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Ignition, the Lawson Criterion, and Perfomance Assessment for ICF implosions
FSC
Review of the National Ignition Campaign
Livermore, CA 9–11 December 2009
R. BettiUniversity of RochesterFusion Science Center andLaboratory for Laser Energetics
“Review of Burning Plasma Physics,” FESAC Report (September 2001).
GTH (keV)
GPHx
E (
atm
× s
)
10–1
10–4
10–3
10–2
10–1
100
101
102
103
100 10–1 102
Jet
ITER
TFTRDIIID
C-mod
LHD
NSTX
LSX
MST
SSPX
OMEGA (2009)Q = 2
IgnitionCTFFRCSpheromakRFPSTStellaratorTokamakTokamakTokamakTokamakITER
NIFThe National Ignition Campaign
Collaborators
K. S. AndersonP. Chang
R. L. McCroryR. Nora
C. D. Zhou
University of Rochester Laboratory for Laser Energetics
B. SpearsJ. Edwards
S. HaanJ. Lindl
Lawrence Livermore National Laboratory
The measurable Lawson criterion and hydro-equivalent curves determine the requirements for an early hydro-equivalent demonstration of ignition on OMEGA and on the NIF (THD)
TC8563b
FSC
Summary
* NIF will begin the cryogenic implosion campaign using a surrogate Tritium–Hydrogen–Deuterium (THD) target. Only a very small fraction of Deuterium (<5%) is used to prevent fusion reactions from affecting the hydrodynamics.
• CryogenicimplosionsonOMEGAhaveachievedaLawsonparameter PxE á 1 atm-s comparable to large tokamaks
• Performancerequirementsforhydroequivalentignition on OMEGA and NIF (THD)*
Hydro-equivalent ignitionGtRHng/cm2
GTiHnkeV
YOC PxE(atm s)
OMEGA (25 kJ)1:60 ~0.30 ~3.4
15%(~3 × 1013 neutrons)
2.5
NIF (THD) (1 MJ) 1:1 ~1.8 ~4.8 ~37% 21
Outline
TC8633a
• A1-DmeasurableLawsoncriterionforICF
• Hydro-equivalentcurvesandhydro-equivalentignition
• The3-DextensionoftheLawsoncriterion
• Comparisonwithmagneticconfinement
FSC
A dynamic model of ignition relates the hot-spot stagnation properties to those of the shell
TC8534a
• R.Bettiet al., Phys. Plasmas 8, 5257 (2001).• R.Bettiet al., Phys. Plasmas 9, 2277 (2002).• J.Sanzet al., Phys. Plasmas 12, 112702 (2005).• Y.Saillard,Nucl.Fusion46, 1017 (2006).• J.GarnierandC.Cherfils-Clérouin,Phys.Plasmas15, 102702 (2008).• C.D.ZhouandR.Betti,Phys.Plasmas15, 102707 (2008).
Deceleration Alphadeposition
Dynamic model of hot-spot formation and ignition
Stagnation
Thermalinstability
IgnitionPeak implosionvelocity V = Vi
The initial conditionsare those at peakimplosion velocity
FSC
Hotspot
Dstagnation
Shell
Ion temperatures, areal densities, and neutron yields are parameters of the fuel assembly that can be measuredwithexistingdiagnostics
TC8533a
• Neutronyieldsandneutronrates Ndt
dNneutron
neutron
are measured with scintillators
• Iontemperature(neutron averaged) is Ti neutronmeasured with the neutron time-of-flight detectors
• Totalarealdensity(neutron averaged) R dr0neutron neutron/t t3#
is measured with magnetic recoil spectrometer measuring the downscattered neutron fraction.
