LLNL-PRES-412216
Lawrence Livermore National Laboratory
Theoretical and Computational Approaches
to Hot Dense Radiative Plasmas
Institute for Pure and Applied Mathematics, UCLA
Computational Kinetic Transport and Hybrid Methods
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. Weisheit
Lawrence 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
Matter at extreme conditions: High energy density plasmas common to
ICF and astrophysics are hot dense plasmas with complex properties
WDM
1/3
hot
dense
ICF
Za Zbe2  4π n 
Γab 


kT  3 
1 Mbar
λa 
γ
3
n1/cc
 3.131022 TkeV
WDM
TkeV  2ρ1/3
gm/cc
γ
Mbar
P
 45.7T
4
keV
Prad=45.7 Mbar (T4(keV))
Metals
2π  2
m a kT
Ichimaru plasma coupling
Thermal deBroglie
wavelength
1 4π e2 n e
4π Zi2e2 n i


λ 2D
kTe
kTi
i
R
ω
D
ion
P
24

8

9
10
1keV 7.410
2.410
9.01015
1014 10eV 1.510 4 1.410 5 6.01011
n
hot
dilute
Debye length
T
λ
Kremp et al., “Quantum Statistics of Non-ideal Plasmas”, Springer-Verlag (2005)
Lawrence Livermore National Laboratory
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2
Hot dense plasmas span the weakly coupled (Brownian motion like)
to strongly coupled (large particle-particle correlations) regimes
Weakly coupled plasma:
  1
– Collisions are long range and many body
Figure
point
 ei
A
2.6
– Debye sphere is densely populated
B
1.2
– Kinetics is the result of the cumulative
effect of many small angle weak collisions
C
0.58
D
0.26
E
0.10
– Weak ion-ion and electron-ion
correlations
– Theory is well developed 1/nλ D  1
3
Strongly coupled plasma:
 1
– 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
density-temperature trajectory
of the DT gas in an ICF capsule
Lawrence Livermore National Laboratory
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3
Hot, dense radiative plasmas are multispecies and involve a
variety of radiative, atomic and thermonuclear processes
Hydrogen
Hydrogen+3%Au
Characteristics of hot dense
radiative plasmas:
1029 cm-3
• Multi-species
– 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
1027 cm-3
density
– Low Z ions (p, D, T, He3..)
1025 cm-3
1021 cm-3
1023 cm-3
1021 cm-3
10 eV
Weakly
Coupled
102 eV
103 eV
104 eV
Temperature
Iso-contours of ei
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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
Laser
beams
• Transport of matter
• Thermonuclear burn
– Fusion reactivity
σv ~ TiP
– Ion stopping power
• Temperature relaxation
– Electron-radiation coupling
– Electron-ion coupling
σv ~ TiP
… .all in a complex, dynamic
plasma environment ….
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Assumptions of a kinetic theory of radiative transfer and radiationmatter 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
Emission source
of photons

jν  Σ 1  e
A
ν
 hν kT
 B T  σ
ν
ν
Bν T
Planck function
at Telectron
Kirchoff-Planck relation
Weapons and Complex Integration
6
S&T: Scientific motivation
Modeling ICF or astrophysical plasmas, rests on a set of matter- radiation
transport equations coupled to thermonuclear burn and hydrodynamics
3
2
1 I ν (x, Ω, t)
Ω,
t)
d
r
d
 Photon distribution function



 Ω    I ν (x, Ω, t)dn
 σν fTν (r,
B
T

σ
e
ν
e
ν Te  I ν (x, Ω, t)  Compton Scattering
c
t
I ν (r,Ω,
t)  chν f ν (r,Absorption
Ω, t)
Free streamingIntensity
Emission

1
of state
Radiation energy densityU UρC
T (r,t) 2Equation
Ω I ν (r,Ω,
t)
R V  dν  d Material
energy
density
c0
Material heating

1
2
Electron-ion
Material
cooling
due
duet )to radiation
Conductivity
Radiation pressure tensor PR 
dν d Ω Ω ΩI ν (r,Ω,
coupling
to
radiative
losses
c
Source due
 
0
to TN burn


U e
   DeU e   τ ei1 Ui  U e    dν σ ν Te  Bν (T )  d 2Ω I ν (r,Ω, t)  STN
t
i
U i
   DiU i  τ ei1 U e  U i   STN
The temporal
How
does oneevolution
the
of accuracy
plasmas depends
of models
oninthe
regimes
complex
assess
t
difficult
interaction
to access
of collisional,
experimentally
radiative,
and
and
theory
reactive
is difficult
processes
Conductivity
Electron-ion
coupling
Source due
to TN burn
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Kinetic equation I: The Landau kinetic equation is the starting
point for computing electron-ion coupling in hot dense plasmas
3/2
 kT kT 
3μ ab  b  a 
mb ma 

τ ab 
8 2π n b Za2 Z 2b lnΛ ab
Fokker-Planck with
Boltzmann distributions
Ta
1
Tb  Ta 
  ab
t
b

λD
lnΛ  ln
 Max Z 2 e 2 /kT ,λ
5 .0
Q







4 .0
3 .0
2 .0
ln 
1 .0
0 .0
 1 .0
 2 .0
0.01
~ 3.16 10
-10
A
Z a2 Z 2b
Major source of uncertainty
 T 


100
eV


3/2
 1021 cm3 

 sec
 n b lnΛ ab 
Many issues are ignored:
• partial ionization (bound states)
• collective behavior (dynamic screening)
• strong binary collisions/strong coupling
 λ

lnΛ  ln 2 2D 
 Z e /kT 
•quantum effects
•non-Maxwellian distributions
•degeneracy*
0 .1
1 .0
*H. Brysk, Phys. Plasmas 16, 927 (1974)
Temperature (keV)
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The standard model of thermonuclear reaction rates assumes a
Maxwellian distributed weakly coupled plasma
D  T  n 
DDT  p
b
a
X
D  D He  n
T  T    2n
3
Fusion reactivity
v
aX
   dU a dU X f a (U a ) f X (U X ) (U a , U X ) U a  U X
Y
ion distribution
cross section
Non-thermal ion distributions
Gamow
peak
1014
DT cross
section
T=10.4 keV
1015
Boltzmann ion
distributions
Ion
distribution
Bare cross
section
1016


σv
cm3/sec
Dense
plasma
f  f Max
effects
 Z a Z X e2



TD 

2
v Screen
e
v aX
f f
1017
Max
e
 mv2
 δ 
 2kT




1018
2 2


 
m
v
 411 (1997)
f Mod.
f Max Phys.
1   69,
Brown and Sawyer, Rev.

 2kT  
Bahcall

1019 et al., A&A, 383, 291 (2002)
Pollock and Militzer, PRL 92, 021101 (2004)
Temperature (keV)
1020
Velocity (cm/microsecond)
1 .0
10.0
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100 .0
1000 .0
9
S&T: Scientific motivation
A micro-physics approach based on a “bottom-up” approach can
provide insight into the validity of our assumptions
H QED
Galinas and Ott (70)
Degl’Innocenti (74)
Cannon (85)
Graziani (03, 05)
Kinetic Theory
Classical or Wigner Liouville equation
f NEX N 
f NEX Fj f NEX 
  v j 


J
t

r
m

v
j1 
j
i
j 


• Systematic expansion in weakly
3
coupled regime 1/nλ D  1
N-body simulation
• Formal connection to the microphysics (Klimontovich)
• Virtual experiment
• Particle equations of motion are
solved exactly
• All response- and correlation-
• Convergent kinetic theory
functions are non-perturbative
• Multi-physics straightforward
• Closure relations are needed (BBGKY)
• Theory is difficult in strongly coupled
regime
• Approximations are isolated and
understood
• Forces tend to be classical like
• Requires large numbers of particles
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Kinetic equation I: The Landau-Spitzer model of collisional relaxation
rests on the assumptions of a weakly coupled classical plasma
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



f a (v, t)  2π Za2 e 4 
1




2







Z
lnΛ



A
f
(
v
,
t)





D
f
(
v
,
t)


v
ab a
v
v
ab a
2
 b
t
m
2


a

 b

b max
 db/b ~ lnλ
D

/λ th   1
b min
• 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)
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Landau treatment of collisional relaxation with radiation and
burn yields insights into the underlying assumptions
Fokker-Planck treatment of an isotropic, homogeneous
DT plasma with TN burn, Compton and bremsstrahlung
D  T  n 
DDT  p
Michta, Luu, Graziani
D  D 3He  n
T  T    2n
J. S. Chang & G. Cooper 1970, JCP, 6, 1
B. Langdon
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Kinetic equation II: The Lenard - Balescu equation describes a
classical but dynamically screened weakly coupled plasma

