Theory of Ignition and Hydro-Equivalencefor Inertial Confinement Fusion

R. BettiFusion Science Center andLaboratory for Laser EnergeticsUniversity of Rochester

24th IAEA Fusion Energy Conference

San Diego, CA 8–13 October 2012

BurnCompression Hot-spot ignition

Hot spot

~4 mm

FSC

Inertial confinement fusion (ICF) ignition theory is used to assess National Iginition Facility (NIF) experiments and design ignition-scalable implosions on OMEGA

TC10290

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Summary

• ICFignitiontheoryisusedtoderiveperformanceparametersthatcanbe measured in experiments

• ThetheoreticalresultscanbeeasilyrelatedtotheLawsoncriterion

• ApplicationstoNIFindirectdriveimplosionsshowPx up to 18 atm s, and pressures up to ~125 Gbar (~350 Gbar is required for ignition)

• Hydrodynamicscalingisusedtodesigndirect-drivesimilarityexperimentsbetweenOMEGAandtheNIF

• OMEGAimplosionsthatscaletoignitionatNIFenergiesrequire an areal density of ~0.3 g/cm2 and a neutron yield of ~4 × 1013

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ICF IGNITION THEORYFROM THE LAWSON CRITERION

Like in magnetic confinement fusion (MCF),theLawsoncriteriondeterminestheICFignitionconditionwhereignition occurs in the hot spot

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Alpha heating exceeds energy losses

" n P4 2

322vo f xa

Ignition parameter "/T

PP T

P24 ig

2/|vo fx

xx=

a ^ h

Ignition condition " | > 1

Hot spot

Rhsr

Ti

t

P ~ tT ~ constant

Sh

ell

Dense shell

Hotspot

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ICF implosions cannot achieve ~10-keV temperatures through compression alone

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• HighT requires high implosion velocity Vi

• HighVi requires thin shells

• Thinshellsbreakupinflightbecauseofhydrodynamic instabilities

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7.5t (g/cc)

3.5

0.5

–5000

200

400

600

800

1000

–250 0

0

5

10

15

20

Implodingshell T ~ Vi

25

0

Px

(at

m s

)

5 10 15T (keV)

20 25 30

z (nm)

r (n

m)

250 500

Ignition| > 1

NIF

ITERICF must ignite at ~5 keV, requiring V ~ 400 km/s and Px > 25 atm s.

Unlike in MCF, heat conduction losses do not reduce thethermal energy (pressure) in the hot spot of ICF capsules

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Hot spot

Rhs r

Ti

t

P ~ tT ~ constant

Sh

ell

Shell

Heat flux

Enthalpyflux

Massablation

q T T

T T02

heat - d

.

l

l l

=5

]]

gg

Mass ablation recycles the heat losses back into the hot spot.

.m Rm T

0 2/

hsa

i 0 05 2l

=o

q PV25

heat a=

Balanceheatfluxwithablationenthalpy*flux

Mass ablation rate from shell into the hot spot

*R. Betti et al. Phys. Plasmas 8, 5257 (2001).

Thehotspotisconfinedbyadenseshellwiththeconfinement time depending on the shell inertia

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• Thetemperaturedependsmainlyonimplosionvelocity

• TheconfinementtimecomesfromapplyingNewton’slawtotheshell

• The1-Dignitionparameter*P T

P

ig

/|xx^ h isrewritten

in terms of measurable quantities

~TV

R/

..i

02 7

1 20 2

lt^ h

~M R M R P R4sh sh hs22

xr=p ~ P R

M4 hs

shxr

.P TP R

T4 7/

..

igg cm

keV0 81 6

2/ .|xx t^ ch m

*C. D. Zhou and R. Betti, Phys. Plasmas 15, 102707 (2008).

One-dimensional simulations of gain = 1 targets confirm that the 1-D ignition condition depends on ion temperature and total areal density

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• Comparison*ofignitionconditionwithsimulationofgain= 1 targets

• Ignition-relevantparametersinferredfrom measurable quantities

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4

Best simplefit for 3.5 < T < 7 keV

tR = 20/T2

0.5

0.0

1.0

1.5

2.0

5 6 7

Ignition

GtR

Hno

a (g

/cm

2 )n

GTHno a (keV)n

| > 1

| < 1

Ignition parameter

Lawson’sPx

D- .P TP R

T4 7/

..

igg cm

keV0 81 6

2/ .|xx t

1 ^ ch m

atmP R T s8 /.

g cm keV0 8

2 :.x t_ i

All hydrodynamic quantities are calculatedwithoutburn(no alphas).

*C. D. Zhou and R. Betti, Phys. Plasmas 15, 102707 (2008).