FSC
Total tR . shell (tD)stagnation
Hot spot
R r
T t
mo
Sh
ell
Shell
R
R+
Hotspot
Shell
Heat fluxRad. 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
TC8536
FSC
q T T T T
T h200 5
1000/
shelleV keV
g cc
02
3
heat Sp
m 2
- d .l l l
mo
t
=
=on
5] ]
c e
g g
m o
q r R q r R0 0heat rad. .= =+ +` `j j
Heat conduction losses do not affect the hot-spot pressure. They reduce the hot-spot temperature and raise the hot-spot density
TC5778b
• Heat-conductionlossesarerecycledbymassablation into the hot spot and do not change the hot-spot pressure
Ti(1le)
Ti(4le)
t
R (nm) R (nm)
t(g
/cc)
T i(k
eV)
Pre
ssu
re (
Gb
ar)
0
200
0
400
600
800
1000
1200
10 20 30 40 50 100 20 30 40 500
2
4
6
8
10
12700
600500400300200100
0
Ti(0.25le)
T
t
P
FSC
The hot-spot ignition condition is given by the balance ofalphaheatingwiththeexpansionlosses
TC7595a
FSC
P = Phs = Ps
r
Rhs
Rs
tsshell
ths
Hot spot
Isobaric fuelassembly
0>EE 1 1
exphs
hs -x x=a
o
~ vn P1 2hs hsfx va a Alpha heating
(fa = 3.5 MeV)
~ R R1 exp hs hsx p Expansion
M R P R4s2
hs hs hsr=p shell Newton’s law
vT24P exp
2
hs x f v=
a
Lawson criterion
Theexpansionlossesrepresenttheinternalenergylostby the hot spot and transferred to the surrounding dense shell as kinetic energy
TC7914
~ R R1 exp hs hsx p Expansion
M R P R4s2
hs hs hsr=p Shell Newton’s law
FSC
RhsPhs
Hot spot
Shell
Dsts
~M Rs s s s2t D
Shell arealdensity
Ds = shell thickness
~P R1
exp s
hs hsx M
Shell mass
The ignition condition depends on shell areal density, implosion velocity, and hot-spot ion temperature
TC7916a
FSC
vRP
MT
const>si2hs
hs v
Hot-spot pressure and temperature
Hot-spot radius Shell mass
v C T
T keV8for 4 < <i
3+v a
• ~ ~P P R R M V Rs s s s3 3 2 3
hs hs` `j j
• ~M Rs s s s2t D
• Rs ~ Rhs
VT constC
>s s it Da
_ i
Shell areal density Implosion velocity
Eliminating the velocity and energy leads to an ignition condition depending on the shell areal density and hot-spot ion temperature
TC7918a
FSC
T const C. 0.83i2 1 .0 03
2atD aif^ h: D
Hydro-variablesscaling from Ref. 1
~V T E. .ifi L
0 8 0 15a .0 056-
~E VifL s s3 0.18.1 75t aD
-_ i
1C. Zhou and R. Betti, Phys. Plasmas 14, 072703 (2007).
Verify that the theory is right. Simulate many marginally igniting targets and plot the total areal density versus the ion temperature
TC8234a
FSC
1-D simulations show that all marginally ignited targets lie on a single curve in the ,R Ti
no not a a- - plane.
• Simulationdatabase– marginal ignited targets
(Gain = 1)– laser energy
35 kJ to 10 MJ– implosion velocity
170 to 530 km/s
R nn
ot a- = neutron-averaged total areal density
Tino
n
a- = neutron-averaged
ion temperature2
0
1
2
3
4
5
6
3 4 65
Ignition1-D simulations
Fit of simulations
GtR
H n
(g
/cm
2 )n
o-a
GTHno-a (keV)n
C. D. Zhou and R. Betti, Phys. Plasmas 15, 102707 (2008).
The analytic model agrees reasonably well with the simulations;thelattercanbeaccuratelyfitbyasimplepower law tR × T2 > 20 for 3.5 < T < 7 keV
TC8634a
FSC
D-.
7.RT
T4
1 3 57
> < <1
2
g cmno
n
keVno
n keVno
2/| t aa
a--
-f p1-D ignitionparameter
Ignition condition
4
Best fit for3.5 < T < 7 keVtR = 20/T2
0.5
0.0
1.0
1.5
2.0
5 6 7
IgnitionGt
R Hn
o-a
(g/c
m2 )
n
GTHno-a (keV)n
When can Tno-a, tRno-a be measured and the Lawson criterion be assessed?
TC8550b
• SurrogateD2 implosions
• SurrogateTHDimplosions
• DTimplosionsonOMEGA
• DTimplosionsontheNIFwithGain< 0.1
• DTimplosionsontheNIFwith0.1< Gain < 1.0: T > Tno-a
• DTimplosionsontheNIFwithGain> 1:T > Tno-a and tR < tRno-a
For those, no need for a Lawson criterion.Just look at the neutron detector.
FSC
For all of the above T . Tno-a, tR . tRno-a. Use the measurable Lawson criterion.
Outline
TC8633b
• A1-DmeasurableLawsoncriterionforICF
• Hydro-equivalentcurvesandhydro-equivalentignition
• The3-DextensionoftheLawsoncriterion
• Comparisonwithmagneticconfinement
FSC
Hydro-equivalent implosions have the same implosion velocity, in-flight adiabat, and laser intensity leading to equal stagnation density and pressure
TC8635
• Inhydro-equivalenttargets,massandsizedependonlaserenergy
Mass ~ EL Radius = ~R EL1 3 Thickness = ~ EL
1 3D
IPV V350
300780
300Gbar
km sg cc
km s.
.maxi i0 9
2
150 13
stst . .a
t a] ^ ^ ^g h h h= =G G
FSC
shellPPadiabat measure the entropyF&/a =
Vi
a
Vi
a
Hydro-equivalent implosions have similar Rayleigh–Taylor growth factors and nonlinear bubble-front penetration
TC8636
• Hydro-equivalentimplosionshavethe same in-flight aspect ratio (IFAR)*
IR V40
300IFARkm s
.i0
0 6
2
151 4
if./
aD
-^ h= G
• Hydro-equivalentimplosionshaveasimilarlineargrowthfactor
~ ~ ~ ~e t k R Rk gt IFARifif if
t0
2h h c DD D
= c
• Hydro-equivalentimplosionshaveasimilarnonlineargrowth
~ ~ ~h Rgt
IFARif if if
2bubble
:b b
bD D D
0.05
~R ~1
FSC
R0
Dif
1
tp
I15
t
*J.D.Lindl,“InertialConfinementFusion:TheQuestforIgnitionandEnergyGain Using Indirect Drive (Springer-Verlag, New York, 1998), Chap 6, p. 61.
Total areal densities and ion temperatures increase as hydro-equivalent implosions are scaled up in energy
TC8637
FSC
Hydro-equivalent curves show how OMEGA implosions would perform (in 1-D) when scaled up in energy.
~RV
EiL1 3
totta .0 6
.0 06
~V
ET ..
iLi0 07
15
1 25
a .0
C. D. Zhou and R. Betti, Phys. Plasmas 14, 072703 (2007).
Ti
tR
tot
Marginalignition
Hydro-equivalentcurveVi = constanta = constantEL varies
EL
Ignition
NIF
OMEGA
Progress in ICF can be assessed on the tR, Ti plane through hydro-equivalent curves
TC8638b
This diagram measures progress in 1-D compression; it does not include a good measure of implosion uniformity.
FSC
* T. C. Sangster et al., Phys. Rev. Lett. 100, 185006 (2008).
10
0.1
0.2
0.5
1.0
2.0 1.5 MJ
Marginal ignition
2
Tota
l Gt
RH n
(g/c
m2 )
3 4 5 6 7 8
Ignition
1.5 MJ
23 kJOMEGA Goal
0.5 MJ
OMEGA23 kJ2009OMEGA
17 kJ2007*
V = 2.2 × 107
a = 2.5CH ablator
V = 3.5 × 107
a = 2.0CH ablator
GTiHn (keV)no-a
no
-a
Outline
TC8633c
• A1-DmeasurableLawsoncriterionforICF
• Hydro-equivalentcurvesandhydro-equivalentignition
• The3-DextensionoftheLawsoncriterion
• Comparisonwithmagneticconfinement
FSC
In 3-D the fusion yield is reduced by the Rayleigh–Taylor instability that cools down parts of the hot spot
TC8551
• Theyield-over-cleanYOC= 3-D fusion yield;1-Dyieldisapproximatelyequal to the ratio clean volume/1-D volume
FSC
~ ~V R V R<3 Dclean
clean3
1D 1D3
- - -
YOCN
NV
V
neutron1Dneutron3 D
1D
3 Dclean
/ .-
-
-
-
Rclean
Hot spotCold
Shell
R1-DGvvH ≈ 0
Can be measured
~ ~N n v NV
VVneutron
3 D3 D neutron
D
1D
3 Dclean
cleanburni
2 1v x--
-
-
-
YOC withouta-depositionYOCno–a
TheYOCisusedtoextendthemeasurableLawsoncriterion to three dimensions
TC8552b
Back to the 1-D Lawson criterion (simple analytic model)
3-D measurable Lawson criterion
FSC
R T YOC const>2.1 1.2
stno
stno not a a a- - -^ a _h k i
~C v dVv va #
~C v dV CV
VC YOCV
3 D 1D
1D
3 D 1D no3 D
:. .va a aa- -
-
- - --
#
R T Cconst>. .2 1 1 2
stno
stnot a a
a- -^ ah k
clean
The clean volume analysis is validated by comparing 2-D simulations with inner-surface roughness and 1-D simulations having reduced GvvH→GvvHV3-D / V1-D ≈ GvvHYOCno-a
TC8553
• Inthe1-Dsimulations,GvvH is reduced by the YOC (or clean volume fraction) until the hot-spot temperature reaches 10 keV.
FSC
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
clean
Results from a 2-D + pseudo 2-D simulation database are in reasonable agreement with the ignition model
TC8640
FSC
• 3-DmeasurableLawsoncriterion(fitfromsimulations)
.
RT4 7 YOC 1>
.3 Dfit
20 7
stno st
nono| t= a
aa
--
--^ f `h p j
0.00.0
0.2
0.4
0.6
0.8N
orm
aliz
ed g
ain
G/G
1-D
Minimizedspread
1.0
0.5 1.0 1.5 2.0
.YOCR
T4 7st
no stno
no2
0.7| t= -a
aa-
-
^ e ^h o h
• WheredoesOMEGAcurrentlystand?
/0.2 g cmRn
2.t 2.1 keVTni . YOC á 10%
Igniton parameter | = 0.0083
• Ascale1:60(25 kJ:1.5 MJ) hydro-equivalent demonstration of ignition on OMEGA requires
/0.3 g cmRn
2.t 3. keVT 4ni . YOC á 15%
Ignition parameter | = 0.045
• Anearlyscale1:1demonstrationofignitionontheNIF-THD*requires
/1.8 g cmRn
2.t 4.8 keVTni . YOC á 37%
Ignition parameter | = 1
OMEGA (DT) and NIF (THD) campaigns aim toachieve an early hydro-equivalent demonstrationof ignition (on different scales)
TC8662b
FSC
*THD = Tritium–Hydrogen–Deuterium targets
.YOCR T
4 7.2 0 7/| t b l
Measurable ITF
The measurable Lawson criterion leads to the ignition threshold factor (ITF) and the energy margin (M)FSC
TC8728
~ , ~
1
R E T E
E
E R T
22
Use scalings:
3 D D
.
.. .
min
i
i
1 3 0 07
2 2 161 5 1 5
n kin n kin
ignkin n
t
t- -ITF YOC ITF YOC. .= ^ f ^h p h
ITF (3-D) . ITF (1-D) × YOC1.5
MYOCNIF pt. design = 5.3–0.66 . 33% . 37% in 2-D simulations
Margin = ITF - 1
st = stagnationkeVR T YOC 22gcm>
.no no no2 0 7 2st st
2t -a a a- - -^ _ ^h i h
Outline
TC8633d
• A1-DmeasurableLawsoncriterionforICF
• Hydro-equivalentcurvesandhydro-equivalentignition
• The3-DextensionoftheLawsoncriterion
• Comparisonwithmagneticconfinement
FSC
A3-DconfinementtimeforICFcanbederived from the ignition condition (simple model)
TC8729
• StartfromtheLawsoncriterionandthemeasurablecondition
• Rewritethe1-Dconfinementtimeusingtheimplosionvelocity
• Extractthe3-Dconfinementtime.The1-Dconfinementtime is reduced by the Rayleigh–Taylor instability
YOC:D1-Ex=D3- .
E0 6x
/ =1 D-4 P R
MV
RE
ist st
sh stxr R2 Pst
3str=2
21 M Vish
Energy conservation
stst .
vT
P R T4 724
YOC. ..
D2 30 83 2 0 7
0 83.
f vx | t=a
- ^ bh l= G 3-D effects
FSC
SimulationsshowaconfinementtimeforOMEGA of about 17 ps and a pressure of about 70 Gbar, leading to PxE ~ 1 atm × s
TC8722a
• OMEGAcryogenicDTimplosions:simulation/experimentalresults
R V22 m 3 10 cm s YOC 0.1i7
stsim sim exp#. . .n
VR
7YOC 1 ps.E
i
0 6st./x
Pst . 70 Gbar
Pst xE . 1.2 atm × s
FSC
0 302010 40
Den
sity
(g
/cm
3 )
Radius (nm)
40
0
160
120
80
Pre
ssu
re (
Gb
ar)
20
0
100
60
80
40
Simulation of shot 52729
Hot spot
P
t
Sh
ell
Simulations show that NIF (THD) needs to achievea Lawson parameter PxE ~ 20 atm × s for an hydro-equivalent demonstration of ignition
TC8730
800700600500
400300200100 200
400
600
800
1000
1200
1400
0 10 20 30 40 50
Stagnation densityand pressure
P (
Gb
ar)
Den
sity
(g
/cc)
R (nm)
P t
Hot spot
Denseshell
650V
R
P
37YOC ps
Gbar
.E
i
0 6st
st
/
.
.x
P 21atm sEst .x -
• NIFsurrogateimplosion:simulationresults
.85 37R V2 m 3 10 cm s YOC 0.mini7
stsim sim sim#. . .n
A measurable PxE for ICF can be constructed using the ignition model and compared with PxE in MCF
TC8663b *“ReviewofBurningPlasmaPhysics,”FESACReportDOE/SC-0041(September 2001).
FSC
• DefineaxE from the ignition model
Pstag T24E
2.x f voa
|0.83
• UseGvvH ~ CaT3 consistent with the analytic model
• OMEGA: 8.3 10 , 2 keV 1 atmT P sE3
&# #. . .| x-
• NIF(THD): , 4.8 keVT P1 21E&. . .| x atm s#
keVatmP
Ts100
ICFEn
#.x| .0 83
^ ^h h
The Lawson plots show the current performanceand the future directions for OMEGA
TC8678b
FSC
* Figure courtesy of J. P. Freidberg (MIT) ** Assuming the YOC prediction is correct
OMEGA hydro-equivalent ignition:** | á 0.045, GTH á 3.4 keV, pxE á 2.5 atm × s
How to resolve the discrepancywith the thermonuclear Q = Pfus/Pinput?
*Deuterium–Tritium Plasmas
Central ion temperature (keV)
W =energy
0.00010.1 1 10 100
0.001
0.01
0.1
1
10
100
Law
son
fu
sio
n p
aram
eter
, niT
ixE
(102
0 m
–3 k
ev s
)
Ignition
Q ~ 1
Q ~ 0.1
Q ~ 0.01
Q ~ 0.001
Q ~ 0.0001
Q ~ 0.00001
Q ~ 10
Q = WFusion/WInput
OMEGA (2009)OMEGA H. E. I.
Tokamaks
*
GTH (keV)
GPHx
E (
atm
× s
)
10–1
10–4
10–3
10–2
10–1
100
101
102
103
100 10–1 102
Jet
ITER
TFTRDIIID
C-mod
LHD
NSTX
LSX
MST
SSPX
OMEGA (2009)Q = 2
IgnitionCTFFRCSpheromakRFPSTStellaratorTokamakTokamakTokamakTokamakITER
NIF (THD)OMEGAhydro-equivalent
ignition
The discrepancy on Qisresolvedbydefininga physics-based thermonuclear Q
TC8679
FSC
• “Conventional”hot-spotenergybalance
• FindPxE from the energy balance
• DefineaphysicalthermonuclearQ for ICF Measured neutron yield: 6 × 1012 For a measured T = 2 keV,
the previous result is recovered
For NIF:
1 atmTQ
Q sP 245E
2#.x f vo
=+a
TQ
QP 245E
2x f vo
=+a
QWW
ICFphysics
hot spot
fusion=QWW
ICFhysics
laser
fusionp!
24 kJWlaserOMEGA =
440 JWhot spotOMEGA .
From 1-Dsimulations
. %Q 3 8physicsOMEGA .
W W W P23
in losses Ex+ = =a
5 5WWW
WQ
P1 123
fus
inEx
+ = + =a af cp m
,.T Q P T244 8keV E
2"3. .x f vo
=a
22 atm s#
The measurable Lawson criterion and hydro-equivalent curves determine the requirements for an early hydro-equivalent demonstration of ignition on OMEGA and on the NIF (THD)
TC8563b
FSC
Summary/Conclusions
* NIF will begin the cryogenic implosion campaign using a surrogate Tritium–Hydrogen–Deuterium (THD) target. Only a very small fraction of Deuterium (<5%) is used to prevent fusion reactions from affecting the hydrodynamics.
• CryogenicimplosionsonOMEGAhaveachievedaLawsonparameter PxE á 1 atm-s comparable to large tokamaks
• Performancerequirementsforhydroequivalentignition on OMEGA and NIF (THD)*
Hydro-equivalent ignitionGtRHng/cm2
GTiHnkeV
YOC PxE(atm s)
OMEGA (25 kJ)1:60 ~0.30 ~3.4
15%(~3 × 1013 neutrons)
2.5
NIF (THD) (1 MJ) 1:1 ~1.8 ~4.8 ~37% 21