LLNL-PRES-790757
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
Simulation-Guided Design of a MegaJoule Dense Plasma Focus
A. Schmidt, E. Anaya, M. Anderson, J. Angus, S. Chapman, C. Cooper, C. Goyon, D. Higginson, I. Holod, E. Koh, A. Link,
D. Max, Y. Podpaly, A. PovilusSeptember 18th, 2019
LLNL-PRES-790757
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Outline
▪ Introduction to dense plasma focus (DPF)
▪ Neutron generation physics
▪ PIC modeling & benchmarks to measurements
▪ Simulation movies and restrikes
▪ Two ways current is diverted from pinch location:
▪ Restrikes in plasma
▪ Arcing behind gun
▪ Trends in the simulations→ reduced order model
▪ Using the model to improve experiments
▪ Next questions to answer
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The “Mather” DPF: an open ended coaxial gun
DPFs make
• energetic (keV to MeV) beams
• x-rays
• neutrons (for D or DT gas)
LLNL DPF
Cathode
AnodeInsulator
Dense Plasma Focus: A coaxial plasma railgun
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Yield
(DD)
Anode
length
(cm)
1e5-1e6 1e7 1e11-1e12
5
10
30
Accelerator-based
AmBe replacement
Portable active
interrogation
90 kA
200 kA
2-3+ MA
DPFs can be sized for relevant yield
Potential: survivability,
neutron imaging
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MegaJOuLe Neutron Imaging Radiography (MJOLNIR) design & build team
Now: 1 MJ/2.7 MA
Upgrade: 2 MJ/4+ MA
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Stages of a DPF
discharge
1) insulator
flashover
2) run-down
3) run-in
4) pinch
Anode
Anode
Cathode
Insulator
Fill gas
Stages of a DPF discharge
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Anode
Anode
Insulator
Stages of a DPF
discharge
1) insulator
flashover
2) run-down
3) run-in
4) pinch
Stages of a DPF discharge
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Anode
Anode
Insulator
Stages of a DPF
discharge
1) insulator
flashover
2) run-down
3) run-in
4) pinch
Stages of a DPF discharge
Magnetic field fills volume
where gas has been swept up
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Anode
Anode
Insulator
Stages of a DPF
discharge
1) insulator
flashover
2) run-down
3) run-in
4) pinch
Stages of a DPF discharge
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Physics of neutron generation during z-pinch phase (according to 2D PIC simulations)
Bq
I
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EzBq
Beam
𝑌𝑖𝑒𝑙𝑑 = න𝑁𝑏𝑒𝑎𝑚𝑛𝑡𝑎𝑟𝑔𝑒𝑡𝐿𝑡𝑎𝑟𝑔𝑒𝑡 f E σ 𝐸 𝑑𝐸
Target
I
N = number
n = number density
Physics of neutron generation during z-pinch phase (according to 2D PIC simulations)
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Kinetic (particle) code captures anomalous resistivity and beam formation in plasmas
A. Schmidt, V. Tang, D. Welch, PRL 2012
Kinetic 8.6 × 106
Hybrid 3.6 × 104
Fluid 0
Kinetic model needed to get correct neutron yields in dense plasma focus (DPF)
+
-
-
Fluid picture: each “pixel” is a fluid element with a density,
temperature, and velocity
-
- ++
++
++ +
+
+
+
+
-
---
-
-
--
- --
-
--
-
-+
+++
+ +
Kinetic picture: each “pixel” is a collection of particles; density, internal energy, and velocity are derived from
collection
Agrees with experiment
▪ Each “pixel” in the pinch region is really
1,000-10,000 particles
▪ 100-500 million particles per simulation
▪ We resolve electron cyclotron motion
(~femtosecond time-steps)
▪ ~1 million time steps
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Details of LLNL PIC modeling
Fluid to Kinetic Transition
• Maps Local plasma distribution
function to a drifting Maxwellian of
kinetic particles
• Preserve during transition
1. Currents and Fields
2. Plasma Conditions
Simulations are performed in two coupled stages:
1. Single Fluid with MHD which establish the initial conditions and circuit dynamics
2. Kinetic PIC with Full Maxwell’s equations starting 10-20 ns prior to the pinch phase
101
102
103
104
1010
1011
1012
1013
1014
Particle Energy [eV]
dN
/dE
(Io
ns
/eV
)
Distribution Function in the Sheath
Fluid Before Transition
Kinetic After Transition
Ti ≈ 800 eV
Ni ≈ 5x Ambient
Resolution
Dx,Dz = 100-400 mm
Dt = 0.25-250 fs
Physics Models
• Collisions
