Lawrence Livermore National Laboratory Radiation

Transcript Lawrence Livermore National Laboratory Radiation

Lawrence Livermore National Laboratory

LLNL-PRES-412216

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 1 Mbar

Metals WDM n γ T keV  3.13

 10 22 3 T keV  2ρ 1/3 gm/cc P γ Mbar  4 45.7T

keV hot dense

ICF

hot dilute Γ ab  Z a Z b e 2 kT 4π n  1/3 3

Ichimaru plasma coupling

λ a  2π  2 m a kT

Thermal deBroglie wavelength

1 λ 2 D  4π e 2 n e kT e   i 4π Z i 2 e 2 n i kT i

Debye length

n 10 24 10 14 T 1keV 10eV 7.4

1.5

 λ  D 10 10   8 4 2.4

1.4

R   ion 10 10   9 5 9.0

ω P  10 15 6.0

 10 11 Kremp et al.,

“ Quantum Statistics of Non-ideal Plasmas ”

, Springer-Verlag (2005)

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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

–

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

1/nλ 3 D  1 Figure point A B C D E 

2.6

1.2

0.58

0.26

0.10

Strongly coupled plasma:

 –

Large ion-ion and electron-ion correlations

 1 –

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 Option:UCRL#

Hot, dense radiative plasmas are multispecies and involve a variety of radiative, atomic and thermonuclear processes

• • • • •

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

10 29 cm -3 10 27 cm -3 10 25 cm -3 10 21 cm -3 10 23 cm -3 10 21 cm -3 10 eV 10 2 eV 10 3 eV

Temperature Weakly Coupled

10 4 eV

Iso-contours of



ei Lawrence Livermore National Laboratory Option: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

σv ~ T i P

Temperature relaxation

– –

Electron-radiation coupling Electron-ion coupling

σv ~ T i P

… .all in a complex, dynamic plasma environment ….

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Laser beams 5

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 Emission source of photons

j ν  Σ A ν  1  e  hν kT  B ν  σ ν B ν

Kirchoff-Planck relation Weapons and Complex Integration Planck function at T electron

S&T: Scientific motivation Modeling ICF or astrophysical plasmas, rests on a set of matter- radiation transport equations coupled to thermonuclear burn and hydrodynamics

1 c  I ν (x,  t Ω, t) 



Ω  



I ν (x, Ω, t)  σ  ν f

   

e e d  3 r σ d ν 2

 

e I ν (x, Ω, t)  Compton Scattering

Free streaming Intensity

I ν (r, Ω, t)  f ν (r, Ω,  U e  t

Radiation energy density

Radiation pressure tensor coupling

   D e  U e   i τ  1 ei



U i   U ρC R  V 1 c  T(r, 0   d 2 Ω I (r, Ω, t) P R  1 c  0   d Ω Ω ΩI (r,

Material heating

Ω,

Source due

U e



  dν σ ν

 

e  B ν (T)   d 2 Ω I ν (r, Ω,

to TN burn

t)   S TN  U i  t    D i  U i  τ  1 ei  U e  U i   S

Conductivity Electron-ion coupling Source due to TN burn Weapons and Complex Integration

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



T a



 

  1



T b



T a

 5 .

0 lnΛ  ln 4 .

0 3 .

0 ln  2 .

0 1 .

0 0 .

0  1 .

0  2 .

0 0 .

01 λ D Max  Z 2 e 2 /kT, 0 .

Temperature (keV)

λ Q lnΛ  ln λ D Z 2 e 2 /kT     1 .

