A Polywell Fusion Reactor Designed for Net Power Generation

Abstract

A brief history of Polywell progress is recounted. The present PIC simulation explains why the most recent Polywell fusion reactor failed to produce fusion energy. Synchronized variations of multiple parameters would require DC power supplies, not available in historic model testing. Even with DC power, the simulation showed that the trapping of cold electrons would ruin plasma stability during start-up. A theoretical solution to this trapping problem was found in Russian literature describing diocotron-pumping of electrons out of a plasma trap at Kharkov Institute. In Polywell, diocotron-pumping required matching the depth of the potential-well to the electron-beam current falling on a special aperture installed in one of the electromagnets. With diocotron-pumping the reactor was simulated to reach steady-state, maximum-power operation in a few milliseconds of simulated time. These improvements, validated in simulating small-scale DD reactors, were scaled up by a factor of 30 to simulate a large, net-power reactor burning p + 11B fuel. Power-balance was estimated from a textbook formula for fusion power density by numerically integrating the power density. Unity power-balance required the size of the p + 11B reactor to be somewhat larger than ITER.

Full text

1 23
Journal of Fusion Energy
ISSN 0164-0313
J Fusion Energ
DOI 10.1007/s10894-017-0147-9
A Polywell Fusion Reactor Designed for Net
Power Generation
Joel G.Rogers

ORIGINAL RESEARCH
A Polywell Fusion Reactor Designed for Net Power Generation
Joel G. Rogers
1
Springer Science+Business Media, LLC, part of Springer Nature 2017
Abstract
A brief history of Polywell progress is recounted. The present PIC simulation explains why the most recent Polywell fusion
reactor failed to produce fusion energy. Synchronized variations of multiple parameters would require DC power supplies,
not available in historic model testing. Even with DC power, the simulation showed that the trapping of cold electrons
would ruin plasma stability during start-up. A theoretical solution to this trapping problem was found in Russian literature
describing diocotron-pumping of electrons out of a plasma trap at Kharkov Institute. In Polywell, diocotron-pumping
required matching the depth of the potential-well to the electron-beam current falling on a special aperture installed in one
of the electromagnets. With diocotron-pumping the reactor was simulated to reach steady-state, maximum-power operation
in a few milliseconds of simulated time. These improvements, validated in simulating small-scale DD reactors, were scaled
up by a factor of 30 to simulate a large, net-power reactor burning p ?
11
B fuel. Power-balance was estimated from a
textbook formula for fusion power density by numerically integrating the power density. Unity power-balance required the
size of the p ?
11
B reactor to be somewhat larger than ITER.
Keywords Polywell fusion power reactor Scale-model testing Particle-in-cell (PIC) plasma simulation 
Start-up control-sequence
Introduction
Polywell continues to show promise as a design for fusion
power production. Scale model reactors of the Polywell
design have been built and tested in the Energy Matter
Conversion Company (EMC2), founded in 1987 by Poly-
well’s inventor, Dr. Robert W. Bussard. Table 1summa-
rizes Polywell progress since the founding of the company.
Over the years that EMC2 was in operation, many
generations of scale-model Polywell reactors were
designed, built, and tested. The latest of these, named WB-
8, was designed to produce more fusion energy than its
predecessors, WB-6 and WB-7, which had already pro-
duced record levels of fusion output energy. For reasons
not fully explained, WB-8 produced no fusion energy at all.
After the failure of WB-8, EMC2 abandoned the Polywell
principle in favor of a new experimental design [9]. The
new design omitted the high voltage bias on the magnets,
an essential feature of Bussard’s designs. This latest
‘‘Polywell’’ relies on high energy plasma injection instead
of high voltage electron beams for heating the plasma.
Without the bias on the magnets, the cusp losses of elec-
trons are expected to be impractically large.
Following a final report from the Company’s CEO in
2014 [6], no further experimental work on Polywell has
been reported. Theoretical work has continued on several
fronts [7,8,10]. The additional theoretical analysis
reported here seeks to answer the following three ques-
tions: (1) What went wrong with the testing of model
reactor WB-8 that prevented it from producing energy? (2)
What new design features could be added to produce
energy from DD model reactors? (3) What are the pro-
spects for a net power, aneutronic reactor burning proton-
plus-boron (p ?
11
B) fuel?
Particle-in-cell (PIC) simulation [11] is the most rigor-
ous method available for analysis of plasma confinement
devices. The computer code used in this work is OOPIC
Pro, a commercial version [12] of a software program
developed over several decades at Berkeley [13]. A public-
domain version of the software, called XOOPIC, is cur-
rently available from the University of Michigan [14].
&Joel G. Rogers
rogersjg@telus.net
1
Vancouver, Canada
123
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https://doi.org/10.1007/s10894-017-0147-9
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PIC Simulation Results Compared
to Experimental Results
Figure 1shows a cut-away view of the interior of a scale-
model DD fueled reactor, simulated and reported in this
section. The drawing is from a patent specification [8]
disclosing an improved design for a cubic Polywell. Six
magnet modules, identical to the five labeled (282), mount
on the faces of a cubic vacuum tank (220). A new design
feature distinguishes the patented design from WB-6 and
WB-7; hollow legs (404) support the coil magnets. In the
positions shown, the legs are magnetically shielded so that
they intercept none of the circulating plasma particles.
OOPIC simulation in the patent specification predicted that
the electron losses would be improved by this modified
mounting arrangement. Following the patent’s publication,
similar legs were implemented in EMC2 and did indeed
improve WB-8’s electron losses over previous models.
WB-8 was measured to have 6-times higher plasma density
than WB-7 [15].
Even with improved plasma density, WB-8 did not
produce fusion. To determine why fusion was not pro-
duced, a more detailed PIC simulation was performed and
is presented here. The simulation has as an option incor-
porating an improved method of start-up. By turning the
improved method on and off in simulation, it was shown
that WB-8 would be expected to produce zero energy, in
agreement with measurements. The symptom of the failure
Fig. 1 Cut-away drawing of
insides of a scale-model
Polywell reactor [8].
Components are indicated by
reference lines labeled as
follows: 220 square vacuum
tank; 282 rectangular cross-
sections of coil magnets
mounted in metal boxes; 404
hollow legs connecting the tank-
walls to the magnet-boxes; 230
electron-emitter mounted on the
right-hand tank wall; and 225
gas-cell containing fuel gas
confined by differentially
pumped aperture plates
Table 1 Publications marking
Polywell progress Authors [reference]—year Publication type Summary of findings
Bussard [1]—1985 U.S. patent—filed Polywell ‘‘power amplifier’’ concept
Rogers [2]—2008 U.S. patent—filed Non-intercepting magnet supports
Bussard [3]—2008 U.S. patent—applied Testing of WB-1 thru WB-6 in EMC2
Rogers [4]—2011 Conference proceedings Model Proton-Boron reactor via 2D PIC
Kazemyzade [5]—2012 Journal of Fusion Energy Model DD reactor designed via 2D PIC
Park [6]—2014 Conference proceedings DD model tested, produced no fusion
Baker [7]—2014 Conference proceedings Massively parallel 3D PIC simulation
Rogers [8]—2014 U.S. patent—filed Differentially pumped DD fuel source
This article—2017 J Fusion Energ—submitted 25 m diameter, power reactor simulated
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measured in WB-8 was the decay of the potential-well,
which occurred as soon as the plasma’s density rose during
start-up [16]. This same symptom was also observed in
these simulations unless the improved start-up method was
added to the simulation.
The PIC simulation presented here tracks the positions,
velocities, and densities of a large number of electron and
ion particles moving step by step in time. While the sim-
ulation program is running, the computer displays plasma
diagnostics on the screen. Snapshots of the diagnostics
were manually stored from time to time to track the time
evolution of plasma parameters. The simulated-time’s
evolution starts at time-zero when the electron beam is first
turned on. Before time-zero, the tank is cold and empty of
plasma. From time-zero onward the tank gradually fills
with electrons and ions over an interval measuring a few
milliseconds. This ‘‘start-up’’ is followed by an indefinitely
long period of operation called ‘‘steady-state.’’ During
steady-state the plasma temperature and density are held
constant in time. It is the goal of start-up to bring the
plasma temperature and density to values producing max-
imum fusion power during steady-state.
Experience with this simulation demonstrates that the
start-up phase of reactor performance must be controlled
by periodically adjusting the magnetic (B) field strength,
electron injection current, and fuel-gas flow. These three
variables may be thought of as knobs which must be
adjusted simultaneously by the operator or by a control-
computer to supervise the rise of plasma density and
temperature during start-up. The surprising discovery was
that the three knobs must be varied in a precise relationship
to each other. If the knobs are adjusted out of step, for
instance, if the B-field is made too strong compared with
the plasma density, the potential-well does not develop and
the reactor never reaches fusion conditions.