• D-D/T Fusion packages
Circuit Driver
Full Maxwell Equations
Implicit Particle advance
• Direct Implicit Scheme
• Full Matrix Inversion for Field Advance
• Currently resolving wc across most of the
simulation
Note: A typical 3
MA simulation
takes 60-120
days of wall time
A few million
cpuHrs
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How a Dense Plasma Focus produces Neutrons
Beam Target dominates the total yield over thermonuclear
Dense D+
0 5 10 150
0.5
1
1.5
2
2.5x 10
9 Neutron Distribution
E (MeV)
dN
/dE
/dW
(N
eu
tro
ns / M
eV
/ S
R)
0o to 20
o
35o to 55
o
80o to 100
o
Beam
Bu
lk P
lasm
a
Strong Electric FieldsD+ Density
1 cm1 cm
1 cm
1 cm
E (keV)
f(E
)
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Model agrees within a factor of 2 with experimental yields
1 1.5 2 2.5 30
1
2
3
4
5
6
Peak Current [MA]
Yie
ld (
10
11 N
eu
tro
ns)
a = 9e+09 Neutrons/MA4
a = 4.5e+09 Neutrons/MA4
Data
Simulations
Data I4
Sim I4
We have typically observed
overall fairly good agreement
but typical under predict the
yield by roughly a factor of 2x
for higher current MJ DPF
shots
Similar scaling to Experiment but at half the coefficient
0 1 2 3 4 5 6 70
0.5
1
1.5
2
2.5
T (ms)
I (M
A)
Data 30 kV
Data 30 kV
Data 35 kV
Data 40 kV
Data 40 kV
Data 40 kV
Data taken from Gemini experiment (Nevada)
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7.2 7.4 7.6 7.8 8 8.2
x 10-6
0
5
x 105
T (ns)
dN
/dT
(N
eu
tro
ns /
ns)
30 m neutron time-of-flight
7.2 7.4 7.6 7.8 8 8.2
x 10-6
0
5
Energy spread
agreement
6.6 6.8 7 7.2 7.4 7.67.6
x 10-6
0
2
x 106
T (ns)
dN
/dT
(N
eu
tro
ns /
ns)
5 m neutron time-of-flight
6.6 6.8 7 7.2 7.4 7.67.6
x 10-6
0
5
Particle Model
5/16/2013 Shot 13 data
Pulse shape agreement
Down-scattered
neutrons (not in model)
0 5 10 150
1
2
3
4
5
6
7
8x 10
8 Neutron Distribution R30 Dedx
E (MeV)
dN
/dE
/dW
(N
eu
tro
ns / M
eV
/ S
R)
0o to 20
o
35o to 55
o
80o to 100
o
150o to 170
o
20 meter
10 metergammas
We get time
offset?
Near/far nToF agreement indicates that simulated pulse shape/neutron energies are reasonable
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Example of a 200 kA fluid-to-kinetic simulation including restrikes
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Particles accelerated across the gap to >1 MeV
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DPFs show variable measured yields and kinetic simulations reproduce variability/stochasticity
Bures, Physics of Plasmas, 19, 112702 (2012)
Bank capacitance [uF]0.8 1.6 3.2
8 x107
2
6
4
0
Vary driver impedance
Challenge: Find ways to increase average yield and simultaneously improve
shot-to-shot consistency
Yiel
d4e17(1mm) 4e17(2mm) 8e17(1mm) 8e17(2mm)
0
1
2x 10
7
Gas jet density (gas jet radius)
Yie
ld
Predicted yields for DPF with modulated gas jet
Individual Simulation
Average
Individual
Simulation
Average
Data Simulations
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High-yield pinches always exhibit strong m=0 instability in simulations
zoom region
4mm x 4mm
Hig
h-y
ield
pin
ch
ion densitylog(cm-3)
rBθAmps
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Electric field sets up in the gap between two separated blobs of plasma
zoom region
4mm x 4mm
Hig
h-y
ield
pin
ch
electric field
ion density
log(kVcm-1)
log(cm-3)
rBθAmps
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Beam forms on axis when electric field sets up there
zoom region
4mm x 4mm
Hig
h-y
ield
pin
chbeam creation
electric field
ion density
log(kVcm-1)
log(cm-3)
[cm.s-2]
rBθAmps
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In a low yield simulation, a full m=0 never sets up, possibly due to restrike currents
zoom region
4mm x 4mm
Hig
h-y
ield
pin
ch
Lo
w-y
ield
pin
ch
beam creation
electric field
ion density
log(kVcm-1)
log(cm-3)
[cm.s-2]
rBθAmps
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Similar to restrikes, current arcing behind the gun diverts current from pinch
Rb
Lb
Cb
RDPF
LDPF
Rs
Cb 400 nF
Lb 50.3 nH
Rb 80 mW
Vs 80 kV
Rs 1 mW – 1 kW
LDPF 4.5 nH
We can add a
parasitic current
path in the
simulation behind
the gun to simulate
arcing behind the
gunLLNL mini DPF
▪ Mystery: why wasn’t LLNL mini
DPF producing expected yields?
