1. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. Material Properties at High Energy Density from Quantum Molecular Dynamics Symposium on Recent Advances in Plasma Physics --- Ron Davidson Celebration PPPL, Princeton, June 10-12, 2007 Michael Desjarlais Science Campaigns/HEDP Theory Sandia National Laboratories

2. Sandia’s Z Machine is used for several HEDP experimental campaigns Z pinches for Inertial Confinement Fusion Magnetically launched flyer plates for EOS studies (Hugoniots) Ramp wave compression for EOS studies (isentropes) The Z machine delivers more than 20 million Amperes of current ZR will generate more than 25 MA

3. HEDP computer simulations rely on “physics packages”: Conductivities, Equations of State, and Opacities This highly structured portion of phase space is Warm Dense Matter Log T (K) Log  (kg/m 3 ) Z* =1 Melt line Tungsten Conductivity Vapor dome Log  Definitions of Warm Dense Matter are varied, but generally center around strongly coupled ions and moderately degenerate electrons --- many different interactions are comparable. A quantum mechanical treatment is generally necessary. Most wide-range equations of state interpolate through this difficult area.

4. Modifications of the Lee-More conductivity algorithms were made to obtain an improved wide-range model* Copper at 6000 K (LMD) * M. P. Desjarlais, Contrib. Plasma Phys. 41 (2001) 2-3, 267-270 Data from Alan DeSilva’s exploding wire experiments

5. A demanding application: Ultra-high velocity magnetically launched flyer plates (30 km/sec, Multi-Mbar pressures) Anode/Flyer Plate Target

6. We simulated these magnetically launched flyer plates using the modified Lee-More (LMD) conductivities Detailed comparison between simulations and experiments for magnetically launched flyer plates suggested that our improved conductivities were still not sufficiently accurate for the warm dense liquid aluminum. For many of our applications we require conductivities accurate to well within a factor of two. Simulations by Ray Lemke with Sandia’s 3D Rad-MHD ALEGRA code

7. "On two occasions I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." - Charles Babbage (1791-1871) “ The general theory of quantum mechanics is now almost complete … and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. ” - P.A.M. Dirac, Proc. R. Soc. London Ser. A 123, 714 (1929).

8. electron interaction external potential Schrödinger view Formally equivalent DFT view Kohn-Sham particle effective potential (non-interacting) Density Functional Theory (DFT) is a formally exact representation of the N electron Schrödinger Equation “Easy” problem to solve ( scales like N 3 or better) Hard problem to solve (scales like N 5 ) Hohenberg and Kohn proved this (1964 ) Nobel prize in Chemistry for Kohn in 1998

9. We are using Density Functional Theory (DFT) to perform Quantum Molecular Dynamics (QMD) simulations of Warm Dense Matter The simulations are performed with VASP (Vienna Ab initio Simulation Program), a plane wave density functional code (with in-house modifications for our HEDP applications) Exchange and Correlation functionals are Generalized Gradient Approximation (GGA/PBE) We typically use up to 256 atoms, but it varies depending on density and the number of electrons we need to carry (the valence); We use Projector Augmented Wave (PAW) all-electron, frozen core potentials for the atoms We generally perform our simulations in the Canonical Ensemble (N,V,T) using either velocity scaling or a Nosé-Hoover thermostat to regulate the temperature; Fermi statistics for the electrons Typical runs cover one to twenty picoseconds

10. The Kubo-Greenwood formula is used to calculate the frequency dependent electrical conductivity where e and m are the electron charge and mass. The i and j summations are over the N discrete bands of the triply periodic calculation for the cubic supercell with volume . The coordinate index is  and in general we average over  to improve the statistics. F(  i,k ) is the Fermi weight corresponding to the energy for the I-th band at k with wavefunction  i,k. We integrate over the Brillouin zone using the method of special k-points, and average over 10 to 20 configurations selected from the MD run. Other optical properties are obtained through the Kramers-Krönig relations for 

11. At liquid aluminum densities just below solid, the optical conductivity is well fit by the Drude model The agreement with the Drude model indicates ‘nearly free’ electrons Ion cores displayed with iso-surfaces of the mean valence charge density

12. At lower density, where phase separation is pronounced, a gap begins to form at low energy The dc conductivity has dropped by a factor of 25 for a factor of 4 drop in density Note the pronounced separation into liquid and void (vapor) regions

13. At low densities and higher temperatures, a Drude component reemerges in the optical conductivity The peak at about 5.8 eV is characteristic of a 3s->3p transition of the isolated ion. The optical conductivity for 0.025 g/cc at 30000 K We have gone from condensed matter to an ideal plasma!

14. The QMD-KG results are in good agreement with Alan DeSilva’s exploding wire data [Desjarlais, Kress, and Collins, PRE 66, 025401 (2002)]

15. The calculated liquid aluminum conductivities are higher than the improved Lee-More (LMD) model predictions The dashed line shows the 2000 K isotherm from our QMD-tuned wide-range aluminum model Dense solid ( 5 g/cc ) Liquid Melt LMD model

16. Flyer plate simulations with the QMD based conductivities give very good agreement with experiment Conductivities based on the QMD calculations have given us a new predictive capability Simulations by Ray Lemke with Sandia’s 3D Rad-MHD ALEGRA code

17. The absorption has the proper Hagen-Rubens low frequency behavior, consistent with the dc conductivity  Hagen-Rubens Kramer’s free-free ~ -3 (most opacity codes) 4s  4p 3d  4p Free electrons (Drude Z ~ 0.6) We are using these results to improve our low energy opacities

18. Wavefunctions  Eigenvalues  We are using quantum molecular dynamics simulations with density functional theory to build manifestly consistent conductivity, EOS, and opacity models Stainless Steel (Fe, Cr, Ni) Low energy opacities Conductivities

19. Recent and active research areas Principal Hugoniot and reshock properties of deuterium Principal Hugoniot and Isentrope of aluminum, release isentropes QMD based conductivity models for Al, W, Be, and stainless steel Shock melting of Be and diamond (National Ignition Campaign) Electrical and thermodynamic properties of water at high energy densities Liquid-vapor critical points for Al, W, and stainless steel Research on advanced electronic structure methods for HEDP (finite temperature Exact Exchange, finite temperature GW) Special thanks to my close collaborators: Marcus Knudson and Thomas Mattsson