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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 1 Mbar 2π 2 λa m a kT γ 3 n1/cc 3.13 10 22 TkeV WDM TkeV 2ρ1/3 gm/cc P 45.7T 4 keV Prad=45.7 Mbar (T4(keV)) Metals γ Mbar Za Z b e 2 4π n Γ ab kT 3 Ichimaru plasma coupling Thermal deBroglie wavelength 1 4π e 2 n e 4π Zi2e 2 n i λ 2D kTe kTi i n hot dilute T λ Debye length R D ion 24 8 10 1keV 7.4 10 2.4 10 9 1014 10eV 1.5 10 4 1.4 10 5 ω P 9.0 1015 6.0 1011 Kremp et al., “Quantum Statistics of Non-ideal Plasmas”, Springer-Verlag (2005) Lawrence Livermore National Laboratory Option:UCRL# 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 Option:UCRL# 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: 10 29 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 10 27 cm -3 density – Low Z ions (p, D, T, He3..) 10 25 cm -3 10 21 cm -3 10 23 cm -3 10 21 cm -3 10 eV Weakly Coupled 10 2 eV 103 eV 10 4 eV Temperature Iso-contours of ei Lawrence Livermore National Laboratory Option:UCRL# 4 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 …. Lawrence Livermore National Laboratory Option:UCRL# 5 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 B ν 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 σν fTν (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 T(r, t) 2Equation Radiation energy densityU UρC Ω I ν (r, Ω,energy t) R V dν d Material density c0 Material heating 1 2 Electron-ion Material cooling due due Conductivity Radiation pressure tensor PR dν d Ω Ω ΩI ν (r, Ω, t) to radiation coupling to radiative losses c Source due 0 to TN burn U e DeU e τ ei1 U i U e dν σ ν Te Bν (T) d 2Ω I ν (r, Ω, t) STN t i U i DiU i τ ei1 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 Weapons and Complex Integration 7 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 ln 4.0 3.0 2.0 1.0 0.0 1.0 2.0 0.01 ~ 3.16 10-10 Major source of uncertainty A T 2 2 Za Z b 100 eV 3/2 10 21 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) Weapons and Complex Integration 8 The standard model of thermonuclear reaction rates assumes a Maxwellian distributed weakly coupled plasma D T n DDT 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 TD 2 v Screen e v aX f f 10 17 Max e mv 2 δ 2kT 10 18 2 2 mv f Mod. f Max Phys. 1 69, 411 (1997) 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 Weapons and Complex Integration 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 j1 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 Weapons and Complex Integration 10 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) Weapons and Complex Integration 11 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 DDT p Michta, Luu, Graziani D D3He n T T 2n J. S. Chang & G. Cooper 1970, JCP, 6, 1 B. Langdon Weapons and Complex Integration 12 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 2 2 π v b 4 t b ma k ε k v, k f (v, t) f (v, t) - m b v a f a (v, t) vf 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 2 λ 2D • Landau equation recovered Boyd and Sanderson, “Physics of Plasmas”, Cambridge Press (2003) Weapons and Complex Integration 13 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 212 i U a (1) g a 11 dr Σ aHF r1 r1 t1 g a r1 t1 r1 t1 t1 2m a t1 d 1 Σ 1 1 Σ 1 1 g a a a t0 11 d 1 Σ a 1 1 Σ ina 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 14 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 Ze 2 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 3N a • 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 qa qb rab2 pa2 r ab H f (,rab ) exp g(,rab ) Te ln(2)exp 2 ln(2) ab 2m r ee a a ab ab 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 1.6110 24 /cc Te 91.5 eV protons Tp 12.1 eV 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 e 2 n e 4π Zi2e 2 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 Φr, t Ar, 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 dhν 3 3 m e2c3 v e2 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) ρem ν dσ em ν n ν 1 π R 2B dhν Piem hν i hν hν i 1 hν i1 em ds ρ s hν i R E em ds ρ s Fn 0 n Fn P , i 1 1 em i pick a random number 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 Lawrence Livermore National Laboratory Option:UCRL# 23 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 Lawrence Livermore National Laboratory Option:UCRL# 24 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 Lawrence Livermore National Laboratory Option:UCRL# 25 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# 26 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# 27