▪ Hypothesis: current is flowing
behind the gun
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Arcing behind the gun can introduce significant reduction in neutron yield
Case Trigger
Voltage [kV]
Yield (106
Neutrons)
restrike 80 .02
no restrike 8000 0.9
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The earlier the arcing, the worse the yield
0.00E+00
1.00E+04
2.00E+04
3.00E+04
4.00E+04
5.00E+04
6.00E+04
7.00E+04
8.00E+04
9.00E+04
1.00E+05
90 100 110 120 130 140 150 160 170 180 190 200
n y
ield
Rs turn on time (ns)
n yield
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Simulations indicate that anode-to-sheath restriking is common
Trailing mass left behind by hydrodynamic instabilities create lower-inductance
current pathways that divert current from the pinch region
Density rBθ
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Hollow anode mitigates anode-to-sheath restrikes
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Lower yield shots show evidence of parasitic current diversion from pinch region
Time (µs)
Cu
rre
nt (M
A)
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Lower yield shots show evidence of parasitic current diversion from pinch region
Time (µs)
Cu
rre
nt (M
A)
Vary resistance of a 5 nH
current path (represents
restrike near the insulator)
to match low yield shot
current trace in snow-plow
model
Ideal: measure location of restrike
with b-dot or other probe
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Insights from PIC modeling to be applied to lower order model
▪ Beam temperature for a given DPF current doesn’t change much
▪ For a given stored energy class of DPFs, conversion efficiency of gun energy
to beam energy is somewhat constant
→ Main influence on yield is through aerial density and temperature of target
▪ Large radius anode/long implosion time leads to hydrodynamic instabilities that
appear early in run-in
▪ We can mitigate these hydro instabilities with a tapered anode, where the
taper stops at a particular radius. Mass is swept up starting from the
radius where the taper stops (the “implosion radius”).
▪ Plasma target needs to be hot to minimize stopping power (increases aerial
density average cross-section)
▪ Too much mass in the implosion (high gas fill or large implosion radius)
can cause target to be cold
▪ A hollow anode can help mitigate anode-to-sheath restrikes
From analytic shock physics: maximum achievable convergence ratio appears to
be about 10 in a shock-driven cylindrical implosion
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EzBq
Beam
𝑌𝑖𝑒𝑙𝑑 = න𝑁𝑏𝑒𝑎𝑚𝑛𝑡𝑎𝑟𝑔𝑒𝑡𝐿𝑡𝑎𝑟𝑔𝑒𝑡 f E σ 𝐸 𝑑𝐸
Simple Model:
• Assume the hydrodynamic disassembly time of the “target” >> duration of
ion beam and acceleration time of ion beam
• Assume beam can’t miss the target, i.e. partially magnetized
• Getting to sufficient areal density will probably mostly guarantee this
• Larmor Radius for 1 MeV D+ near the pinch is about 0.5-1 mm
• Useful pinch length is ≈ 2 cm long
• Beam spectrum is decaying exponential
Target
I
N = number
n = number density
𝑓 𝐸 = 𝑒 ൗ−𝐸
𝐸𝑏
With a few assumptions, we can make a reduced order model to explore wide parameter space
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Anode design evolution influenced by both kinetic and reduced order models
Reduce anode-to-
sheath restrikes
Hollow anode
Delay
hydrodynamic
instabilities
Tapered
anode
Hotter
plasma
target
Smaller
radius
hollow
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Example of how modeling insights helped improve anode design
▪ First anode fielded on
MJOLNIR was
unsuccessful at high
current
▪ Modeling gave us
insight that plasma
target was not getting
hot enough
▪ We reduced the hollow
radius and recovered
performance at high
currents
0.0E+00
5.0E+10
1.0E+11
1.5E+11
2.0E+11
2.5E+11
3.0E+11
25 30 35 40 45 50
Yie
ld
Charge voltage +/- [kV]
3" hollow anode
0.7" hollow
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Example of how modeling insights helped improve anode design
▪ First anode fielded on
MJOLNIR was
unsuccessful at high
current
▪ Modeling gave us
insight that plasma
target was not getting
hot enough
▪ We reduced the hollow
radius and recovered
performance at high
currents
0.00E+00
5.00E+10
1.00E+11
1.50E+11
2.00E+11
2.50E+11
3.00E+11
0 5 10 15 20 25 30
Yie
ld
Pressure (torr)
Yields from first 2 anodes on MJOLNIR
3" hollow lowcurrent
3" hollow highcurrent
0.7" hollow lowcurrent
0.7" hollowhigh current
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Key future questions
▪ In MJOLNIR, where do restrikes occur?
▪ At base of DPF/insulator
▪ In A-K gap
▪ From anode-to-sheath
▪ How do we avoid restrikes?
▪ Operating pressure
▪ Could be problematic since high pressure is needed
for high yields
▪ Increase A-K gap
▪ Increases head inductance, lowers peak current –
what is effect on pinch current?
▪ Anode shape
▪ What limits performance at higher pressures?