   


f a (v, t)  2π Za2e 4 
1
k
k
δ
k
 v  k  v

2
3
3


 


Z
d
v
d
k


b 
2
π
 v
  2
4
t
m

b
a


k ε k  v, k



   f (v , t) f (v, t) - m
 
 

b
v a

 
f a (v, t) vf a (v, t)
mb

a
Requires a model for the dielectric
function of the electron gas
• 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
 

ε
k
,0  1  1/ k 2λ 2D
• Landau equation recovered
Boyd and Sanderson, “Physics of Plasmas”, Cambridge Press (2003)
Weapons and Complex Integration
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The quantum kinetic equations of Kadanoff-Baym and Keldysh
provide the basis for describing strongly coupled complex plasmas
Dense strongly coupled plasma properties:
•Mutual ion-ion and electron-ion interactions are strong
   212

 i

 U a (1) g a  11   dr Σ aHF r1 r1 t1  g a  r1 t1 r1 t1 
 t1 2ma

t1


  d 1 Σ 1 1   Σ 1 1  g

a

a
  
a
t0
11   d 1 Σ a  1 1   Σina 1 1 g a 11  g a 11
t1
t0
Time diagonal K- B equation describes the Wigner distribution
Quantum Landau
RPA self energy with a
statically screened potential
Quantum Lenard-Balescu
RPA self energy (dynamic screening)
• 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
Kremp et al., “Quantum Statistics of Non-ideal Plasmas”, Springer-Verlag (2005)
Weapons and Complex Integration
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More advanced treatments of the electron-ion coupling avoid the
divergence problems of earlier theories
Divergenceless models of
electron ion coupling
Quantum kinetic theory
Gericke-Murillo-Schlanges
Convergent kinetic theory
Brown-Preston-Singleton
 

  2
2 
λ  R ion 
1
ln   ln1   2D

2   λ th Ze2  

 

8π
kT  


 λD  1
lnΛ  ln   ln16π   γ  1
 λ th  2
Short distance Boltzmann
Long distance Lenard-Balescu
Dimensional regularization
Although finite, these theories make assumptions regarding
correlations and hence are still approximate…..
Weapons and Complex Integration
15
N-body simulation techniques based on MD, WPMD or Wigner offer a nonperturbative technique to understanding plasma dynamics
Molecular dynamics
Classical like forces
with effective 2-body
potentials
Wigner equation
Wave packet MD
2π  2
Ze 2
Λa 
sets the short range length scale, not
m a kT
kT
How do we use a particle based simulation to capture short
distance QM effects and long distance classical effects?
Weapons and Complex Integration
16
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:
1
r (t  t)  r (t)  v(t)t  a(t)t 2
2
1
v(t  t)  v(t)  a(t  t)  a(t)t
2
• “data” accumulated with no
thermostat
m
 relaxation phase
Ta (t)  a
3Na
• time step ~0.02/pe
The forces include pure Coulomb,
diffractive, and Pauli terms:

• separate velocity-scale thermostat for
each species during equilibration
phase (~20,000 steps)
 establish two-temperature system
v
2
j,a
j

q q 
 2

pa2

H 
  a b  f (,rab )  exp  rab
  g(,rab )  Te ln(2)exp  rab

ab 
 ln(2)ee2 


a 2ma
ab  rab




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
Weapons and Complex Integration
17
MD has recently been used to investigate electron ion
coupling in hot dense plasmas and validate theoretical models
1
log()
Temperature (eV)
electrons
 pe

ln 
J LS
n  1.61 1024 /cc
protons
Te  91.5eV
Tp  12.1eV
Time (fs)
Temperature (eV)
J.N. Glosli et al., Phys. Rev. E 78 025401(R) 2008.
G. Dimonte and J. Daligault, Phys. Rev. Lett. 101,
L.S. Brown, D.L. Preston, and R.L. Singleton, Jr., Phys.
135001 (2008).
Rep. 410, 237 (2005).
B. Jeon et al., Phys. Rev. E 78, 036403 (2008).
D.O. Gericke, M.S. Murillo, and M. Schlanges, Phys.
Lawrence Livermore National Laboratory
Option:UCRL#
Rev. E 65, 036418 (2002)
18
The MD code predicts a temperature relaxation very different
than what LS or BPS predict…and it should be measurable!
LANL has built an experiment to measure
temperature relaxation in a plasma
SF6 gas jet
53K electrons
6K F
1K S
e heated by laser to 100 eV
ions are heated to 10 eV
1 4π e2 n e
4π Zi2e2 n i