The need for synchronized control of the knob settings is
a discovery which explains the failure of WB-8 to produce
fusion. All the testing of WB-8 was made with pulsed
electron-drive power and gas-puffed fuel injection. The
time variation of the power and gas-flow were not suffi-
ciently controlled to provide the required relationship of
one knob setting to another. This lack of control was a
deficiency caused by EMC2’s limited budget. In the future,
more expensive DC power supplies will be required to
control the B-field and electron drive current in correct
relationship to each other. In addition, the ion density must
be controlled by providing the reactor with vigorous vac-
uum pumping to remove un-ionized fuel gas. Unless
removed by pumping, un-ionized gas interacts with elec-
trons to create ions outside the core. These ‘‘exterior’’ ions
do not enter the core but rush to the tank walls where they
drain energy from the plasma and spoil the power-balance.
The simulation has been used to search the parameter
space of reactor operation to find operating regimes with
the maximum power-balance. Power-balance is the ratio of
fusion power output to drive power input. The drive power
input is the sum of the magnets’ drive power plus the
electrons’ drive power. The magnets’ power is reliably
predicted by the manufacturer of standard coil magnets. In
contrast, the electrons’ drive power is more difficult to
predict. Electron drive power depends on the detailed
losses of electrons to internal structures such as the magnet
boxes and tank walls.
Simulating the Start-Up of a Scale-Model
Polywell Reactor
When the simulation program is initiated on the computer
the operator is asked for the name of a text file containing
lines of source code. A partial listing of a typical file [17]is
shown in Fig. 2. The language of the source is defined in an
OOPIC User’s Manual, available on the web [18]. The
program reads the file, checks it for syntax errors, and
compiles it into an executable program in memory. Then
an operator’s control panel appears on the screen. The
control panel enables the operator to start and stop the
execution of the program and also enables the display of
one or more of a large number of diagnostic graphs. The
control panel also displays the elapsed simulation time
updated at each time-step.
Figure 3shows several diagnostic plots made after
150 ls of simulated time. Figure 3a is a snapshot of elec-
trons’ positions in the central plane of the reactor. Each
black dot represents the position of one electron
macroparticle having the mass and charge of 1e9 electrons.
In addition to the electrons, Fig. 3a shows the components
of the reactor as solid lines. The vacuum tank is repre-
sented by a surrounding square with tick marks, labeled in
meters (m). Inside the tank are shown eight rectangles
representing the conducting surfaces of magnet-boxes.
Each segment of the magnet-boxes is an equipotential
surface biased at a fixed voltage of 20 kV. In a real reactor,
the magnet boxes would be electrically connected to the
high-voltage terminal of a DC power supply dialed to
20 kV.
A current-carrying wire (not shown) is simulated at the
center of each rectangular box. The wires generate eight
magnetic fields according to the electrostatics formula [19]
for the field from a straight wire oriented perpendicular to
the simulated plane. The vector sum of the eight fields is
tabulated at each PIC-cell as a pair of vector-components,
the x- and y-components of field at each position in the
central plane.
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In Fig. 3a the electron emitter is shown as a vertical
orange bar located just inside the right-hand tank wall and
surrounded by a small red circle. Newborn electrons are
accelerated away from the emitter by the positive bias on
the magnets. The electric field of the bias causes electrons
to accelerate leftward along a horizontal line from the
emitter into the center of the tank. The inertia of the
electrons entering the core provides heat-energy directly to
the trapped electrons. Inertia also concentrates electrons at
the center of the core; they form a potential-well which
heats newborn ions.
Electrons are trapped by the combined effect of the
B-field impressed by the currents in the wires, plus the
electric field imposed by a 20 kV bias voltage applied to
the magnet-boxes. While the simulation is running, the dots
representing electrons move on the screen with each time-
step. Trapped electrons flow in and out of the central region
along eight cusp-lines. When an electron approaches a tank
wall, it slows under the influence of the electric field of the
bias voltage. At the wall, it reverses its direction and falls
back into the core. The inward and outward motion of the
electrons along the cusp-lines is called ‘‘recirculation.’’ As
they recirculate, electrons scatter from each other.
Scattering causes electrons to deviate from their injec-
tion energy. Electrons which deviate very much from their
original energy are removed from the plasma by two dif-
ferent mechanisms. Up-scattered electrons are removed by
hitting the tank wall. Down-scattered electrons hit a narrow
aperture designed to remove them. The removed electrons
are replaced one-by-one by electrons of the correct
(&20 keV) energy so as to maintain a narrow energy
distribution centered on the energy provided by the bias
voltage.
The final element of the internal structure shown in
Fig. 3a is a gas-cell, marked at the point of the vertical
arrow. The gas-cell is delimited by two vertical aperture
plates located in the bore of the left-hand magnet’s coil.
The cell’s position was relocated from the position shown
Fig. 2 A sample portion of the input-file text compiled to simulate the DD scale-model Polywell reactor. Individual text lines are labeled by
leading reference numbers referred to in the text
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in Fig. 1. The relocation was made to correct a fatal flaw
discovered in the patented design [8]. The new position of
the gas-cell in Fig. 3a enabled diocotron-pumping of cold
electrons [20]. Diocotron-pumping is essential to avoid the
runaway trapping of cold electrons along the cusp-lines
near the magnet coils. Cold, secondary electrons are born
as an undesired side-effect of ionizing fuel gas in the gas-
cell. Cold electrons also occur separately, caused by down-
scattering of primary electrons in the bulk of the plasma.
Energetic electrons from the emitter produce ions con-
tinuously in the gas-cell. At each time step of the simula-
tion, each electron has a computed probability of ionizing
the gas in the PIC-cell it occupies at that time-step. Each
electron’s probability-of-ionizing is proportional to the
cross section for electron-atom ionization times the density
of gas in the PIC-cell. An electron ionizes the gas or not,
depending on a throw of a weighted, random number in
software. Due to the random nature of the process, this
simulation method was named Monte Carlo Collisions
(MCC) by its inventor [13]. The simulation’s input file,
listed in Fig. 2, describes the gas-cell and gas-pressure in
each PIC-cell in a block of code starting with the ‘‘MCC’’
operator in line ‘‘142’’ of the listing. Inside the block of
code following ‘‘MCC’’, the function ‘‘analyticF’’ on the
left hand side of line ‘‘150’’ specifies the density of fuel gas
in each PIC-cell of the simulated central plane.
Figure 3b shows color-coded gas density in each PIC-
cell. The positions of each of 4096 (= 64 964) PIC-cells
are represented by black squares outlined in a color rep-
resenting the gas density in that PIC-cell. Five magenta-
colored PIC-cells are shown at the head of the arrow in
Fig. 3b. These five cells contain simulated deuterium gas at
pressure 1e-4 Torr, the gas being partially confined by
aperture plates. Outside the gas-cell, PIC-cells are color-
coded red, representing background gas at 1000-times
lower density than the gas in the cell. Un-ionized gas leaks
out of the gas-cell to be continuously pumped away by
vacuum pumps. The pressure of the background gas was a
(a) (b) (c)
(d) (e)
Fig. 3 Snapshots of PIC diagnostic displays. Individual panels of the
figure are as follows: aBeam-electrons’ positions in the central plane.
Magnet-boxes (rectangles) are biased to 20 kv. Tank-walls, shown as
a surrounding square, are held at ground potential (0 V). Beam
electrons originate in an (orange) emitter, marked by a small (red)
circle. The arrow points to a gas-cell with aperture-plates. bPositions
of 4096 PIC-cells, each represented by a small black square. Each
PIC-cell’s gas density is coded by the color of cell’s border, red for
background density and magenta for 1000-times higher density inside
the gas-cell. cPositions of secondary-electron, defined as a unique
specie produced as a byproduct of fuel ionization. dIons’ positions.
eCurves plotting the time-variations of the particle-counts in the
previous (a,c, and d) panels (Color figure online)
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defined variable of the simulation chosen to mimic the
effect of differential pumping [8]. The vacuum pumps are
not explicitly simulated; only the resulting background-
pressure is simulated.
Ions born outside the gas-cell are undesirable because
they do not accelerate to the optimum energy for fusing.
The position of the gas-cell was chosen so that ions born at
that position gain the maximum energy available from the
electrostatic potential-well. Previous simulations [8]
showed that a maximum background density of 0.1% of the
gas-cell’s density could be tolerated without ruining the
functioning of the reactor. With the intent of making the
vacuum pumps as inexpensive as possible, this highest-
permissible background pressure was simulated. Lower
pressure would require extra pumping.
The energies of electrons produced by ionizing the
deuterium gas are very low compared to the energies of the
beam-electrons from the electron-emitter. Low-energy
electrons become trapped near the magnets and degrade
performance. The build-up of low-energy electrons causes
a fatal problem unless they are removed from the plasma
by some means.
In order to investigate the magnitude of the cold-elec-
tron-trapping, ionization electrons were tracked separately
from electrons from the emitter. The ionization electrons
were defined to belong to unique specie. In the parlance of
the OOPIC compiler, this was accomplished with the
‘‘Specie’’ block of code starting at line ‘‘127’’ in Fig. 2.
This ‘‘Specie’’ block defines the charge and mass of
‘‘MCCelectron’’ to be the same as the charge and mass of
‘‘electrons,’’ defined in the preceding ‘‘Specie’’ block of
code. Following the MCCelectrons specie definition, the
same name was assigned to the compiler-defined name,
‘‘eSpecies,’’ in line ‘‘151’’ of the ‘‘MCC’’ block of code.
This assignment primes the MCC operator to create one of
these special, low-energy electrons with each ionization
event.
Figure 3c shows the positions of the MCCelectron par-
ticles as black dots. These secondary electrons are most
concentrated at the position of the gas-cell. The potential-
well designed to accelerate ions into the core acts in the
opposite direction on the oppositely-charged electrons. The
potential traps electrons near the gas-cell. By the time of
the snapshot shown in Fig. 3c, some of the MCCelectrons
had escaped the trap and moved away from their birth-
place. These more distant, secondary electrons had been
heated by the primary electrons passing through the gas-
cell. The MCCelectrons tend to be concentrated in the
narrow portions of the cusps, both in the magnets’ bores
and in the gaps between the magnet boxes. This visible
concentration of the MCCelectrons in the cusps in Fig. 3c
provided a clue, useful for designing a means to remove
them. The same method was found useful to remove down-
scattered primary electrons which also become trapped in
the cusps. All eight of the simulated cusps demonstrated
electron trapping, not just the cusp with the gas cell. Prior
to the results reported in this article, cusp-trapping was an
ancient, little recognized problem with Polywell.
It is important to note that trapped electrons pose a
problem; they weaken the potential-well, causing it to
become more shallow. Confronted by the concentration of
cusp-trapped electrons, incoming beam-electrons are slo-
wed by the repulsion of the cusp-trapped electrons. The
central electron density tends to be low due to the beam-
electrons not reaching the center of the reactor. Low cen-
tral-electron-density causes a shallow potential-well which
translates to low ion energy which in turn translates to low
fusion energy output.
Figure 3d shows a snapshot of ion positions, taken at the
same time as the other diagnostic plots of Fig. 3. Ions are
born in the gas-cell with approximately zero kinetic energy
and approximately 10 keV of potential energy. Initially,
the newborn ions accelerate horizontally to the right,
‘‘falling’’ down the inner slope of the potential-well. The
ions’ positions appear randomly distributed in Fig. 3d, but
their energies are not random. Each ion accelerates on its
trips toward the center and decelerates on its trips away
from center. Their energies at the edge of the well are small
compared with their energies at center. At the center the
ions have enough energy to have a substantial probability
for fusing.
Figure 3e shows the early-time sequence of the filling of
the reactor with three species of particles. The vertical axis
shows the logarithm of the particle counts, as a function of
‘‘Time,’’ shown on the horizontal axis, labeled in seconds
(s). At the latest time shown, 0.5 ms after time-zero, the
counts of all three particle species have become constant in
time. This is a sign that steady-state reactor operation has
been reached. Steady-state operation means that the rate of
loss of each particle type is equal to the rate of production
of the same type.
The final goal of successful start-up is an optimum
density at which the ratio of plasma pressure to magnetic
pressure, traditionally called Beta (b), is equal to unity. A
sharp surface enclosing the confined electrons is clearly
visible in Fig. 3a. This surface remains stable as long as the
electron-current and gas-pressure are held constant at val-
ues producing steady-state. The condition b=1 defines
this surface.
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Raising the Plasma Density to Reach
Maximum-Power Operation
Figure 3shows diagnostics early in start-up. To obtain
maximum power output from the model reactor, the density
of ions at the center must continue to rise until it approx-
imately equals the density of electrons. A slight excess
density-of-electrons is needed to maintain the potential-
well, but density above a few parts-per-million of the ion-
density would represent a waste of drive power. In a real
reactor, this change would be affected by gradually
adjusting one or more knobs controlling electron drive
current, gas-flow, etc. In the simulation, gradual variation
of knobs is mimicked by stopping the simulation using the
control panel, dumping the particle positions and velocities
to disk, editing the input file to insert the new knob values,
and then restarting the simulation with altered knobs but
with the same particle positions/energies. Simulated time is
suspended while the program is stopped. This method of
adjusting parameters changes the values between the time
steps simulated. The technique mimics arbitrarily slow
variation of knobs in a real-world reactor.
In a real reactor, there are practical limitations to the
permissible rate-of-change of knob values. For example,
the valves and regulator controlling the gas flow to the gas-
cell have inertia and thus can only be changed on time-
scales that are longer than milliseconds. Magnets also have
inductance and cooling requirements that limit their rate of
change. These features of valves and magnets would cause
the time needed to reach steady-state to be longer in the
real world than the 0.5 ms after time-zero shown in Fig. 3e.
In an actual reactor, different diagnostics from the simu-
lated ones shown in Fig. 3would be monitored, for
instance, gas-pressure and magnet-temperature. In the
simulation, steady-state operating points were found by
trial and error adjustments of parameters. Temporarily
stopping the simulation at steady-state operating points
mimicked the slow variation of real world parameters
restricted by the hardware’s inertia.
Key Features of Polywell Operation—Wiffleball
Formation
By Bussard’s definition, a Wiffleball [21] is a quasi-
spherical concentration of plasma which has been inflated
by diamagnetic exclusion of the central B-field. Lavrentev
discovered the same effect experimentally in a scale-model
reactor he tested at Kharkov Institute. Lavrentev called the
Wiffleball ‘‘the volume of superseded magnetic field.’’ In
his words [22], ‘‘The reduction of electron and ion losses
from the trap with growth of plasma density is the result of
replacement of the magnetic field in the central region of
the trap. …With the growth of plasma density the volume
of the superseded magnetic field increases….’’ (End of
Lavrentev quote).
Wiffleball formation is an essential feature of Polywell
start-up. Unfortunately, the OOPIC Pro software was not
capable of rigorously simulating it. To obtain results in a
reasonable amount of time the simulation was, by neces-
sity, executed in electrostatic mode. In this mode the total
B-field arises only from the static current in the electro-
magnets. The electrostatic mode omits additional diamag-
netic fields which would realistically arise from the
plasma’s interior currents. To correct for the electrostatic
approximation, the supersession of the central B-field was
added artificially. The applied B-field was manually mod-
ified to impose a central region of zero B-field. The size of
the central region was increased with simulated time as if it
were controlled by a knob value. This technique created the
field-free core of the Wiffleball while avoiding the extra
computer time to simulate it rigorously.
Figure 4shows snapshots made at three selected time-
steps leading toward Wiffleball formation during start-up.
Nine diagnostics from the simulation are arranged in rows
and columns of three panels each. Panels in the same
column are snapshots made at the same time of plasma
evolution. Earlier time is in the left-hand column and later
time is in the right-hand column. The top row shows
electrons’ two-dimensional (2D) distributions in the central
plane of the reactor. The middle row shows the horizontal
component, Bx, of the impressed B-field along a horizontal
path through the center of the reactor. The selected path is
indicated by a solid black line in Fig. 4d. The bottom row
of Figures shows the electrostatic potential along the same
selected path.
The B-field inside the reactor is simulated by assignment
of arithmetic expressions for the two non-zero components
of the 3-vector. These assignments for x- and y-compo-
nents are in lines ‘‘118’’ and ‘‘119’’ of the source code
shown in Fig. 2. These expressions assign two components
of field to each of the 4096 cells of the PIC simulated
central plane. The expressions contain the cell-coordinates
themselves, ‘‘x1’’ and ‘‘x2’’, as well as a variable ‘‘dia2’’
giving the half-diameter (i.e. the radius) of the field-free
region at the center of the reactor. The value of ‘‘dia2’’ is
set in line ‘‘16,’’ chosen in simulation by editing the input
file. In actual reactor operation, the diameter of the Wif-
fleball would expand spontaneously, as described by Lav-
rentev [22]. In simulation, the expanding diameter is
controlled in sequential steps, increasing the assigned value
of the ‘‘dia2’’ variable while the program is stopped for that
purpose.
