lawrence livermore national laboratory f. graziani, j. bauer, l. benedict, j. castor, j. glosli, s....
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Lawrence Livermore National Laboratory
F. Graziani, J. Bauer, L. Benedict, J. Castor, J. Glosli, S. Hau-Riege, L. Krauss, B. Langdon, R. London, R. More, M. Murillo, D. Richards, R.
Shepherd, F. Streitz, M. Surh, J. WeisheitLawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551
This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
Theoretical and Computational Approaches to Hot Dense Radiative Plasmas
Institute for Pure and Applied Mathematics, UCLA
Computational Kinetic Transport and Hybrid Methods
LLNL-PRES-412216
2Option:UCRL#
Lawrence Livermore National Laboratory
Matter at extreme conditions: High energy density plasmas common to ICF and astrophysics are hot dense plasmas with complex properties
kTm
2πλ
a
2
a
1/32ba
ab 3
n 4π
kT
eZZΓ
Ichimaru plasma coupling
Thermal deBroglie wavelength
11106.05101.44101.510eV1410
15109.09102.48107.41keV2410
Pω
ionR
DλTn
i i
i22
i
e
e2
2D kT
ne Z4π
kT
ne 4π
λ
1Debye length
Pra
d=
45
.7 M
ba
r (T
4 (k
eV
))
hot dilute
Metals
1 Mbar
WDM
Kremp et al., “Quantum Statistics of Non-ideal Plasmas”, Springer-Verlag (2005)
ICF
4keV
γMbar
1/3gm/cckeV
3keV
22γ1/cc
45.7TP
2ρT
T103.13n
WDM hot dense
3Option:UCRL#
Lawrence Livermore National Laboratory
Hot dense plasmas span the weakly coupled (Brownian motion like) to strongly coupled (large particle-particle correlations) regimes
Figurepoint
A 2.6
B 1.2
C 0.58
D 0.26
E 0.10
ei
density-temperature trajectory of the DT gas in an ICF capsule
Weakly coupled plasma:
– Collisions are long range and many body
– Weak ion-ion and electron-ion correlations
– Debye sphere is densely populated
– Kinetics is the result of the cumulative effect of many small angle weak collisions
– Theory is well developed
Strongly coupled plasma:
– Large ion-ion and electron-ion correlations
– Particle motions are strongly influenced by nearest neighbor interactions
– Debye sphere is sparsely populated
– Large angle scattering as the result of a single encounter becomes important
11/nλ3D
1
1
4Option:UCRL#
Lawrence Livermore National Laboratory
Hot, dense radiative plasmas are multispecies and involve a variety of radiative, atomic and thermonuclear processes
-329 cm 10
-327 cm 10
-325 cm 10
-323 cm 10
-321 cm 10eV 104eV 10 eV 102 eV 103
-321 cm 10
den
sity
Temperature
HydrogenHydrogen+3%Au
Strongly
Couple
d
Moder
atel
y Couple
d
Wea
kly
Coupled Stro
ngly C
oupled
Moder
atel
y Couple
d
Weakly Coupled
Characteristics of hot dense radiative plasmas:
• Multi-species
– Low Z ions (p, D, T, He3..)
– High Z impurities (C, N, O, Cl, Xe..)
• Radiation field undergoing emission, absorption, and scattering
• Non-equilibrium (multi-temperature)
• Thermonuclear (TN) burn
• Atomic processes
– Bremsstrahlung, photoionization
– Electron impact ionization
Iso-contours of ei
5Option:UCRL#
Lawrence Livermore National Laboratory
Transport and local energy exchange are at the core of understanding stellar evolution to ICF capsule performance
The various heating and cooling mechanisms depend on :
• Transport of radiation
• Transport of matter
• Thermonuclear burn
– Fusion reactivity
– Ion stopping power
• Temperature relaxation
– Electron-radiation coupling
– Electron-ion coupling
Laser beams
… .all in a complex, dynamic plasma environment ….