R3-D

Hot spotCold

Shell

R1-DGvoH ≈ 0

0.00.0

0.2

0.4

0.6

0.8

No

rmal

ized

gai

n G

/G1-

D

Minimizedspread

1.0

0.5 1.0|3-D

1.5 2.0

Three-dimensional effects1,2 are included through the yield over clean (YOC) and its relation to the hot-spot clean volume

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YOCY

YV

V

n1Dn3 D

1D

3 D/ .

-

-

-

-D- .

YOCRT4 7/

..

3fit

g cmno keV

no

no0.40 8

1 6

2| t= aa

aa ek o

3-D ignition condition

1 P. Y. Chang et al., Phys. Rev. Lett. 104, 135002 (2010). 2 R. Betti et al., Phys. Plasmas 17, 058102 (2010).

• Fusionreactionsoccurin the clean volume (red)

The LLNL ignition threshold factor (ITFx) from fitting of LASNEXresultsisthecubicpower oftheLawsonparameter

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• RewritetheLawsoncriterionbyusingthe1-DyieldYN in the YOC1

• Theignitioncriterioncanbewrittenintermsofenergy

• CompareLLNLperformanceparameter2 ITFx withcubicpowerof| for MDT = 0.17 mg

.R

MY0 24

/.

.

3 D g cmno

fuelmg

N0 616 0 34

2.| t-a_ fi p

D- / . .ITFxY R0 32 1 5

.

NIFN16 2 3

1 tc m . .Y R0 35 1 5

.

3 DN316 1 8

/|t

- c m

1 R. Betti et al., Phys. Plasmas 17, 058102 (2010). 2 B. K. Spears et al., Phys. Plasmas 19, 056316 (2012).

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~ ~ ~ ~P P PRM

P PRR R E /sh 3

1 3x ~ITFE

EPP

ig igmin

33.

xx |=^ h> H

TheLawsonparameterisusedtoestimate P and Pxfor NIF indirect drive and OMEGA direct drive

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• Thehot-spotpressureisinferredfrom Px and the burn duration xburn

D-3 .GbarPT

27 4 7 .

burnns

ikeV

0 8.

x|^ eh o

NIF shot N120321 N120131 N110914 N111215

Px atm/s 18 12 15 15

| / Px/|Px|ig 0.48 0.37 0.45 0.44

P Gbar* 124 76 111 94

|3/ITFx 0.11/0.12 0.05/0.046 0.09/0.09 0.09/0.09

OMEGA shot 67290 67289 67288 67291

Px atm/s 3.6 3.5 3.1 3.3

| / Px/|Px|ig 0.086 0.082 0.080 0.083

P Gbar 34 33 29 31

*Similar values are inferred by C. Cerjan (LLNL), private communication.

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SCALING FROM OMEGA TO DIRECT-DRIVE NIF(assume symmetric drive and similar laser smoothing)

OMEGA results can be scaled to NIF energies

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1.3 to1.8 MJ

Direct-driveNIF 1.8 MJ

Indirect-drive NIF

3.6 mm

2.2 mm

Hydrodynamic scaling

0.86 mm

OMEGA 26 kJ

Scale 1:70in energy

Scale 1:1

Scale 1:1

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• TheshellimplodeswithViandexpandswithCs

• TheMachnumberCVi

s is the only dimensionless parameter

• UseisentropicimplosionconditionPa ~ at5/3

• 1-DsimilarityrequiresequalMachnumbers

~MachIP

V/ /

a L

i23 5 2 5

2

a ^ h

One-dimensional implosion similarity requiresequal Mach numbers

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D

RVi

Multidimensional implosion similarity imposesadditional requirements on entropy and velocity

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• Themultidimensionalbehaviorisdeterminedprimarily by the Rayleigh–Taylor (RT) instability

• Numberofe-foldingsofRTgrowth*forwavenumbers ,k R.

– –d d d, ,N t kg kV tRR

R VV

3 3eRT

RT

t

a

t

i

a

0 0 0

1i i

c x= = = t

tp

t_ fi p# # #

• SimilarimplosionshavethesamedimensionlessR: R RR

0=t

• 3-Dsimilarityrequiressame" ~I

IVV

Vm

P

mV/

/

i

a

i

a

a L

a L

i3 5

3 5

ta=

o o

^^hh

FSC

*H. Takabe et al., Phys. Plasmas 28, 3676 (1985).