Figure 4a shows the electron positions with no dia-
magnetic effect, i.e. for ‘‘dia2’’ equal to zero. This is the
electron distribution that would arise after 25 lsof
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simulated time using the maximum strength of B-field
available from the selected coil magnets. The value of the
horizontal component of this B-field is graphed in Fig. 4b.
The magnitude of the field is zero at the center of the tank
and maximum at the centers of the bores of the left and
right magnets. The magnets simulated are commercially
available coils of outside-diameter 0.345 m, selected from
the GMW catalog [23]. The magnets were selected to be
approximately the size of those used by Bussard in his
successful WB-6 model reactor.
The maximum field capability of the magnets is the
main factor determining the maximum power capability of
the reactor. Because of its importance, a consistency check
was executed to verify that the tabulated values of field
were realistic. The maximum value of the field in the left-
hand and right-hand magnets’ bores is shown in Fig. 4bas
0.08 Teslas (T), about half the maximum field listed in the
GMW catalog [23]. A reduction from the catalog value is
expected since the field in each magnet is the superposition
of its own strong inward field added to weaker outward
fields from the other 3 magnets. To determine if the
reduction is quantitatively correct, a test-run of the simu-
lation was performed with the four magnets moved far
apart. This was accomplished by temporarily setting the
value assigned to ‘‘magCornerGap’’ in line ‘‘13’’ to be
much larger than the specified diameter of the coils. In this
configuration the field in each bore was generated by just 2
wires, those on either side of the bore. The other 6 wires
were moved so far away that they contributed negligible
field compared to the 2 nearer ones. In this configuration
the Bx diagnostic similar to Fig. 4b showed a peak value
increased to approximately equal the specified catalog
value. This test served to validate the expressions for the
two components of B-field.
The plasma’s spatial distribution in Fig. 4a has a visible
flaw which would make it defective in its power output.
The diameter of the plasma ball is seen to be much smaller
than the space between the magnets. This shrunken size
would result in a reduced fusion energy output from the
plasma, compared to what it would have if the plasma filled
(a)
(c)
(d) (g)
(e)
(b)
(f) (i)
(h)
Fig. 4 Snapshots of Wiffleball formation sequence during reactor
start-up. Panels are as follows: aElectrons’ positions with maximum
magnetic field. bMaximum magnetic field’s x-component as a
function of position along a selected line. cElectrostatic potential
function along the same line, the solid-line shown in the next panel.
dColor-coded electron-density with reduced B-field. eReduced
magnetic-field’s x-component. fElectrostatic potential showing
symmetric potential-well. gElectrons’ 2D density with Wiffleball.
hMagnetic field showing central region of diamagnetic exclusion
(between arrows). iPotential-well with Wiffleball (Color
figure online)
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the space. The fusion power is proportional to the volume
integral of the ions’ power-density which is shrunken in
proportion to the shrunken size of the Wiffleball. Smaller
volume translates to smaller integrated fusion power, a
fatal flaw if not corrected by improved start-up method.
In addition to the defective volume, the electron distri-
bution in Fig. 4a produced a defective potential-well.
Figure 4c shows a potential barrier lower on the right of
center than the left of center. Newborn ions starting at the
position indicated by the dashed line acquire too much
energy on their initial fall to the center. Their momentum
carries them over the lower peak on the right. They hit the
tank wall and are lost after only one pass through center.
The lost ions take energy out of the plasma without having
a chance to fuse. To maintain the plasma’s temperature, the
lost energy must be made up by increasing the current of
the incoming electron beam. This increase in input power
further ruins the power-balance.
To cure the shrinkage problem, the initial magnitude of
the B-field was reduced from the maximum capability of
the selected magnet [23]. Raising the B-field back to
maximum without shrinking the plasma ball required
gradually increasing the size of the field-free region at the
center of the reactor. This was accomplished in steps by
changing the input file each time the reactor reached a
steady-state stopping point. The maximum allowed B-field
was attained by the latest snapshots in Fig. 4, as shown in
Fig. 4h. The B-field is zero between the vertical arrows and
reaches the specified maximum at the centers of the mag-
nets, the same maximum values shown in Fig. 4b, but
without the shrunken plasma ball.
Key Features of Polywell Operation—Diocotron
Pumping
The concept of diocotron-pumping was first described in
English by Dolan [20]. In his section 2.5.4, Dolan wrote
‘‘ …electrons produced by ionization of neutral gas tend to
become electrostatically trapped in the anode region
[cusps]…but the diocotron instability may help remove
cold trapped electrons without seriously impairing hot
electron confinement.’’ Dolan’s statement is misleading in
its use of the word ‘‘instability.’’ Dolan’s ‘‘instability’’
refers to the random wanderings of electron beams recir-
culating along the cusps, not to the density of the plasma
itself. Ironically, instability in the beams is essential to
create stability in the plasma.
A general formula for diocotron-pumping in electro-
static traps was published by Yushmanov in 1984 [24]. In
1994, Dolan [20] applied the general formula to the
geometry of a proposed net power reactor of cylindrical
shape. He found that the power-balance depends critically
on the diameter of ‘‘anode gaps,’’ the term he used for the
recirculation openings through surrounding magnets.
Dolan’s ‘‘anode gaps’’ are analogous to the aperture shown
by the arrow in Fig. 4a. Dolan optimized his ‘‘anode gap’’
to produce maximum power-balance from the reactor. The
optimum value was found to be a compromise between
too-narrow a gap, which cut out too many hot electrons,
and too-wide a gap, which produced a shallow potential-
well allowing ions to escape. With the optimum value of
gap, Dolan found a power-balance equal to ten for a
hypothetical reactor with magnets 8 m inside-diameter.
The present simulation discovered that cold electrons
did also become trapped in cusps, as shown previously in
Fig. 3c. It was further found that the cold electrons could
be selectively removed from the plasma by an electrode
placed near the gas-cell. Figure 5a shows the electrons’
position-distribution under conditions of successful dio-
cotron-pumping. Cold electrons were selectively pumped
onto the aperture marked by the arrow in Fig. 5a. For
current on the aperture measuring approximately one-half
the injected electron current, the plasma reached a stable,
hot, high-density state following start-up. Stable operation
also required judicious control of the electrostatic potential,
using knobs to adjust injection current and gas pressure.
The curve in Fig. 5b shows the electrostatic potential along
the horizontal midline of the simulation plane. The dif-
ference in potential between the position of the gas-cell and
the center is about one-half the applied voltage, or about
10 kV. The energy imparted to the singly-charged deuteron
ions falling through this potential difference was 10 keV.
Figure 5c shows the rise of the plasma volume with
time, starting at time-zero. All three particle types reach
constant count by the latest time shown, 352 ls after time-
zero. Reaching such a steady-state condition is an essential
step required to mimic real-world magnets’ characteristics.
Reaching steady-state in simulation implies that the anal-
ogous, real-world-magnets’ drive could be increased over
an arbitrarily long time interval, rather than the fraction of
a millisecond in the simulation.
Without diocotron-pumping no such steady-state con-
dition would be possible. To see what happens without
diocotron-pumping, the simulation was run with zero
electron-current hitting the aperture. Figure 5d shows the
simulated electron distribution which resulted from
changing the input file in only one assignment; that
assignment opened the diameter of the aperture indicated
by the arrow. This one change made a vast difference in the
development of the plasma. Instead of the uniform internal
electron volume seen in Fig. 5a, d shows a central
depression almost devoid of electrons. The electrons have
become concentrated along cusp-lines in all eight magnet
gaps, and especially concentrated in the right-hand mag-
net’s bore.
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The potential function displayed in Fig. 5e is defective.
It is asymmetric, lower on the right than the left. This
asymmetry allows newborn ions to escape and hit the right
tank wall. The one change in the input file resulted in the
time sequence shown in Fig. 5f. At the latest time simu-
lated, 163 ls, the electron counts are still rising while the
ion counts are falling with time. The simulation was not
continued further since the time-variations of the parame-
ters would not lead to a useful plasma configuration.
The Physical Principle Behind Diocotron Pumping
Figure 6shows diagnostic displays from a special simu-
lation designed to exhibit the characteristics of the dio-
cotron instability. Figure 6a is a snapshot of beam-electron
particle positions made early in start-up, while the electron
density is still low. To make the beam visible against the
sparse background plasma, the beam emitter was specified
as having zero-width. The beam originates from a point-
source placed at the same position as the extended source
shown previously in Fig. 3a. The incoming beam in Fig. 6a
is visible as a straight line between the emitter and the
center of the reactor, indicated by the dashed arrow. To the
left of center, the beam’s path was seen to deviate ran-
domly upward and downward from its original straight-line
direction. Complete changes in the beam’s direction were
observed to happen over approximately 10 time-steps,
about one nanosecond of simulated time.