PiT~σv
PiT~σv
6Weapons and Complex Integration
Assumptions of a kinetic theory of radiative transfer and radiation-matter interactions rest on a “top-down” approach
Kinetic description of radiation:
• Basis is a phenomenological semi-classical Boltzmann equation
– Radiation field is described by a particle distribution function
– QM processes occur through matter-photon interactions
• Inherent limitations of semi-classical kinetic approach
– Photon density is large so fluctuations can be ignored
– Interference and diffraction effects are ignored
– Polarization, refraction and dispersion are neglected
Pomraning (73)Degl’Innocenti (74)
Matter: Local Thermodynamic Equilibrium (LTE):
• Atomic collisions dominate material properties
• Thermodynamic equilibrium is established locally (r,t)
• Electron and ion velocity distributions obey a Boltzmann law
TB σTB e1Σj νννkThνA
νν Emission source of photons
Planck function at Telectron
Kirchoff-Planck relation
7Weapons and Complex Integration
Modeling ICF or astrophysical plasmas, rests on a set of matter- radiation transport equations coupled to thermonuclear burn and hydrodynamics
S&T: Scientific motivation
t)Ω,(r,f chνt)Ω,(r,I νν Intensity
23ν dr d t)Ω,(r,fdn Photon distribution function ScatteringCompton t)Ω,(x,I TσTBTσt)Ω,(x,I Ω
t
t)Ω,(x,I
c
1νeνeνeνν
ν
AbsorptionEmissionFree streaming
t)Ω,(r,ΩI Ω Ωddνc
1P
t)Ω,(r,I Ωddνc
1U
ν
0
2R
ν
0
2R
Radiation energy density
Radiation pressure tensor
TNν2
νeνeii
1eiee
e St)Ω,(r,I Ωd(T)BTσ dνUUτUDt
U
TNie1
eiiii SUUτUD
t
U
t)T(r,ρCVU
Conductivity Electron-ion coupling
Conductivity Electron-ion coupling
Equation of stateMaterial energy density
Material heating due to radiationMaterial cooling due
to radiative losses Source due to TN burn
Source due to TN burn
The temporal evolution of plasmas depends on the complex interaction of collisional, radiative, and reactive processes The temporal evolution of plasmas depends on the complex interaction of collisional, radiative, and reactive processes How does one assess the accuracy of models in regimes difficult to access experimentally and theory is difficultHow does one assess the accuracy of models in regimes difficult to access experimentally and theory is difficult
8Weapons and Complex Integration
abb
aba TTt
T
1
sec lnΛn
cm10
eV 100
T
ZZ
A103.16 ~
lnΛ Z Zn 2π8
mkT
mkT
3μ
τ
abb
3213/2
2b
2a
10-
ab2b
2ab
3/2
a
a
b
bab
ab
Many issues are ignored:
• partial ionization (bound states)
• collective behavior (dynamic screening)
• strong binary collisions/strong coupling
•quantum effects
•non-Maxwellian distributions
•degeneracy*
Major source of uncertainty
0.5
0.4
0.3
0.2
0.1
0.0
0.10.2
0.11.001.0Temperature (keV)
ln
*H. Brysk, Phys. Plasmas 16, 927 (1974)
/kTeZ
λlnlnΛ
22D
Kinetic equation I: The Landau kinetic equation is the starting point for computing electron-ion coupling in hot dense plasmas
Fokker-Planck with Boltzmann distributions
Q22D
λ/kT,eZMax
λlnlnΛ
9Weapons and Complex Integration
The standard model of thermonuclear reaction rates assumes a Maxwellian distributed weakly coupled plasma
Y
Xab
nTT
nHeDD
pTDD
nTD
2
3
XaXaXXaaXaaXUUUUUfUfdUdUv ),()()(
DT cross section
Ion distribution
Gamow peak
Velocity (cm/microsecond)
T=10.4 keV
Fusion reactivity
ion distribution cross section
Boltzmann ion distributions
Bare cross section
Non-thermal ion distributions
/seccm σv 3
0.100.1 0.100 0.1000
Temperature (keV)
1410
1610
1710
1810
1910
2010
1510
22
2kT
mvδ
Maxeff
22
Max 21ff
kT
mv
Maxff
Dense plasma effects
D
XaT
eZZ
aX
Screen ev
v
2
Brown and Sawyer, Rev. Mod. Phys. 69, 411 (1997)Bahcall et al., A&A, 383, 291 (2002)Pollock and Militzer, PRL 92, 021101 (2004)
10Weapons and Complex Integration
A micro-physics approach based on a “bottom-up” approach can provide insight into the validity of our assumptions
S&T: Scientific motivation
Jv
f
m
F
r
fv
t
f N
1j j
EXN
i
j
j
EXN
j
EXN