0 τ ab  8 3μ ab   kT b m b 2π n b Z 2 a  kT m a a   3/2 Z 2 b lnΛ ab

Major source of uncertainty

~ 3.16

 10 10 A Z a 2 Z 2 b 100 T eV 3/2   10 n b 21 cm lnΛ  3 ab   sec

Many issues are ignored:

•

partial ionization (bound states)

•

collective behavior (dynamic screening)

•

strong binary collisions/strong coupling

•

quantum effects

•

non-Maxwellian distributions

•

degeneracy*

*H. Brysk, Phys. Plasmas 16, 927 (1974) 8

Weapons and Complex Integration

The standard model of thermonuclear reaction rates assumes a Maxwellian distributed weakly coupled plasma





n D





  

p D



 3



n T



   2

a X b Y Fusion reactivity



v aX

 

dU a dU X f a

(

U a

)

f X

(

U X

)  (

U a

U X

)

U a



U X

ion distribution cross section Gamow peak T=10.4 keV Ion distribution DT cross section Velocity (cm/microsecond) Boltzmann ion distributions Bare cross section Non-thermal ion distributions

10  14 σv  3  10  15 10  16 10  17 f  f

Max

Screen



v aX



  f 

Z a Z

X e

2  δ



mv 2   2 2kT e Max 10  18 Brown and Sawyer, 10 10  

A&A

, f

383

 f   1   

2   2 Pollock and Militzer, PRL 92 , 021101 (2004)

Temperature (keV)

20   1 .

0 10 .

0 100 .

0 1000 .

0 9

Weapons and Complex Integration

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) Classical or Wigner Liouville equation

 f N  t EX  j N   1    v j   f EX N  r j  F m i j   f N  v EX j     J

Kinetic Theory N-body simulation

•

Systematic expansion in weakly coupled regime

1/nλ 3 D  1 •

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

Weapons and Complex Integration

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 (  t  v , t)    2π Z a 2 e 4 m 2 a    b Z 2 b lnΛ    v    A ab f a (  v , t)   1 2  v    v     D ab f a (  v , t)   b b  max db/b min ~ ln  λ D /λ th   1 •

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) Weapons and Complex Integration

Landau treatment of collisional relaxation with radiation and burn yields insights into the underlying assumptions





 







p D T



 3



   2

Fokker-Planck treatment of an isotropic, homogeneous DT plasma with TN burn, Compton and bremsstrahlung

Michta, Luu, Graziani J. S. Chang & G. Cooper 1970, JCP, 6, 1 B. Langdon

Weapons and Complex Integration

Kinetic equation II: The Lenard - Balescu equation describes a classical but dynamically screened weakly coupled plasma

 f (  v , a  t t)    2π Z 2 a e 4 m 2 a    v   b Z 2 b  d 3 v    1 π  d 3 k k k δ k 4 ε



 k



  k  v    v ,  k  k



 2  v 



   f b  ( v  , t)  v f a (  v , m t) m b a f a (  v , t)  v  f a (  v , t)  

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 Landau equation recovered

ε    1  1 / k 2 λ 2 D Boyd and Sanderson,

“ Physics of Plasmas ”

, Cambridge Press (2003) 13

Weapons and Complex Integration

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

  i    t 1   2  1 2 2m a  U a (1)   g a     t 1 0 t  d 1



Σ a      a



g a      dr Σ HF a  r 1 r 1 t 1  g a     r 1 t 1 r 1  t 1   t 1 0 t  d 1



Σ a     Σ in a



 g  a  g a 

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

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

ln   1 2 ln     1       λ 2 D λ 2 th 8π   R 2 ion Ze 2 kT         

Convergent kinetic theory Brown-Preston-Singleton

lnΛ  ln   λ D λ th    1 2  ln  γ  1 

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

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 Wigner equation Wave packet MD

Λ a  2π  2 m a kT sets the short range length scale, not Ze 2 kT

How do we use a particle based simulation to capture short distance QM effects and long distance classical effects?

Weapons 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:

(

 

) 

(

) 

(

) 

t v

(

 

) 

(

)   1 2

(

) 

2 1 2 

(

 

) 

(

)  

•

separate velocity-scale thermostat for each species during equilibration phase (~20,000 steps)



establish two-temperature system

•

“data” accumulated with no thermostat



relaxation phase

T a

(

) 

m a

N a



j v j

2 ,

The forces include pure Coulomb, diffractive, and Pauli terms:

 

a p a

2 2

m a

•

time step ~0.02/



 



 

q a q b r ab

 

(  ,

r ab

)  exp  

 

(  ,

r ab

) 

T e

ln(2)exp  

 ln(2)  2

       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

MD has recently been used to investigate electron ion coupling in hot dense plasmas and validate theoretical models

electrons  1

 ln 

J LS

protons n  1.61

 10 24 /cc T T p e  91.5

eV  12.1

Time (fs)

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).