The wandering of the beam has the beneficial effect of
increasing the rate of energy transfer of the beam in a
limited region of space near the gas-cell. Cold electrons
congregate near the gas-cell because of the local maximum
of the electrostatic potential. Figure 6b shows a section of
this potential along a horizontal path through the center of
the reactor. The gas-cell aperture is located at the peak of
this potential, at the position of the vertical dashed line
connecting Fig. 6b and a. To the negatively charged elec-
trons, the peak appears as a valley, trapping cold electrons.
Unless heated, cold electrons will continue to accumu-
late in the cusps. Diocotron oscillation of the beam
increases the heating of the cold electrons near the aper-
ture, compared to what it would have been if the beam had
continued on its straight-line-trajectory through the gas-
cell. The undisturbed beam interacts with the electron
plasma as a whole, whereas the wandering beam interacts
locally. The wandering beam imparts transverse momen-
tum to the trapped electrons at positions where it turns
aside, c.f. at the head of the horizontal arrow in Fig. 6a.
Figure 6c is a snapshot of electrons plotted in horizon-
tal(x)-velocity versus x-position space. Each dot plotted in
Fig. 6a has a corresponding dot plotted in Fig. 6c. The
electron beam is visible as a curved, solid line. At the point
of the vertical arrow, the beam heads from the emitter to
(a)
(b)
(c)
(e)
(f)
(d)
Fig. 5 Diagnostics with and
without diocotron-pumping.
Individual panels are as follows:
aElectron-particle positions
with pumping onto the aperture
indicated at the arrow.
bHorizontal section of the
electrostatic potential function
showing symmetric ion
confinement. cTime course of
rising densities leading to the
steady-state snapshots shown in
(a) and (b). dElectron positions
showing suppressed central
density caused by wide-open
aperture (arrow). ePotential
function showing depression of
right-hand peak from cusp-
trapped electrons in the right-
hand cusp (above). fTime
course of particle-counts
showing signs of unstable ion
confinement, i.e. rising electron
density and falling ion density
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the left. Left-going electrons have negative velocities.
Between x-positions marked and `, the beam-electrons
show a unique relationship between x-velocity plotted on
the vertical scale, and the potential voltage shown in
Fig. 6b. In this region, the beam-electron’s velocity u is
related to its potential Uby the relationship Ue=mu
2
,
where m and e are the mass and charge of the electron. At
position `the beam electrons have reached their minimum
velocity, corresponding to the potential minimum voltage
marked by the down-pointing arrow in Fig. 6b. Once the
beam crosses to the left of position ´its direction becomes
unstable and its velocity varies randomly above and below
the velocity determined by the potential. Between positions
ˆand ˜the beam has reflected smoothly from the left-
hand wall and returns to the right, i.e. with positive x-ve-
locity. Traveling to the right at position ˜, random
oscillations again increase until the beam disappears into
the background plasma at position Þ.
By trial and error the simulation showed that 80% of the
bias voltage was the largest useful choice for potential-
well-depth. This depth is shown in Fig. 6b as the vertical
distance between the dashed and dotted lines. At a later
time in the simulation the well-depth had increased to 90%
of the injection energy. Figure 6d shows a snapshot made
at that later time. The incoming electrons from the emitter
approach the center at marker þ, but the beam does not
cross the center like it did in Fig. 6c. Rather, it reflects
from a repulsive core of negative charges at reactor center
and oscillates in velocity near position ¼, causing an
increase in the density of dots to the right of the center. If
the beam does not reach the gas-cell, it cannot pump out
the cold electrons there. With any potential-well deeper
than the one in Fig. 6c, the beam energy is wasted in its
(a) (c)
(b)
(d)
Fig. 6 Diagnostics characterizing diocotron-pumping. Panels are as
follows: aSnapshot of trapped electrons in 2D position-space. The
incoming beam travels right to left, and then wanders randomly
(arrow) after passing tank center (dashed arrow). bSnapshot of the
electrostatic potential function as a function of x-position. DU is the
depth of the potential-well, 80% of the applied bias voltage.
cSnapshot of electrons’ locations in 2D velocity-position space,
made at the same simulated time as previously. dSnapshot of ions in
velocity-position space made at a slightly later time, after DU had
increased to 90% of the bias voltage. Circled numbers and arrows are
explained in the text
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heating to the right of center. The cold electrons needing to
be heated are concentrated across the well near the gas-cell.
Steady-State, Maximum Power Operation
Bringing the reactor to a state of maximum-power opera-
tion was accomplished by synchronized manipulation of
the knobs controlling electromagnets’ drive current, elec-
tron drive current, and fuel gas pressure. Figure 7shows
the simulation diagnostics characterizing the maximum
power state. Figure 7a shows the time course of the plas-
ma’s three main components during the first 0.5 ms of
steady-state running, i.e. from ‘‘Time’’ 1.0 to 1.5 ms after
time-zero. The Figure shows that counts of the number of
electrons, ions, and MCCelectrons have all become con-
stant in time. For this reason the simulation was stopped at
1.5 ms after time-zero. The plasma conditions shown in
Fig. 7could be maintained indefinitely, but continuing
beyond 1.5 ms would have been a waste of computer time.
Steady-state, in a real reactor, would be accompanied by
continuous outflow of fusion energy in the form of high
energy particles reaching the tank walls. Charged particles
are steered out of the core by the magnets; they flow out
along the same cusp-lines as the electrons, through the
bores and between the magnets. Charged particles would
all reach the tank walls where their kinetic energy would be
converted to power, either directly on grids [25] or through
heat exchangers. The total energy of the fast particles from
each fusion event is known from the measured energy yield
[19] of the DD fusion reaction, approximately four million
electron-volts (Mev). The outflow of fusion power was not
simulated directly in PIC, but rather it was calculated by
numerically integrating the textbook [26] formula for
energy density. The power-balance was maximized by
searching for the right combination of knob settings.
(a)
(b)
(c)
(d)
(e)
Fig. 7 Snapshots of scale-model’s diagnostics in steady-state opera-
tion optimized for maximum power output. Panels are as follows:
aThree components of plasma’s density shown as functions of time.
Constant values indicate steady-state has been reached. bParticle
densities of electrons and ions along a selected central path. The
curves show that the central densities of electrons and ions are equal
except for statistical noise. cElectrostatic potential along the same
path as (b). The arrow shows the initial potential voltage of newborn
ions. dElectron-particles’ positions in 2D. eIon-particles’ positions.
The dashed (red) line marks the edge of a central region of uniform
density of electrons and ions, the Wiffleball (Color figure online)
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An essential feature of successful start-up is the near-
equality of the central ions’ and electrons’ densities. Fig-
ure 7b shows diagnostics of these two particles’ densities
along the usual horizontal path through the center of the
reactor. As simulated time progressed during start-up, the
ion density rose until it approximately equaled the electron
density at the center. This equality was the signal that the
simulated start-up was complete. The ion density was
stabilized at that time by experimentally reducing the rate
of fuel ionization in the gas-cell until the ion density
stopped rising.
The curves in Fig. 7b are subject to statistical variation
due to the limited number of simulated particles in each
PIC-cell. This statistical noise made it difficult to accu-
rately judge the difference between electron and ion den-
sities, except to say they are approximately equal at the
center. The adjustment of the drive conditions was
accomplished more accurately by monitoring the depth of
potential in a diagnostic like that shown in Fig. 7c. The
start-up phase was stopped when the depth of potential
reached the desired 80% of the drive voltage. Raising the
density beyond this value caused the potential to become
too shallow for effective diocotron-pumping. The potential
function in Fig. 7c is smoother than the densities in Fig. 7b
because the potential depends, via Maxwell’s equations, on
the charges of all the particles in the simulation, not just the
ones along the central section. In contrast, the curves in
Fig. 7b are determined from the particle counts in a single
row of PIC-cells.
To check that the variations in the ions’ density in
Fig. 7b are realistic, an estimate of the ions’ expected
statistical noise was made using standard counting theory.
