First-principles thermal conductivity of warm-dense deuterium plasmas for inertial confinement fusion applications

Dynamic and transient processes in warm dense matter

In this paper, we discuss some of the key challenges in the study of time-dependent processes and non-equilibrium behaviour in warm dense matter. We outline some of the basic physics concepts that have underpinned the definition of warm dense matter as a subject area in its own right and then cover, in a selective, non-comprehensive manner, some of the current challenges, pointing along the way to topics covered by the papers presented in this volume. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Cylindrical and Spherical Nucleus-Acoustic Solitary and Shock Waves in Degenerate Electron-Nucleus Plasmas

The basic characteristics of cylindrical as well as spherical solitary and shock waves in degenerate electron-nucleus plasmas are theoretically investigated. The electron species is assumed to be cold, ultra-relativistically degenerate, negatively charged gas, whereas the nucleus species is considered a cold, non-degenerate, positively charged, viscous fluid. The reductive perturbation technique is utilized in order to reduce the basic equations (governing the degenerate electron-nucleus plasmas under consideration) to the modified Korteweg-de Vries and Burgers equations. The latter are numerically solved and analyzed to detect the basic characteristics of solitary and shock waves in such electron-nucleus plasmas. The nonlinear nucleus-acoustic waves are found to be propagated in the form of solitary as well as shock waves in such degenerate electron-nucleus plasmas. Their basic properties as well as their time evolution are significantly modified by the effects of cylindrical as well as spherical geometries. The results of this study is expected to be applicable not only to astrophysical compact objects, but also to ultra-cold dense plasmas produced in laboratory.

Finite-temperature density-functional-theory investigation on the nonequilibrium transient warm-dense-matter state created by laser excitation

We present a finite-temperature density-functional-theory investigation of the nonequilibrium transient electronic structure of warm dense Li, Al, Cu, and Au created by laser excitation. Photons excite electrons either from the inner shell orbitals or from the valence bands according to the photon energy, and give rise to isochoric heating of the sample. Localized states related to the 3d orbital are observed for Cu when the hole lies in the inner shell 3s orbital. The electrical conductivity for these materials at nonequilibrium states is calculated using the Kubo-Greenwood formula. The change of the electrical conductivity, compared to the equilibrium state, is different for the case of holes in inner shell orbitals or the valence band. This is attributed to the competition of two factors: the shift of the orbital energies due to reduced screening of core electrons, and the increase of chemical potential due to the excitation of electrons. The finite-temperature effect of both the electrons and the ions on the electrical conductivity is discussed in detail. This work is helpful to better understand the physics of laser excitation experiments of warm dense matter.

Particle-in-cell simulation method for macroscopic degenerate plasmas

Nowadays hydrodynamic equations coupled with external equation of states provided by quantum mechanical calculations is a widely used approach for simulations of macroscopic degenerate plasmas. Although such an approach is proven to be efficient and shows many good features, especially for large scale simulations, it encounters intrinsic challenges when involving kinetic effects. As a complement, here we have invented a fully kinetic numerical approach for macroscopic degenerate plasmas. This approach is based on first principle Boltzmann-Uhling-Uhlenbeck equations coupled with Maxwell's equation, and is eventually achieved via an existing particle-in-cell simulation code named LAPINS. In this approach, degenerate particles obey Fermi-Dirac statistics and nondegenerate particles follow the typical Maxwell-Boltzmann statistics. The equation of motion of both degenerate and nondegenerate particles are governed by long range collective electromagnetic fields and close particle-particle collisions. Especially, Boltzmann-Uhling-Uhlenbeck collisions ensure that evolution of degenerate particles is enforced by the Pauli exclusion principle. The code is applied to several benchmark simulations, including electronic conductivity for aluminium with varying temperatures from 2 eV to 50 eV, thermalization of alpha particles in a cold fuel shell in inertial confinement fusion, and rapid heating of solid sample by short and intense laser pulses.

Multiple scattering theory for dense plasmas

Dense plasmas occur in stars, giant planets, and in inertial fusion experiments. Accurate modeling of the electronic structure of these plasmas allows for prediction of material properties that can in turn be used to simulate these astrophysical objects and terrestrial experiments. But modeling them remains a challenge. Here we explore the Korringa-Kohn-Rostoker Green's function (KKR-GF) method for this purpose. We find that it is able to predict equation of state in good agreement with other state-of-the-art methods, where they are accurate and viable. In addition, it is shown that the computational cost does not significantly change with temperature, in contrast with other approaches. Moreover, the method does not use pseudopotentials-core states are calculated self consistently. We conclude that KKR-GF is a very promising method for dense plasma simulation.

