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

1. LLNL-PRES-412216 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 Theoretical and Computational Approaches to Hot Dense Radiative Plasmas Institute for Pure and Applied Mathematics, UCLA Computational Kinetic Transport and Hybrid Methods 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

2. Matter at extreme conditions: High energy density plasmas common to ICF and astrophysics are hot dense plasmas with complex properties WDM hot dense Ichimaru plasma coupling ICF 1 Mbar Thermal deBroglie wavelength Metals Debye length Prad=45.7 Mbar (T4(keV)) WDM hot dilute Kremp et al., “Quantum Statistics of Non-ideal Plasmas”,Springer-Verlag (2005)

3. Hot dense plasmas span the weakly coupled (Brownian motion like) to strongly coupled (large particle-particle correlations) regimes • 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 density-temperature trajectory of the DT gas in an ICF capsule

4. 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: • 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 Strongly Coupled Moderately Coupled Strongly Coupled density Moderately Coupled Weakly Coupled Weakly Coupled Temperature Iso-contours of Gei

5. 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 ….

6. 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 Planck function at Telectron Emission source of photons Kirchoff-Planck relation

7. S&T: Scientific motivation Modeling ICF or astrophysical plasmas, rests on a set of matter- radiation transport equations coupled to thermonuclear burn and hydrodynamics Photon distribution function Equation of state Material energy density Intensity Material heating due to radiation Electron-ion coupling Material cooling due to radiative losses Conductivity Source due to TN burn Free streaming Emission Absorption Radiation energy density Radiation pressure tensor Electron-ion coupling Source due to TN burn Conductivity 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 difficult

8. 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 Major source of uncertainty • 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. Plasmas16, 927 (1974) Temperature (keV)

9. The standard model of thermonuclear reaction rates assumes a Maxwellian distributed weakly coupled plasma Fusion reactivity b a X Y ion distribution cross section Non-thermal ion distributions Gamow peak Dense plasma effects DT cross section T=10.4 keV Boltzmann ion distributions Ion distribution Bare cross section Brown and Sawyer, Rev. Mod. Phys.69, 411 (1997) Bahcall et al., A&A, 383, 291 (2002) Pollock and Militzer, PRL92, 021101 (2004) Temperature (keV) Velocity (cm/microsecond)

10. S&T: Scientific motivation A micro-physics approach based on a “bottom-up” approach can provide insight into the validity of our assumptions Classical or Wigner Liouville equation Galinas and Ott (70) Degl’Innocenti (74) Cannon (85) Graziani (03, 05) Kinetic Theory N-body simulation • 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. 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 • 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)

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

13. Kinetic equation II: The Lenard - Balescu equation describes a classical but dynamically screened weakly coupled plasma 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 Boyd and Sanderson, “Physics of Plasmas”,Cambridge Press (2003)

14. 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 Time diagonal K- B equation describes the Wigner distribution Quantum Landau Quantum Lenard-Balescu RPA self energy with a statically screened potential 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)

15. 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 Short distance Boltzmann Long distance Lenard-Balescu Dimensional regularization Although finite, these theories make assumptions regarding correlations and hence are still approximate…..

16. N-body simulation techniques based on MD, WPMD or Wigner offer a non-perturbative technique to understanding plasma dynamics Molecular dynamics Wigner equation Classical like forces with effective 2-body potentials Wave packet MD How do we use a particle based simulation to capture short distance QM effects and long distance classical effects?

17. 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: • 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/pe The forces include pure Coulomb, diffractive, and Pauli terms: 125M particles on 131K processors • 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

18. MD has recently been used to investigate electron ion coupling in hot dense plasmas and validate theoretical models electrons Temperature (eV) log(L) protons 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, 135001 (2008). B. Jeon et al., Phys. Rev. E 78, 036403 (2008). 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)

19. 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 Te - Thomson Scattering Ti – Doppler Broadening Dominant for Ti/Te>>1 Dominant for Te/Ti>>1 Glosli, et al, PRL submitted

20. 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 Radiation: Classical EM fields (Maxwell eqs) • Lienard-Wiechert Potentials • Normal mode expansion • Problem: Planckian spectrum is not produced in LTE Dipole emission

21. Emission and absorption of radiation is the aggregate of many binary encounters Modeling matter + radiation: Molecular dynamics coupled to quantum mechanical radiation fields Spectral intensity 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 absorption emissivity

22. Algorithm: Molecular dynamics coupled to either classical or quantum mechanical radiation fields Step 0: Begin with the Kramers formulas for emission and absorption Step 1: Tag a close encounter event and determine probability of any radiative process Integrated Kramers cross sections Step 2: If a radiative event occurs, test to decide emission or absorption Emission and absorption of radiation is the aggregate of many binary encounters

23. Algorithm: Molecular dynamics coupled to either classical or quantum mechanical radiation fields 1 Step 3: Identify energy of photon emission (absorption) R Fn 0 i=1 i=n Emit to frequency group i Step 4: Update electron energy and photon population

24. 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 Photon Energy (eV) A Maxwellian plasma of 3 keV electrons produces a BB spectrum at 3 keV

25. Three temperature relaxation problem for a hot hydrogen plasma agrees well with a continuum code 512e+512p V = 512 Å3 =1024 cm-3 Photon Energy (eV) The dynamics of the spectral intensity are consistent with the lower groups coupling faster Glosli et al, J. of Phys. A, 2009 Glosli et al, HEDP, 2009

26. 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

27. 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