The standard-deviation of counts in a PIC-cell is the
reciprocal of the square-root of the average number of ion-
macroparticles per cell. From the tick mark where the
‘‘ions’’ graph intersects the vertical axis in Fig. 7a, the
number of ions in the entire simulation is 6e12. The
number of macroparticles is this number divided down by
the weight of one macroparticle, specified as 1e9 in Fig. 2,
line ‘‘20.’’ This leaves 6000 as the total number of ion-
macroparticles in the simulation. Figure 7e shows this
number of particles to be uniformly distributed inside the
Wiffleball. The fractional area of the Wiffleball is the
square of its fractional diameter; Wiffleball diameter is
shown by the dashed line to be the same as the diameter of
the magnets. Magnet diameter is specified in line ‘‘10’’ of
the source listing. The fractional area occupied by ions is
(0.345/0.86)
2
=0.16. The number of ion-occupied cells is
this fraction multiplied by the total number of cells in the
simulation, 0.16 964
2
=660. The average number of ion
particles per cell is thus 6000/660 =9. From counting
theory, the fractional statistical noise on a count of 9 is the
reciprocal of the square root of 9, which is 1/3. Each count
in the ions’ density profile has an uncertainty of plus or
minus 1/3 of its value. This theoretical uncertainty agrees
approximately with the visible variations from cell to cell
in Fig. 7b. Such statistical noise cannot be avoided except
by lowering the particle weight in line ‘‘20’’ which would
slow the simulation.
In Fig. 7b the electrons’ density exhibits peaks on either
side of the central ion-peak. These side-peaks in electrons’
density occur at the same positions as the side-peaks in the
electrostatic potential in Fig. 7c. The concentration of cold
MCCelectrons at the same position in Fig. 3c suggests that
these side-peaks are caused by trapping of cold, down-
scattered beam-electrons. Cold electrons eventually
migrate to the left-hand cusp, the one containing the
aperture, where they are pumped out by diocotron-pump-
ing. The height of the side-peaks is a measure of the rel-
ative rates of down-scattering and diocotron-pumping. As
long as the peaks stop growing at the end of start-up, the
pumping rate is keeping up with the down-scattering rate.
The constant density of ‘‘electrons’’ in Fig. 7a assures this
is the case.
The units of the densities shown in Fig. 7b are particles
per square-meter, the natural units of the 2D PIC simula-
tion. In a real reactor, the density would be measured in
units of particles per unit volume, not per unit area. The
simulation was used to predict the expected 3D density
from the simulated 2D density by imposing the condition
that the plasma pressure equals the magnetic pressure at the
surface of the Wiffleball. The surface is marked by the
vertical dashed line connecting the electron surface in
Fig. 7d to the ion surface in Fig. 7e.
As an aside, it should be explained why the areal density
of ion-dots is obviously less than the areal density of
electron-dots in Fig. 7d and e. This might seem to con-
tradict the curves in Fig. 7b where the 2D densities are
shown to be equal at center. The scarcity of central dots
plotted in Fig. 7e is an artifact due to a feature of the
simulation adopted to speed execution. The maximum
legal time-step is a time just short enough so that the fastest
electrons cannot skip over a cell between time-steps. The
speed of execution would be faster for longer time steps,
but the simulation program crashes if the time step is
longer than the maximum just stated. Being heavier and
slower, ions stay in a single cell for a large number of time-
steps, as computed by the ‘‘subcycle’’ parameter in line
‘‘138’’ of the Fig. 2. By skipping the calculation of new
ion-positions for this number of time-steps, the program
runs faster. The number of dots plotted in Fig. 7e is less
than the number in 7d by a factor of ‘‘subcycle.’’ This is an
artifact of 2D plotting, not a sign of lower ion density,
which would violate plasma quasi-neutrality.
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Scaling the Simulation to Predict the Size
of a Full-Scale Power Reactor
In order to be practical for commercial power production,
the full-scale reactor size should ideally be smaller than
ITER, the main competitor for how to do fusion. ITER will
be about 30 times larger than the model reactor just sim-
ulated. For this reason the simulation was repeated with the
size of the magnets scaled up by a factor of 30 from the
size simulated above.
DD fuel might not be the best choice for a full-scale
reactor. The drawback to DD fuel is the presence of a
2.45 MeV neutron among the reaction products. These
neutrons would activate the copper of the magnets, which
fill much of the solid-angle around the core of the reactor.
The activation of copper would create a waste-disposal
problem. In addition, neutrons escaping through the mag-
net openings would pose a health hazard to operators,
necessitating heavy shielding outside the tank. To avoid the
neutron radiation, p ?
11
B fuel is a better choice than DD
for net power. In addition to being neutron-free (i.e.
aneutronic), p ?
11
B has a much higher fusion energy
yield. Omitting the neutron energy, the Q value for DD
reactions is only 2.9 meV compared to 8.7 meV for
p?
11
B. This larger fusion-energy yield leads to a smaller
reactor, other factors being equal.
The weight of shielding needed for reactors used in
spaceflight is a special concern. A proposal by Raymond
Sedwick has recently been funded by NASA in the US to
develop a Polywell-type reactor for a mission to Mars.
[27,28]. A poster of Sedwick’s proposal [29] was pre-
sented at a NASA-sponsored conference prior to its being
funded. The reactor will burn p ?
11
B fuel and use direct
conversion [25] of the reaction products to produce elec-
tricity. Sedwick’s poster [29] proposes, ‘‘Eventually, the
need to neutralize the ion space charge in the core of the
device led to the idea of the continuous grid, whereby
embedded permanent magnets generate a passive cusped
field, similar to that of Polywell. However, ions are allowed
to pass in and out of the core rather than having to be
trapped within the well’’.
The OOPIC simulation was used to estimate the power-
balance of an ITER-sized reactor burning p ?
11
B fuel.
The simulation was designed to test if the Polywell would
be a candidate for practical power production. If the sim-
ulated power-balance is greater than unity, the reactor is
predicted to produce more energy than it consumes. This is
the first and foremost requirement of practical design.
Power-balance is represented by the quotient Pout/Pin,
where Pout is the fusion power output and Pin is the drive
power input. Pout and Pin were separately estimated by
simulation. Fusion power Pout was simulated as a first step
in computing the power-balance.
Simulating the Fusion Power Output (Pout)
for a Net Power Reactor
The magnets chosen for the net-power reactor are crucial.
The simulation of the DD scale model showed the need to
vary the magnets’ fields as a function of time during start-
up. This is one of two ways of forming a Wiffleball
according to Bussard [30]. An alternate method in which
the B-field stays constant but the electron drive current
varies was also conjectured as a possibility. This other
method [30] might use permanent magnets, but it has not
been tested even in simulation. This simulation discovered
a requirement for varying B-field as a function of time
during start-up. For this reason, electromagnets are
believed to have an advantage over permanent magnets.
An electromagnet selected for further simulation, as
reported here, is the Bitter magnet invented by Francis
Bitter in 1939. These electromagnets also use water cool-
ing, like the coil magnets [23], but with the water pushed
through narrow gaps between flat copper plates. Due to
more efficient cooling, Bitter magnets produce fields at
least a factor of 4 higher [31] than coil magnets of the same
size.
Figure 8shows a partial listing of the source program
used for the full-scale p ?
11
B simulation. The magnet
current specified in line ‘‘15’’ contains the extra factor of 4
to raise the field of the catalog magnet to that of a Bitter
magnet. The current also contains as a factor the 2nd power
of the scale factor, ‘‘radiusScale,’’ assigned the value 30 in
line ‘‘8.’’ Raising the current by the square of the scale-
factor raises the magnetic field by the scale-factor. In
general, magnetic field rises in direct proportion to the
rising-size specified by ‘‘radiusScale.’’
Figure 9shows diagnostics from simulating the full-
scale reactor. Figure 9a is the now-familiar snapshot of
electron positions inside the square magnet enclosure in the
central plane. The outer dimensions of the tank have grown
to 25 m, approximately the same size as ITER. Figure 9b
shows the electrostatic potential plotted along the usual
horizontal path through the center of the reactor. This is the
path taken by newborn ions on their first trip through
center. Ions are trapped in this potential-well for many
bounces. As seen in the top label of the vertical axis, the
scale of the potential has been increased from 20 kV in the
previous simulation to 200 kV in this simulation. Two-
hundred kilovolts is the highest energy the simulation
could accurately handle. At higher voltages the electron
energies approached the rest-mass of the electron, which
meant the electrons’ velocities approached the speed of
light. The electrostatic approximation used in simulation is
Journal of Fusion Energy
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inherently non-relativistic; therefore, higher voltages were
avoided to maintain simulation accuracy.
Figure 9c shows the simulated positions of boron
macroparticles, each ion’s position plotted as a black dot.