Classical or Wigner Liouville equation
N-body simulationKinetic Theory
• Systematic expansion in weakly coupled regime
• Formal connection to the micro-physics (Klimontovich)
• Convergent kinetic theory
• Multi-physics straightforward
• Closure relations are needed (BBGKY)
• Theory is difficult in strongly coupled regime
• Virtual experiment • Particle equations of motion are solved exactly
• All response- and correlation-
functions are non-perturbative
• Approximations are isolated and
understood
• Forces tend to be classical like
• Requires large numbers of particles
11/nλ3D
Galinas and Ott (70)Degl’Innocenti (74)Cannon (85)Graziani (03, 05)
QEDH
11Weapons and Complex Integration
Classical weakly coupled plasma properties:
• Collisions are long range and many body
• Mutual ion-ion and electron-ion interactions are weak
• Fully ionized
Charged particle scattering is the result of the
cumulative effect of many small angle weak collisions
1 /λλln~db/b thD
b
b
max
min
b
aabvvaabv2b2
a
42aa t),v(fD
2
1 t),v(fA lnΛ Z
m
e Z2π
t
t),v(f
• Brownian motion analogy• Static Debye shielding • Particle, momentum and kinetic energy conservation• Markovian• H-Theorem (Maxwellian static solution)• Short and long distance divergence (Coulomb logarithm)
Kinetic equation I: The Landau-Spitzer model of collisional relaxation rests on the assumptions of a weakly coupled classical plasma
12Weapons and Complex Integration
Landau treatment of collisional relaxation with radiation and burn yields insights into the underlying assumptions
J. S. Chang & G. Cooper 1970, JCP, 6, 1B. Langdon
Michta, Luu, Graziani
Fokker-Planck treatment of an isotropic, homogeneous DT plasma with TN burn, Compton and bremsstrahlung
nTT
nHeDD
pTDD
nTD
2
3
13Weapons and Complex Integration
Kinetic equation II: The Lenard - Balescu equation describes a classical but dynamically screened weakly coupled plasma
b
avab
aavb24
332bv2
a
42aa t),v(ft),v(f
m
m - t),v(ft),v(f
k,vkεk
vkvkδk k kd
π
1vd Z
m
e Z2π
t
t),v(f
• Dynamic screening of the long range Coulomb forces
– plasma dielectric function provides cutoff
• Particle, momentum and kinetic energy conservation
• Markovian
• H-Theorem (Maxwellian static solution)
• Short distance cutoff still needed
• Landau equation recovered 2D
2λk/11,0kε
Boyd and Sanderson, “Physics of Plasmas”, Cambridge Press (2003)
Requires a model for the dielectric function of the electron gas
14Weapons and Complex Integration
The quantum kinetic equations of Kadanoff-Baym and Keldysh provide the basis for describing strongly coupled complex plasmas
11g11g 11Σ11Σ 1d11g 11Σ11Σ 1d
tr trg tr rΣdr 11g (1)U2mt
i
aainaa
t
t
aaa
t
t
1111a111HFaaa
a
21
2
1
1
0
1
0
• Quantum diffraction, exchange and degeneracy effects
• Interacting many body conservation laws obeyed (total energy)
• Formation and decay of bound states included
• Dynamical screening
• Non-Markovian
RPA self energy with a statically screened potential
Quantum Landau
RPA self energy (dynamic screening) Quantum Lenard-Balescu
Time diagonal K- B equation describes the Wigner distribution
Dense strongly coupled plasma properties:
•Mutual ion-ion and electron-ion interactions are strong
Kremp et al., “Quantum Statistics of Non-ideal Plasmas”, Springer-Verlag (2005)
15Weapons and Complex Integration
More advanced treatments of the electron-ion coupling avoid the divergence problems of earlier theories
Quantum kinetic theoryGericke-Murillo-Schlanges
Divergenceless models of electron ion coupling
Convergent kinetic theoryBrown-Preston-Singleton
kTZe
8πλ
Rλ1ln
2
1ln 22
th
2ion
2D 1γ16πln
2
1
λ
λlnlnΛ
th
D
Short distance Boltzmann Long distance Lenard-BalescuDimensional regularization
Although finite, these theories make assumptions regarding correlations and hence are still approximate…..Although finite, these theories make assumptions regarding correlations and hence are still approximate…..