Hydrodynamic similarity leads to geometricand energy scaling of implosion performance

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Hydrodynamic equivalence:

Fixed Vi, a, IL, " EL ~ R3, PL ~ R2, M ~ R3, D ~ R, t ~ R

• 1-Dhydrodynamicsimilarity" ~MachIP

V/ /

a L

i23 5 2 5

2

a ^ h

• 3-Dhydrodynamicsimilarity" ~I

IVV

P

mV

/

/i

a

a L

a L

i

3 5

3 5ao

^^hh

• Laser-energyrequirement

• ThreeconstraintsforVi, a, IL " hydrodynamic equivalence requires same Vi, a, IL

I I IP R E R VR

P V34 4 3a L L

ia L i L0

302 0

$. .r h h r h= ^ h

1

2

3

OMEGA 26 kJ Direct-driveNIF 1.8 MJ

3.6 mm

0.86 mm

00

5

10

15

20

1Time (ns)

Po

wer

(T

W)

2 00

100

200

300

4Time (ns)

Po

wer

(T

W)

8

Targets and laser pulses are designed for OMEGAto reproduce direct-drive NIF hydrodynamics

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Same Vi , a, IL and EL ~ R3, PL ~ R2, M ~ R3, D ~ R, t ~ R

~ ~R E /L1 3D

~ EP /L L

32

~Time E /L

31

Hydrodynamic scaling

Ignition theory and hydro-similarity provide the energyscaling of critical parameters

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• Energyscaling*ofarealdensity,iontemperature,andfuelmass for hydro-equivalent implosions

• Energyscalingforneutronyieldwithoutburn(no alphas)

• EnergyandYOCscalingforignitionparameterandITFx

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tR ~ E0.33 T ~ E0.07 Mfuel ~ E

~Y E .nno 1 5a

|3-D ~ E0.37 YOC0.4 ~ITF YOCx E .1 3

5.EE . .

kJNIF

MJ

26

1 8 0 37

X

f p .EE

220. .

kJNIF

MJ

26

1 8 1 3

X

f p

*C. D. Zhou and R. Betti, Phys. Plasmas 14, 072703 (2007).

Hydro-equivalent ignition on OMEGArequires | / Px/(Px)ig á 0.16

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• TheenergyandYOCscalingis|3-D ~ E0.37 YOC0.4

• ExpectYOCimprovementof2× on NIF versus OMEGA because of larger hot-spot size, more beams, and equal ice roughness (see back-up slides for details)

YOCNIF ~ 2 × YOCX

• ApplyOMEGA-to-NIFscaling

kJkJ

261800 2 1.

..

DDMJ kJ1 8 26

0 370 4. .| |X b l

FSC

.0 16eq ignition .| -X

Hydro-equivalent implosions are designed usingcurrent OMEGA targets (butwithbetterperformance)

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CHDT ice

DT gas

OMEGA

378 nm

41 nm 10.5 nm

430 nm

CHDT ice

DT gas

NIF

1542 nm

185 nm 39 nm

1766 nm

EL kJ 26

Vi km/s 350

amin 1.5

IFAR2/3* 22

GtRHn g/cm2 0.3

YN (1-D) 1.8 × 1014

EL kJ 1800

Vi km/s 350

amin 1.5

IFAR2/3* 22

GtRHn g/cm2 1.3

YN (1-D)/gain 3.4 × 1019/52

*In-flightaspectratio

The time evolution of implosion velocityand IFAR are the same for the NIF and OMEGA

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0.0

–300

–200

–100

0

0.4

Normalized time (tV/R)

Imp

losi

on

vel

oci

ty (

km/s

)

0.8

NIF 1.8 MJOMEGA 26 kJ

NIF 1.8 MJOMEGA 26 kJ

0.6

20

10

0

30

40

50

60

IFAR2/3 = 22

0.8

Normalized time (tV/R)

Same 1-D hydro Same RT growth (3-D)

IFA

R1.0

0.00.0

0.5

1.0

1.5

0.8 1.0

Normalized time (tV/R)

tR ~ E1/3

Are

al d

ensi

ty (

g/c

m2 )

1.2 1.0

2

1

0

3

4

1.1

Normalized time (tV/R)

Comparison of scaled tR

Neu

tro

n r

ate

(102

6 /s)

1.2

NIF 1.8 MJOMEGA 26 kJ × RE

0.33NIF 1.8 MJOMEGA 26 kJ × RE

1.5

Comparison of scaled Yn

Yn ~ E3/2

The 1-D areal density and no-burn neutron ratescale as predicted

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R EENIF

E /X

• | á 0.16 is required for hydro-equivalent ignition at 26 kJ

• UsetR = 0.3 g/cm2, MDT = 0.021 mg in .