λ 2D
kTe
kTi
i
Te - Thomson Scattering
Ti – Doppler Broadening
Lawrence Livermore National Laboratory
Option:UCRL#
Dominant for
Ti/Te>>1
Dominant for
Te/Ti>>1
Glosli, et al, PRL submitted
19
Modeling matter + radiation: Molecular dynamics coupled to classical
radiation fields is straightforward but is not relevant for hot dense matter
Radiation:
2-electron + 2-proton+radiation
Classical EM fields (Maxwell eqs)
 Lienard-Wiechert Potentials

 

  v  Bi  
1 A
 E i  Φ 
Fi  q i  E i 
c 
c t



qj
q jv j
Φr, t   
Ar, t   


r

r
t
j
j r  rj t ret 
j ret

Normal mode expansion
 


dα k, t
i  
 iωα k, t 
J k, t
dt
2 Ωk

 
 
Problem: Planckian spectrum is not produced
in LTE
Dipole emission
Lawrence Livermore National Laboratory
Option:UCRL#
20
Modeling matter + radiation: Molecular dynamics coupled to
quantum mechanical radiation fields
Photons:

Isotropic and homogeneous spectral
intensity

Kramer’s for emission and
absorption + detailed balance
Spectral
intensity

4 3
I ν t   h ν
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
n t 
ν
1 dI ν t 
 ρ κ ν t  I ν t   ε ν t 
c dt
absorption
• Planckian spectrum in equilibrium

c2
emissivity
RB
Emission and absorption of
radiation is the aggregate of
many binary encounters
Lawrence Livermore National Laboratory
Option:UCRL#
21
Algorithm: Molecular dynamics coupled to either classical or
quantum mechanical radiation fields
Step 0: Begin with the Kramers formulas for emission and absorption
2
2 6
dσ em
32π
Z
e
ν

dhν  3 3 me2c3 ve2 hν
Step 1: Tag a close encounter event and determine
probability of any radiative process
P
σ emiss  σ abs
π R 2B
Integrated Kramers
cross sections
Step 2: If a radiative event occurs, test to decide
emission or absorption
σ em
Pem  em
σ  σ abs
Pabs
σ abs
 em
σ  σ abs
RB
Emission and absorption of
radiation is the aggregate
of many binary encounters
Lawrence Livermore National Laboratory
Option:UCRL#
22
Algorithm: Molecular dynamics coupled to either classical or
quantum mechanical radiation fields
Step 3: Identify energy of photon emission (absorption)
ρ em
ν
 dσ em

ν

n ν  1 π R 2B

 dhν  
Piem hν i  hν  hν i 1  

hν i1
em
ds
ρ
 s
hν i
R

E
em
ds
ρ
 s
Fn
0
n
Fn   P ,
i 1
1
em
i
pick a randomnumber R  0,1  h  F-1 R 
0
i=1
i=n
Emit to
frequency group i
nν
Step 4: Update electron energy and photon population
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LTE test Case: A 3 keV Maxwellian electron plasma produces a
black-body spectrum at 3 keV
Neutral hydrogen plasma
Protons, electrons and photons
Trad=3 keV
I t 
Photon Energy (eV)
A Maxwellian plasma of 3 keV electrons produces a BB spectrum at 3 keV
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Three temperature relaxation problem for a hot hydrogen
plasma agrees well with a continuum code
512e+512p
V = 512 Å3
=1024 cm-3
I t 
Photon Energy (eV)
Glosli et al, J. of Phys. A, 2009
Glosli et al, HEDP, 2009
The dynamics of the spectral
intensity are consistent with the
lower groups coupling faster
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Our initial approach to coupling particle simulations to quantum
radiation fields has both strengths and weaknesses
 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
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Conclusion
We are developing an MD capability that allows us to model the
micro-physics of hot, dense radiative plasmas
 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
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