16Weapons and Complex Integration
N-body simulation techniques based on MD, WPMD or Wigner offer a non-perturbative technique to understanding plasma dynamics
Molecular dynamics
Classical like forces with effective 2-body potentials Wave packet MD
Wigner equation
How do we use a particle based simulation to capture short distance QM effects and long distance classical effects?
kT
Zenot scale,length rangeshort thesets
kTm
2πΛ
2
a
2
a
17Weapons and Complex Integration
The MD code is massively parallel and it is based on effective quantum mechanical 2-body potentials
Newton’s equations for N particles are solved via velocity-Verlet:
The forces include pure Coulomb, diffractive, and Pauli terms:
H pa2
2ma qaqb
rabf (,rab ) exp rab ab
g(,rab )Te ln(2)exp rab
2
ln(2)ee2
ab
a
• separate velocity-scale thermostat for each species during equilibration phase (~20,000 steps) establish two-temperature system
• “data” accumulated with no thermostat relaxation phase
• time step ~0.02/peTa (t)
ma3Na
v j ,a
2
j
r (t t)
r (t)
v(t)t 1
2
a(t)t 2
v(t t)
v(t) 1
2
a(t t)
a(t) t
Ewald approach breaks problem into long range and short range parts
Short range explicit pairs are “easy” to parallelize: local communication.
Long range FFT based methods are hard to parallelize: global communication.
Solution: Divide tasks unevenly, exploit concurrency, avoid global communication
125M particles on 131K processors
18Option:UCRL#
Lawrence Livermore National Laboratory
MD has recently been used to investigate electron ion coupling in hot dense plasmas and validate theoretical models
J.N. Glosli et al., Phys. Rev. E 78 025401(R) 2008.
G. Dimonte and J. Daligault, Phys. Rev. Lett. 101,
135001 (2008).
B. Jeon et al., Phys. Rev. E 78, 036403 (2008).
LSpe J
ln1
eV 12.1T
eV 91.5T
/cc101.61n
p
e
24
electrons
protons
Tem
per
atu
re (
eV)
Temperature (eV)Time (fs)
log
()
L.S. Brown, D.L. Preston, and R.L. Singleton, Jr., Phys.
Rep. 410, 237 (2005).
D.O. Gericke, M.S. Murillo, and M. Schlanges, Phys.
Rev. E 65, 036418 (2002)
19Option:UCRL#
Lawrence Livermore National Laboratory
The MD code predicts a temperature relaxation very different than what LS or BPS predict…and it should be measurable!