In a real reactor, both protons and boron ions could be
produced in a gas cell containing a mixture of hydrogen
(H
2
) and diborane (B
2
H
6
) gases. In the simulation, the
boron ions were generated in the gas cell described in the
MCC code shown beginning in line ‘‘141’’ of Fig. 8. The
position and size of the gas cell was scaled up from the DD
simulation using the same ‘‘radiusScale’’ used to scale the
size of the magnets. The ion specie produced in the gas-cell
is associated with the specie-name ‘‘MCCborons,’’ in line
‘‘152.’’ The specie ‘‘MCCborons’’ was previously defined
to have mass 11 and charge 5. In order to make the
resulting plasma neutral, it was necessary to simultane-
ously create artificial macroparticles. These new
macroparticles were named ‘‘MCCelectrons’’ in the
‘‘Species’’ block starting in line ‘‘127.’’ This artificial
specie was defined with a negative charge of 5 electrons.
After being born together, MCCelectrons and MCCborons
were tracked separately along with electrons from the
emitter. As in the previous simulation, MCCelectrons were
born cold and extracted by diocotron-pumping.
The above described assignments simulate production of
plasma composed of fully stripped boron ions and associ-
ated electron macroparticles. Protons were not produced by
the simulated ionization due to a restriction of the MCC
operator to produce only one ion-specie. Protons were
assumed to be present as a fraction of the plasma, but not
explicitly tracked. In computing the fusion power output,
the protons were assumed to have a particle density five
times larger than that of the boron ions. This ratio of
densities was previously found favorable for minimizing
electron-boron bremsstrahlung losses [4]. In the real world,
the fraction would be adjusted by controlling the partial-
pressures of hydrogen and diborane gasses fed to the gas-
cell.
Pout was computed by integrating the textbook formula
[26] for power-density, n
1
n
2
\rv[Q, where n
1
n
2
is the
product of ion-densities, \rv[is the average ion-reac-
tivity, and Q is the well-known energy yield of one fusion
Fig. 8 A sample portion of the input-file used to simulate a full-scale p ?
11
B fueled Polywell reactor
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event. Using the plasma’s quasi-neutrality and assuming a
ratio of 5-to-1 for proton-to-boron densities, n
1
n
2
reduces
to (1/2)(1/10)n
2
, where n is the 3D electron density from
the b=1 condition at the surface of the Wiffleball. From
pg. 29 of the Plasma Formulary [19], the b=1 equation
rearranged is n =B
2
/[(4e-11)T], where B and T are the
surface B-field and electron energy in Gaussian cgs units.
The B-field at the surface of the Wiffleball was deter-
mined from the diagnostic shown in Fig. 9d. A convenient
surface point was chosen to evaluate the surface B-field.
The point is at the intersection of the dashed lines in
Fig. 9a. The selected point was chosen to simplify the
graphical determination of B, due to the observation that
the B-field vector at that point has only one non-vanishing
component, namely the component pointing vertically,
parallel to the y-axis. This By component was available as
a 2D diagnostic plot from the simulation. Figure 9disa
horizontal section of the By values made along the dashed
horizontal line in Fig. 9a. The value of By at the sample
point was read from the intersection of the vertical dashed
line with the graph of By in Fig. 9d. The component is
shown by the horizontal arrow to be By =-6.5T which
makes the magnitude of the 3-vector B =6.5e4 Gauss, in
the appropriate cgs units.
The uniform ion density in Fig. 9c suggests a useful
approximation to the volume integral needed to convert
fusion-power-density to fusion power. Due to the cubic
symmetry of the reactor, it was only necessary to integrate
the reactivity \rv[over a sphere of the same diameter
as the circle in Fig. 9c. The uniform ion-density inside the
circle contributes a constant factor to the volume integral
and can be brought outside the integral sign. With this
simplification, the volume integral reduces to (n
2
/20) Q
$dr(4pr
2
)\rv[, where r is a dummy integration vari-
able and the term (4pr
2
)dr is the volume of a spherical shell
of radius r and thickness dr. The limits of integration are
(a)
(b)
(d)
(e)
(c)
Fig. 9 Snapshots of a full-scale reactor’s steady-state diagnostics.
Panels are as follows: aElectron-particle positions, each particle
indicated by a black dot. A selected surface point is marked by the
intersection of the two dashed lines bElectrostatic potential along the
section indicated by the (red) arrow from a). cIon-particle positions.
The circle shows the integration limits of a spherical volume of
trapped ions. The vertical line marks an integrand-sampling-point at
half the radius of the circle. dVertical component By of magnetic
field along the horizontal, dashed line shown in (a). A horizontal
arrow marks the magnitude of the B-field, which was read from the
vertical axis. eElectrostatic potential along the same horizontal line
shown in (a). A horizontal arrow marks the numerical value of the
electrons’ kinetic energy at the selected surface point (Color
figure online)
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from r =0tor=R, where R is the radius of the circle.
The integral was approximated using a 3-point Simpson’s
Rule [32]. The integrand vanishes at the origin (r =0) and
at the circular boundary (r =R). This leaves only the
central sample of the integrand at r =R/2 contributing to
the integral. Incorporating the weighting factors of Simp-
son’s Rule [32], the integral reduces to (n
2
/30)(pR
3-
Q) \rv[
1
, where \rv[
1
is the average reactivity
evaluated at the r =R/2 position, marked by the dashed
line connecting Fig. 9b and c.
The brackets in the term \rv[
1
indicate the enclosed
expression is to be averaged over the randomly distributed
directions of the fusing ions’ velocities for all the ions
residing at the marked R/2 position. Bussard [33] showed
that angle-averaging reduces the value of the peak cross
sections only slightly. This reduction could be partially
compensated for by lowering the plasma energy, to take
advantage of the 1/T dependence of the b=1 electron
density n. Compensating factors of rising density and
falling cross section would make the above expression for
fusion power stationary and maximum at one particular
energy. For simplicity, it was assumed the rising and fall-
ing terms multiplied together have the same value at the
peak that they would have at this lower energy.
Thus, the \rv[
1
term was evaluated at the peak
p?
11
B cross section and n was evaluated by adjusting the
simulation to produce the ion-energy where the cross sec-
tion peaks. This evaluation was equivalent to the assump-
tion that the rising density at lower energies compensated
for the reduced cross section at the peak due to angle
averaging. For these reasons of convenience, \rv[
1
was
evaluated at the maximum, non-averaged cross section and
corresponding proton energy. The measured cross-section
data [25] show the maximum cross section, 1.2b (1.2e-
24 cm
2
), occurs for a fusing proton energy of 550 keV.
Using the standard expression for v as a function of energy
(E) and mass (M), the corresponding proton-velocity
v=(2E/M)

=c(2 9550e3/956e6)

=2e9 cm/s. In
this expression, the proton’s mass is expressed in units of
its rest-energy, Mc
2
= 956 million-electron-volts (MeV).
This leaves only the scaling of the surface electron
energy T needed to evaluate the surface electron density n.
Figure 9e is a section of the electrostatic potential function
along the same horizontal line through the selected surface
point. The full-scale density was determined by scaling-up
the ‘‘T =135 kV’’ value labeled in Fig. 9e by the ratio of
the full-scale-bias-voltage to the simulated-bias-voltage.
As seen in Fig. 9b, a proton at r =R/2 would have
acquired 75 keV kinetic energy falling from the edge of the
Wiffleball inward through the simulated voltage difference,
labeled DU. The proton energy at which the fusion cross
section is maximum, 550 keV, would be obtained by
raising the bias voltage enough to produce DU =550 keV
at the R/2 position. The shape of the potential function does
not change with applied voltage; thus, the applied voltage
would need to rise by the ratio of 550/75 which is a factor
of 7.3
The scaled electron energy would rise by the same
factor. From Fig. 9e the surface electron energy would rise
from 135 keV by a factor of 7.3, which makes
T=1 MeV. Substituting this scaled-up energy into the
b=1 expression yields the electron density n =B
2
/[(4e-
11)(T)] =(6.5e4)
2
/[(4e-11)(1e6)] =1.0e14 cm
-3
. The
radius of the circle containing the ions in Fig. 9c was
determined graphically to be R =698 cm. The final factor
needed to compute Pout is the fusion yield,
Q=8.7 MeV =1.4e-12 W s [4,19]. Evaluating the
entire expression for output power, Pout =(n
2
/30) (pR
3
)-
Q\rv[
1
. Substituting the values determined in the
previous paragraphs yields (1e14)
2
(1/30) (3.14) (698)
3
(1.4e-12)(1.2e-24) (2e9) watts; Pout =600 megawatts
(MW).
This simulated value for the power output,
Pout =600 MW, forms the numerator of the power-bal-
ance Pout/Pin. The denominator Pin was computed from
the simulation as shown in the next section. To make a
reactor of 25 m diameter practical would require that the
denominator be less than the numerator. Pin would need to
be less than 600 MW.