Lawrence Livermore National Laboratory Option:UCRL# Temperature (eV)

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) 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 SF 6 gas jet 53K electrons 6K F 1K S e heated by laser to 100 eV ions are heated to 10 eV

i - Thomson Scattering – Doppler Broadening Lawrence Livermore National Laboratory Option:UCRL#

λ 1 2 D  4π e 2 n e kT e Glosli, et al, PRL submitted

Dominant for Ti/Te>>1

  i 4π Z i 2 e 2 n i kT i

Dominant for Te/Ti>>1

Modeling matter + radiation: Molecular dynamics coupled to classical radiation fields is straightforward but is not relevant for hot dense matter



Radiation: Classical EM fields (Maxwell eqs)



Lienard-Wiechert Potentials

 F i  q i    E i Φ     j   v   B i c    E i r  r j q j   ret  A   Φ   1   A  j c r  t  q r j  j v t j   ret

Normal mode expansion

d  dt  iω   i 2 Ω k  J  2-electron + 2-proton+radiation 

Problem: Planckian spectrum is not produced in LTE

Dipole emission

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   

Modeling matter + radiation: Molecular dynamics coupled to quantum mechanical radiation fields

Photons: Isotropic and homogeneous spectral Spectral intensity

I ν   h 4 ν 3 c 2  n ν

intensity Kramer’s for emission and absorption + detailed balance

1 dI ν c dt  ρ κ ν  ε ν

absorption emissivity

•

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

R B

Emission and absorption of radiation is the aggregate of many binary encounters Lawrence Livermore National Laboratory Option:UCRL#

Algorithm: Molecular dynamics coupled to either classical or quantum mechanical radiation fields

Step 0: Begin with the Kramers formulas for emission and absorption

dσ d em ν

 

 32π 2 3 3 Z 2 e 6 m 2 e c 3 v 2 e hν

Step 1: Tag a close encounter event and determine probability of any radiative process

P  σ emiss  π R 2 B σ abs

Integrated Kramers cross sections

R B

Step 2: If a radiative event occurs, test to decide emission or absorption

P em  σ em σ em  σ abs P abs  σ em σ abs  σ abs

Lawrence Livermore National Laboratory Option:UCRL# Emission and absorption of radiation is the aggregate of many binary encounters

Algorithm: Molecular dynamics coupled to either classical or quantum mechanical radiation fields

Step 3: Identify energy of photon emission (absorption) 1

P i em



hν i  ρ em ν hν     dσ d hν i

  

 1 em ν



   hν n i  1 ν  1 ds ρ s



em hν  i   B E 0  ds ρ em s F n  i n   1 P i em , pick a random number R     h   F 1

 

F n 0

n ν

R i=1 Emit to frequency group i i=n Step 4: Update electron energy and photon population Lawrence Livermore National Laboratory Option:UCRL#

LTE test Case: A 3 keV Maxwellian electron plasma produces a black-body spectrum at 3 keV

Neutral hydrogen plasma Protons, electrons and photons T rad =3 keV



 

Photon Energy (eV) A Maxwellian plasma of 3 keV electrons produces a BB spectrum at 3 keV Lawrence Livermore National Laboratory Option:UCRL#

Three temperature relaxation problem for a hot hydrogen plasma agrees well with a continuum code

512e+512p V = 512 Å 3  =10 24 cm -3



 

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 Lawrence Livermore National Laboratory Option:UCRL#

  

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 Lawrence Livermore National Laboratory Option:UCRL#

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 Lawrence Livermore National Laboratory Option:UCRL#

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