RM

Y0 24/

..

g cmno

fuelmg

N0 6116 0 34

2.| t a_ fi p• Requiredneutronyieldat26kJis3.7× 1013

• 1.8-MJtargetsignite(gain = 1) with44%of1-Dno-burnneutronyield (use YOC model*)

• Requiredyieldat26kJis~22% of the 1-D yield (Y1-D = 1.8 × 1014); i.e., required yield is 4 × 1013

Sim

ula

tio

ns

The requirements for hydro-equivalent ignitionat 26 kJ are confirmed by simulations

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Hydro-equivalent ignition parameters for OMEGA: tR á 0.3 g/cm2 and YN á 4 × 1013

* P. Y. Chang et al., Phys. Rev. Lett. 104, 135002 (2010).

Th

eory

CHDT ice

DT gas

26 kJ

378 nm

41 nm 10.5 nm

CHDT ice

DT gas

1.8 MJ

1542 nm

185 nm 39 nm

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Summary/Conclusions

Inertial confinement fusion (ICF) ignition theory is used to assess National Iginition Facility (NIF) experiments and design ignition-scalable implosions on OMEGA

• ICFignitiontheoryisusedtoderiveperformanceparametersthatcanbe measured in experiments

• ThetheoreticalresultscanbeeasilyrelatedtotheLawsoncriterion

• ApplicationstoNIFindirectdriveimplosionsshowPx up to 18 atm s, and pressures up to ~125 Gbar (~350 Gbar is required for ignition)

• Hydrodynamicscalingisusedtodesigndirect-drivesimilarityexperimentsbetweenOMEGAandtheNIF

• OMEGAimplosionsthatscaletoignitionatNIFenergiesrequire an areal density of ~0.3 g/cm2 and a neutron yield of ~4 × 1013 (a 1.6× improvement in areal density and 3× improvementinyieldwithrespect to best implosion to date)

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Hydrodynamicscalingindicateslower3-Ddegradationin performance on NIF versus OMEGA

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• Yield-over-clean(YOC): the required YOC on OMEGA is difficult to estimate. Use simple clean volume analysis:

D D=- - –R3 RT1R DR ~R GRT RT0vD G GRTNIF

RT. X

RT spike amplitude Growthfactor Hydro-equivalency

– –YOC 1 1 YOCE

E/

NIF/

NIF

NIF

L

L

0

0 1 3

31 3

.v

vX

X

Xf ap kR

T

SSSS

V

X

WWWW

Initial seed

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AYOCof30%onOMEGAextrapolates(approximately) toa60%YOConNIF

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• SeedsfortheRTcomefromtheiceroughnessandthe

laser nonuniformities: ice laser02 2.v v v+

• BetalayeringmakesNIFtargetsassmoothasOMEGA’s: ice iceNIF.v vX

• Lasernonuniformitiesgrowwithsize E /L1 3_ i and are reduced by a larger

number of overlapping beams (Nb) ~ E N/ /laser b

1 2ZL1 3v

E

E

N

N1

1/NIF NIF

NIFb

b

L

L0

02 3

2

2.

v

v

v

v

+

+X

X

X

X

X

f fp p

ˆ

ˆˆ

ice

L/vv

vX X

X

1.0

0.8

0.6

0.4

0.2

0.00 1

YO

CN

IF

2 3

NIF YOC extrapolatedfrom measured

30% YOC on OMEGA

vXˆ

Hot spot

R r

T t

mo

Sh

ell

Shell

R

R+

Hotspot

Shell

Heat fluxRadiation flux

Radiation flux . 0Heat flux . 0

mo < Dshell

The heat and radiation energy lost by the hot spot does not propagate through the dense cold shell; no heat orradiationfluxatthehot-spotboundary

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q T T T T

T h200 5

1000/

shelleV keV

02

3

heat Sp

mg cc

2

- d .

mo

t

l ll=

=on

5] ]

c e

g g

m o

q r R q r R0 0heat rad. .= =+ +` `j j

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The clean volume analysis is validated by comparing 2-D simulationswithinner-surfaceroughnessand1-Dsimulations having reduced GvoH→GvoHVclean/V1-D ≈ GvoHYOCno a

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• Inthe1-Dsimulations,GvoH is reduced by the YOC (or clean volume fraction) until the hot-spot temperature reaches 10 keV.

YOCno a (%) YOCno a (%)

Gai

n

Gai

n

Direct-drive 1.5-MJ all-DT target design

Direct-drive surrogate of NIF 1.0-MJ indirect drive

0 604020020

25

454035302520151050

20

15

10

5

06040 8080 100 100

1-D code2-D code

1-D code2-D code

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Physics other than hydrodynamics (based on purely theoretical extrapolations)

• Laser-energy collisional absorption • Laser plasma instabilities and hot electron preheat • Cross-beam energy transfer • Radiation preheat

OMEGA NIF

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better

worse

worse

better