Simulating the Drive Power (Pin) and the Power
Balance Pout/Pin
The drive power Pin is the sum of the power consumed to
maintain the B-field plus the power consumed to replace
electrons lost from the plasma. The magnet power is known
to rise as the 3rd power of the scale-factor ‘‘radiusScale.’’ If
the electron power can be shown to rise with scale-factor
raised to a smaller exponent than 3, the magnet power will
dominate the electron power, which can then be ignored in
computing the power-balance. The scaling of the electron
power with reactor size was simulated first, in the hope that
it could be ignored.
Figure 10 shows simulations made for two different
sized reactors, on the left an intermediate-scale sized one
and on the right the full-scale reactor analyzed in the
previous subsection. Figure 10a and b show the positions
of electrons in the two different sized reactors. The labels
on the axes indicate the tank dimensions, 12.6 m on the left
and 25.2 m on the right. The B-fields, not shown, were
found to be proportional to size; the larger magnets pro-
duced twice the field of the smaller magnets. This is a
fundamental characteristic of coil magnets, to be expected
whenever magnets are scaled-up by the same factor in all
three dimensions. The two simulations were matched by
trial and error to have approximately the same depth-of-
Journal of Fusion Energy
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potential. Figure 10c and d show the horizontal section of
the potentials through the reactor centers. By matching the
potential-well-depths, ions were made to fall through the
same potential difference between their birthplace and the
center of the reactor; therefore, both sets of ions have the
same kinetic energy at center.
The currents of electrons pumped onto the aperture at
the gas cell were displayed and determined to be 4.2 and
3.9 A. The two electron-loss-currents were approximately
equal for both sizes of reactor. This was a surprise. Logi-
cally, it was expected that the larger reactor would require
a larger current to pump out its down-scattered electrons.
The larger volume of plasma has many more electrons to
down-scatter and these contribute to cusp-trapping. How-
ever, other factors, besides the number of electrons,
apparently come into play in determining the required
balance between cusp-trapping and diocotron-pumping.
These other factors are not fully understood, but fortu-
nately, the other factors have the beneficial effect of
reducing the power required to stabilize the electron
density.
In each case, the power consumed by the drive current is
the magnitude of the current times the voltage on the
aperture; the voltages were adjusted to be the same as
described in the previous paragraph. This also made the
power consumed by the two reactors the same. Figure 8
shows the source code for the larger of the two reactors.
For the smaller reactor, the source code was the same as in
Fig. 8, except the ‘‘radiusScale’’ assigned in line ‘‘8’’ was
reduced from 30 to 15, and the electron current assigned in
line ‘‘38’’ was reduced from 5 to 4.5 A.
The overall conclusion from the diagnostics of Fig. 10
was that the rise in electrons’ drive power with size occurs
at a much slower rate than the rise in magnet power. This
allowed the electrons’ power to be ignored in computing
the power-balance of the full-scale reactor. The electron
drive power was assumed to be swamped by the magnet
drive power which is known to rise as the 3rd power of
size. The power-balance was computed as the ratio of the
fusion power output to the magnet power.
According to the specification in the GMW catalog [23],
each small-scale magnet consumes 1.75 kW at full power.
This magnet power increases from the catalog value in
proportion to the volume of copper, which grew as the 3rd
power of the scale factor, or by a factor of 30
3
. In addition
to this factor, the small-scale magnets’ power was multi-
plied by two additional factors in estimating the power of
the larger reactor. The input file listing of Fig. 8shows the
catalog magnets’ current scaled by an extra factor of 4 to
account for the extra current capability of hypothetical
Bitter magnets. Like any resistor, electromagnets’ power
scales as the 2nd power of its current; thus, the extra factor
of 4 in current multiplies the catalog value of magnet
power by a factor of 4
2
, or 16. Finally, six identical mag-
nets are equally powered to cover the six faces of the cubic
reactor, therefore introducing an additional factor of 6 into
the scaled-up power. Including all these factors, the esti-
mated power consumed by the magnets of the full-scale
reactor is Pout =1.75 kW 930
3
94
2
96=4.5 giga-
watts (GW).
The power-balance is, by definition, the ratio of output
to input power, 600 MW, divided by 4.5 GW. In other
words, Pout/Pin =0.13. This power-balance of 0.13 is
(a) (b)
(c) (d)
Fig. 10 Snapshots of
diagnostics simulating two
different-sized reactors burning
11
B fuel. The sizes differ by a
factor-of-two, a half-sized
reactor on the left and full-scale
reactor on the right. Individual
panels are as follows:
aElectron-particle positions in
a 12.6 m reactor. bElectron-
particle positions in a 25.2 m
reactor. cElectrostatic potential
along the horizontal center line
of the 12.6 m reactor.
dElectrostatic potential along
the horizontal centerline of the
25.2 m reactor
Journal of Fusion Energy
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much less than the desired value of unity. However, it is
not too small to be useful. The small power-balance can be
increased to unity by only a modest increase in the reactor
size. The new reactor design has many advantages over the
tokamak ITER, to which it will naturally be compared.
Implications of the Power-Balance Simulated
for Net Power Reactors
The size of a reactor with power-balance of unity will be
somewhat larger in size than the 25 m reactor simulated.
Pout scales as the 7th power of size. Pin scales as the 3rd
power of size. The quotient, power-balance, therefore
scales as the 4th power of size. To increase the power-
balance by a factor of 1/0.13 (= 8) would require an
increase in size by only a factor of 8
1/4
which equals 1.7. A
reactor 1.7 times larger than the simulated 25 m reactor
would break even. This would indeed be a large-sized
reactor, but not out of the question for powering cities or
spaceships. If the super-conducting magnets planned for
ITER are successful, they could be adapted to this Polywell
design and would thus vastly shrink the reactor size.
Incorporating super-conducting magnets into Polywell
would greatly improve the commercial viability of the
design.
Summary and Conclusions
PIC-simulation was used to investigate why the latest
scale-model Polywell reactor tested at EMC2 Inc. failed to
produce fusion energy [6]. It was discovered that suc-
cessful start-up of Polywell reactors requires synchronized
variation of the B-field, electron-drive current, and fuel
supply on a millisecond time scale. Such controlled vari-
ation would be difficult or impossible to provide with
pulsed power, the only type of power available at EMC2.
Controllable DC power supplies are required to power the
electromagnets and to drive the electron beam into the
reactor.
The simulation can complete start-up in milliseconds of
simulated time, but real-world magnets must be turned on
more gradually. With DC power supplies gradually adjus-
ted by operator or computer, the plasma density can be
controlled to rise at an arbitrarily slow rate. The diameter
of a B-field-free central region (i.e. the Wiffleball) grows
gradually with the density of the plasma. At the end of
start-up the diameter of the field-free region has expanded
to almost fill the whole volume inside the magnets. At the
same time, the magnetic field rises to the maximum field
capability of the electromagnets, as governed by their
power consumption and cooling. A scenario of successful
start-up was simulated for a small-scale-model reactor
burning D ?D fuel and also for a full-scale reactor
burning p ?
11
B fuel.
In addition to the need for DC power, an entirely new
parameter was discovered which must be controlled to
maintain plasma stability. Unless extracted, cold electrons
become trapped in the reversed-sign potential-well they see
midway through the magnet bores. Cold electrons arise
continuously as a necessary byproduct of fuel gas ioniza-
tion and also arise from down-scattering of primary beam-
electrons. If not removed, they become more and more
concentrated with time during start-up and eventually kill
the potential-well needed to accelerate ions. The key to
removing the cold electrons is diocotron pumping.
Diocotron pumping required careful adjustment of the
depth of potential at the center of the reactor and a corre-
sponding adjustment of the diameter of an aperture at the
outside edge of the fuel-gas-cell. To successfully pump out
the cold electrons from the gas-cell, both the potential and
aperture needed to be adjusted to make a controlled portion
of the primary beam hit the aperture.
The simulation was extended to predict the size of a net-
power reactor when the small-scale design is enlarged
enough to have a power-balance near unity. The magnets
of the small scale model were enlarged by a factor of 30 in
all three dimensions. The factor of 30 was enough to make
the large reactor come close to break-even, even with the
advanced fuel p ?
11
B.
The simulation did not reveal any barriers to developing
a practical Polywell reactor for terrestrial and space
applications. The logical path to development is to build
and test small scale models burning D ?D, followed by
gradually increasing the size to validate the real-world
scaling of the power-balance. Once the actual scaling
relationship has been verified experimentally, a validated
PIC simulation can be used to predict the size of a net-
power Polywell reactor burning p ?
11
B fuel.
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Journal of Fusion Energy
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