Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551

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#


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#


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#


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#


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


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


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


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)


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


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


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


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


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)


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…..


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


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#


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#


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#


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#


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#


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#


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#


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#


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#


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#


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

Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551

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Transcript of Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551