boundaries within the device the potential along the line will subsequently tend toward the potential applied to the boundary. In this
way, we may account for the uniform potential distributions within
MCVC-0 at elevated magnetic field strengths. Due to the large size of
the MCVC-0 coils relative to the vacuum chamber, a majority of field
lines terminate directly on the grounded chamber walls. The increase
in the degree of ionization and plasma density at elevated magnetic
field strengths coupled with the increasing disparity between rk and
r? means that all points within the plasma volume converge to the
potential applied to the chamber wall, that is, zero. Potential slices,
directly analogous to those in Figs. 8–10, are given in Figs. 12–14. We
notice a marked similarity between the form of the simulated potentials and their experimentally determined counterparts, providing a
strong indication that the anisotropic conductivity model accurately
describes the behavior of the MCVC-0 device. We do note, however,
that there is a four order of magnitude discrepancy between the magnetic field strength required to saturate the simulated potentials
(11–12 lT) and those measured during the experiments (100
mT). It is therefore apparent that our estimates for the electron collision frequencies vastly underestimate the collisions within the MCVC0 plasma, indicating errors in our estimates of the electron temperature, electron density, or the system pressure.
The estimates of Te and ne used in the computation of the conductivity tensor were obtained from the biased Langmuir probe measurements discussed in Sec. II B. Because similar measurements were
not possible while biasing the grid at high voltages (>500 V), it is possible that the obtained values of ne and Te differ somewhat from the
actual values while operating in the high voltage mode. While additional work is required to obtain more accurate values for these
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Published under license by AIP Publishing
parameters during high voltage operation, incremental adjustments to
ne and Te will not sufficiently alter the values of ee or ei to account
for the large differential between the simulated and experimental
results. We therefore consider the estimated electron–neutral collision
frequency to be the leading cause of the observed discrepancy.
The computation of eH2 O ¼ 6:7 kHz, through Eq. (17),
assumed a neutral gas pressure of 1 107 Torr. While the vacuum
FIG. 12. Simulated floating potential, for varying magnetic field strength, as computed radially in the central plane of the MCVC-0 device. The sampled positions
are indicated by circular markers in Fig. 5 and directly correspond to the measurements presented in Fig. 8. The solid red line represents the computed vacuum
potential while the dashed vertical line denotes the major radius of the field coil.
28, 042702-9
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FIG. 13. Simulated floating potential, for varying magnetic field strength, as computed in the vertical direction along the central axis of the MCVC-0 device. The
sampled positions are indicated by triangular markers in Fig. 5 and directly correspond to the measurements presented in Fig. 9. The solid red line represents the
computed vacuum potential while the dashed vertical line denotes the central plane
of the field coil.
FIG. 14. Simulated floating potential, for varying magnetic field strength, as computed in the vertical direction at a radial distance of 40 mm from the central axis of
the MCVC-0 device. The sampled positions are indicated by square markers in Fig.
5 and directly correspond to the measurements presented in Fig. 10. The solid red
line represents the computed vacuum potential while the dashed vertical line
denotes the central plane of the field coil.
system maintained a constant pressure of 1 107 Torr while idle, it
is likely that plasma sputtering of adsorbed water from the electrode
surfaces during a shot resulted in transiently higher pressures for the
duration of the measurement. A significant increase in neutral density
could account for the discrepancy between the magnetic field sensitivity of our simulated potentials compared to the measured values.
Exploratory variation of the value of eH2 O used in the simulations
found that excellent agreement is obtained between the simulated and
experimental potentials when eH2 O 330 Mhz, consistent with a
gas pressure of approximately 5 103 Torr. While the hot cathode
Bayard–Alpert pressure gauge used in this work had sufficiently fast
temporal response to measure such pressure changes over the timescale of a plasma shot, perturbation of the gauge reading by the large
fluxes of charged particles from the plasma render instantaneous measurement of the pressure impossible. Gating the vacuum system
shortly before the shot may allow the integrated pressure rise to be
measured; however, the system is not currently configured in such a
way as for this experiment to be carried out.
Regardless of the factors discussed above, the striking similarities
between the functional form of the experimental and simulated data
provide a strong indication that the non-isotropic conduction model
accurately describes the behavior of the plasma within MCVC-0. Care
must be taken in future experiments to obtain an instantaneous measurement of the system pressure such that the experimental and computational work may be fully reconciled.
high impedance floating Lamgnuir probe, which was demonstrated to
be an adequate indicator of plasma potential to within a few tens of
volts. Contrary to expectation, electrostatic potential well formation,
resulting of magnetic confinement of electrons within the core, was
not observed. The electric potential throughout the device was instead
found to flatten with increasing magnetic field, eventually saturating at
an approximately uniform value of zero volts.
In order to account for the unexpected behavior of the MCVC-0
device, a computational model was developed based on the Poisson
equation within an inhomogeneous, non-isotropic conductor. It was
found that electric conduction along magnetic field lines from
grounded Dirichlet boundaries was sufficient to explain the observed
potential profiles within MCVC-0. The form of the simulated and
experimental datasets shows excellent agreement, differing only in the
precise sensitivity of the electric potential to the magnetic field
strength. The simulated potentials were found to match their experimental counterparts at applied magnetic fields strengths four orders of
magnitude smaller than the experimental values. This disparity is
attributed to poor estimation of a single free parameter, namely, the
electron–neutral collision frequency, itself a function of the system
pressure, as well as the electron density, temperature and available
cross-section data. Refinement of our estimate for eH2O is best
achieved by obtaining a real-time measure of the neutral gas pressure
during a plasma shot and this will become the topic of future work. In
this way, it is hoped that the model may be tuned for predictive
purposes.
The described findings have significant implications for hybrid
electric/magnetic fusion devices. In systems where biased electrodes
are used to produce an electric field, those boundaries must not be
intersected by magnetic field lines. Any field line that intersects a
V. CONCLUSIONS
In Secs. II–IV, we have presented both experimental and computational data describing the electrostatic potential profiles within the
MCVC-0 device. Potential measurements were obtained by way of a
Phys. Plasmas 28, 042702 (2021); doi: 10.1063/5.0040792
Published under license by AIP Publishing
28, 042702-10
Physics of Plasmas
Dirichlet boundary will tend toward a uniform electric potential along
the field line, as prescribed by the voltage on the boundary.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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