Introduction:first­principles methods for warm dense matter (WDM)

First-Principles Equations of State Calculations of First- and Second-Row PlasmasK. P. Driver1, F. Soubiran1, S. Zhang1, and B. Militzer,1,2

1Department of Earth and Planetary Science, University of California, Berkeley2Department of Astronomy, University of California, Berkeley

100101102103104105106107108109101010111012

100 101 102 103 10410­110­210­310­4

Tem

pera

ture

 (K)

CondensedMatter

CondensedMatter

Warm DenseMatter/PlasmaWarm Dense

Matter/Plasma

Hot DenseMatter/Plasma

Hot DenseMatter/Plasma

RHIC/LHC – quark­gluon plasma

NIF

Sun’s core

Analytic Plasma Methods

Analytic Plasma Methods

Earth’s coreJupiter’s core, DARHT

NIF ignition

Lightningdischarge

WhiteDwarfZ­machine

[1] B. Militzer and D.M. Ceperley, Phys. Rev. E 63, 066404 (2001).[2] B. Militzer, Phys. Rev. B 79, 155105 (2009).[3] K.P. Driver and B. Militzer Phys. Rev. Lett. 108, 115502 (2012)[4] L.X. Benedict, K.P. Driver, et al., Phys. Rev. B, 89, 224109 (2014)[5] K. P. Driver and B. Militzer, under review in Phys. Rev. B[6] K. P. Driver, F. Soubiran, Shuai Zhang, and B. Militzer, J. Chem. Phys. 143, 164507 (2015)[7] K. P. Driver and B. Militzer, Phys. Rev. B 91, 045103 (2015)[8] B. Militzer and K. P. Driver, Phys. Rev. Lett. 115, 176403 (2015)

●Financial support provided by the DOE, NSF, and UC Berkeley.●Computational support provided by NCAR and NERSC

PlasmaModels

No ab­initio method exists beyond He.(PIMC, OF­DFT)

StandardKohn­ShamDFT

Simulation Method

Nitrogen Dissociation

Comparison of Pressure, Energy, and Hugoniot Curves 

All-electron studies reveal the L-shell is pressure ionized, but the K-shell is not (up to 100g/cm3).A more rapid shift in the Fermi energy relative to the K-shell energy as density increases results in adecrease in the K-shell temperature-ionization fraction.

The blue region marks the range of consistent overlap between DFT-MD and PIMC.The slope of the pressure and energy curves softens near 106 K, corresponding to the onset of K-shell ionizationand shifts with increasing Z due to increased binding energy.

Temperature­Pressure­Density Conditions of N, O, Si Simulations

Conclusions

Acknowledgements

We have computed the EOS of H[1], He[2], C[3,4], N[5], O[6], Ne[7] and Si[8] in the liquid, WDM, and plasma regimes for a wide range of temperatures and densities.PIMC simulations of first-row elements use a free-particle nodal surface. Valid when 1s are fully occupied and 2s states are partially occupied.PIMC simulations of second-row elements (Si) required development of new localized nodal surfaces in order to properly treat bound and partially ionized states.

Hugoniot curve maxima shift with increasing Z due toincreased binding energy.

Density (g/cm3)

In WDM, bonding, ionization, dissociation, and quantum degeneracy are all important.DFT-MD and PIMC together can produce a coherent EOS bridging the WDM regime.WDM is important for inertial confinement fusion, planetary cores, and solar physics.

We have a PIMC method that produces accurate results for WDMfor first and second row elements.

PIMC and DFT-MD together form a coherent equation of state from condensed matter to the plasma limit. PIMC pressures, internal energies,and pair correlation functions agree with DFT-MD near 106 K.

Nitrogen Oxygen Silicon

105

First­Principles Methodology

DFT­MD: Kohn and Pople nobel prize (1998)

PIMC: Based on Feynman's path integral formalism of quantum statistics (1940)

Nitrogen Phase Diagram Nitrogen Hugoniot

K­shell Ionization in Warm, Dense Oxygen

First­order liquid­liquid dissociation transition; doubly shocked cooling phenomena.Hugoniot curve shows sharp increase in compressibility the dissociation regime; DFT agrees well with experiments.

N(r

)

Oxygen Oxygen

K­shell

L­shell

ETOT=T [n]+ Eion [n]+ EH [n]+ EXC [n]

H ψi=[12

∇2+ V eff (r )]ψi=ϵi (r )

V eff =V ion+V H +V XC

Maps many-body problem to single-particle framework (Hohenberg-Kohn)Computational efficiency decreases with temperature due to occ. ortibals.Exchange-Correlation functionals: LDAs, GGAs (Designed for T=0 K).Solve Newton equations of motion for Molecular Dynamics

ρF (R ,R ' ;β)=1N !

∑℘

(−1)℘ ∫

R→℘R ,ρT

e−S [R t] d R t

ρ=e−β H =[e−τ H ]M

Z=Tr [ ρF ]Partition Function:

Density matrix:

Temperature explicitly included in the formulation.Computational efficiency increases with temperature (shorter paths).Historically, PIMC focused on H and He; we now extend it to 2nd row.

ρT(R , R ' ;β)>0

Fermion Sign problem: permutation summation instability of positive and negative terms.Solution: restrict the simulation to a uniform nodal cell of a trial density matrix.

(Choose either free particle[5] and localized nodal surface[6])

DFT­MDdissociation

References: