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

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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?