SF6 gas jet
LANL has built an experiment to measure temperature relaxation in a plasma
e heated by laser to 100 eVions are heated to 10 eV
Te - Thomson ScatteringTi – Doppler Broadening
i i
i22
i
e
e2
2D kT
ne Z4π
kT
ne 4π
λ
1
Dominant for Ti/Te>>1
Dominant for Te/Ti>>1
Glosli, et al, PRL submitted
53K electrons6K F1K S
20Option:UCRL#
Lawrence Livermore National Laboratory
Radiation: Classical EM fields (Maxwell eqs)
Lienard-Wiechert Potentials
Normal mode expansion
Problem: Planckian spectrum is not produced in LTE
Modeling matter + radiation: Molecular dynamics coupled to classical radiation fields is straightforward but is not relevant for hot dense matter
2-electron + 2-proton+radiation
Dipole emission
j retj
jj
j retj
j
ii
iii
trr
vqtr,A
trr
qtr,Φ
t
A
c
1ΦE
c
BvEqF
t,kJΩ 2
it,kαiω
dt
t,kαd
k
21Option:UCRL#
Lawrence Livermore National Laboratory
Photons: Isotropic and homogeneous spectral
intensity
Kramer’s for emission and absorption + detailed balance
• Planckian spectrum in equilibrium
e-i radiation only (neglect e-e, i-i quadrupole emission)
Monte-Carlo tests decide emission or absorption of radiation
• Close collisions are binary
• Each pair only gets one chance to emit, absorb per close collision
Modeling matter + radiation: Molecular dynamics coupled to quantum mechanical radiation fields
tn c
νhtI ν2
34
ν
tεtI t κρdt
tdI
c
1ννν
ν
emissivityabsorption
Spectral intensity
Emission and absorption of radiation is the aggregate of many binary encounters
BR
22Option:UCRL#
Lawrence Livermore National Laboratory
Algorithm: Molecular dynamics coupled to either classical or quantum mechanical radiation fields
Emission and absorption of radiation is the aggregate of many binary encounters
BR
Step 0: Begin with the Kramers formulas for emission and absorption
hνvcm
eZ
33
32π
hνd
dσ2e
32e
622emν
Step 1: Tag a close encounter event and determine
probability of any radiative process
Step 2: If a radiative event occurs, test to decide
emission or absorption
2B
absemiss
R π
σσP
Integrated Kramers
cross sections
absem
abs
absabsem
em
em σσ
σP
σσ
σP
23Option:UCRL#
Lawrence Livermore National Laboratory
Algorithm: Molecular dynamics coupled to either classical or quantum mechanical radiation fields
Step 3: Identify energy of photon emission (absorption)
RF h 0,1Rnumber random apick
,PF
ρdsρ dshνhνhνP
1-
n
1i
emin
ems
E
0
ems
hν
hν
1iiemi
1i
i
R π1nhνd
dσρ 2
Bν
emνem
ν
0
1
i=1 i=n
Fn
R
Emit to frequency group i
Step 4: Update electron energy and photon population
νn
24Option:UCRL#
Lawrence Livermore National Laboratory
LTE test Case: A 3 keV Maxwellian electron plasma produces a black-body spectrum at 3 keV
Neutral hydrogen plasmaProtons, electrons and photons
Photon Energy (eV)
tI
A Maxwellian plasma of 3 keV electrons produces a BB spectrum at 3 keV
Trad=3 keV
25Option:UCRL#
Lawrence Livermore National Laboratory
Three temperature relaxation problem for a hot hydrogen plasma agrees well with a continuum code
Photon Energy (eV)
The dynamics of the spectral intensity are consistent with the lower groups coupling faster
tI
512e+512p V = 512 Å3
=1024 cm-3
Glosli et al, J. of Phys. A, 2009Glosli et al, HEDP, 2009
26Option:UCRL#
Lawrence Livermore National Laboratory
Strengths• Easy to implement in an existing MD code• Radiation that obeys detailed balance
Weaknesses
• Kramers cross sections
Isolated radiative process assumed
• Multiple electrons within radius not treated correctly
• Low frequency radiation is ignored
Alternative approaches
• Hybrid methods
• WPMD with radiation-almost complete
• Langevin equation for the charged particles in a QM radiation field
• Normal mode formulation that incorporates stimulated and spontaneous emission
Our initial approach to coupling particle simulations to quantum radiation fields has both strengths and weaknesses
27Option:UCRL#
Lawrence Livermore National Laboratory
It is possible to do MD simulations including radiative processes• Charged particles• Radiation that obeys detailed balance• Radiation that relaxes to a Planckian spectrum
There’s a rich variety of micro-physics to explore: • Impurities
Partial ionization (Atomic physics)
• High energy particles (e.g. fusion products)
• Micro-physics of energy and momentum exchange processes
• Reaction kinetics
We are developing an MD capability that allows us to model the micro-physics of hot, dense radiative plasmas
Conclusion