Model of electron transport in dense plasmas spanning temperature regimes

We present a new model of electron transport in warm and hot dense plasmas which combines the quantum Landau-Fokker-Planck equation with the concept of mean-force scattering. We obtain electrical and thermal conductivities across several orders of magnitude in temperature, from warm dense matter conditions to hot, nondegenerate plasma conditions, including the challenging crossover regime between the two. The small-angle approximation characteristic of Fokker-Planck collision theories is mitigated to good effect by the construction of accurate effective Coulomb logarithms based on mean-force scattering, which allows the theory to remain accurate even at low temperatures, as compared with high-fidelity quantum simulation results. Electron-electron collisions are treated on equal footing as electron-ion collisions. Their accurate treatment is found to be essential for hydrogen, and is expected to be important to other low-Z elements. We find that electron-electron scattering remains influential to the value of the thermal conductivity down to temperatures somewhat below the Fermi energy. The accuracy of the theory seems to falter only for the behavior of the thermal conductivity at very low temperatures due to a subtle interplay between the Pauli exclusion principle and the small-angle approximation as they pertain to electron-electron scattering. Even there, the model is in fair agreement with ab initio simulations.

Correlations between conduction electrons in dense plasmas

Most treatments of electron-electron correlations in dense plasmas either ignore them entirely (random phase approximation) or neglect the role of ions (jellium approximation). In this work, we go beyond both these approximations to derive a formula for the electron-electron static structure factor which properly accounts for the contributions of both ionic structure and quantum-mechanical dynamic response in the electrons. The result can be viewed as a natural extension of the quantum Ornstein-Zernike theory of ionic and electronic correlations, and it is suitable for dense plasmas in which the ions are classical and the conduction electrons are quantum-mechanical. The corresponding electron-electron pair distribution functions are compared with the results of path integral Monte Carlo simulations, showing good agreement whenever no strong electron resonance states are present. We construct approximate potentials of mean force which describe the effective screened interaction between electrons. Significant deviations from Debye-Hückel screening are present at temperatures and densities relevant to high-energy density experiments involving warm and hot dense plasmas. The presence of correlations between conduction electrons is likely to influence the electron-electron contribution to the electrical and thermal conductivity. It is expected that excitation processes involving the conduction electrons (e.g., free-free absorption) will also be affected.

Testing thermal conductivity models with equilibrium molecular dynamics simulations of the one-component plasma

Equilibrium molecular dynamics simulations are used to calculate the thermal conductivity of the one-component plasma via the Green-Kubo formalism over a broad range of Coulomb coupling strength, 0.1≤Γ≤180. These simulations address previous discrepancies between computations using equilibrium versus nonequilibrium methods. Analysis of heat flux autocorrelation functions show that very long (6×10^{5}ω_{p}^{-1}) time series are needed to reduce the noise level to allow ≲2% accuracy. The simulations provide accurate data for Γ≲1. This enables a test of the traditional Landau-Spitzer theory, which is found to agree with the simulations for Γ≲0.3. It also enables tests of theories to address moderate and strong Coulomb coupling. Two are found to provide accurate extensions to the moderate coupling regime of Γ≲10, but none are accurate in the Γ≳10 regime where potential energy transport and coupling between mass flow and stress dominate thermal conduction.

Heat-release equation of state and thermal conductivity of warm dense carbon by proton differential heating

Warm dense carbon is generated at 0.3-2.0 g/cc and 1-7 eV by proton heating. The release equation of state (EOS) after heating and thermal conductivity of warm dense carbon are studied experimentally in this regime using a Au/C dual-layer target to initiate a temperature gradient and two picosecond time-resolved diagnostics to probe the surface expansion and heat flow. Comparison between the data and simulations using various EOSs and thermal conductivity models is quantified with a statistical χ^{2} analysis. Out of seven EOS tables and five thermal conductivity models, only L9061 with the Lee-More model provides a probability above 50% to match all data.

Ab Initio Studies on the Stopping Power of Warm Dense Matter with Time-Dependent Orbital-Free Density Functional Theory

Electronic transport properties of warm dense matter, such as electrical or thermal conductivities and nonadiabatic stopping power, are of particular interest to geophysics, planetary science, astrophysics, and inertial confinement fusion (ICF). One example is the α-particle stopping power of dense deuterium-tritium (DT) plasmas, which must be precisely known for current small-margin ICF target designs to ignite. We have developed a time-dependent orbital-free density functional theory (TD-OF-DFT) method for ab initio investigations of the charged-particle stopping power of warm dense matter. Our current dependent TD-OF-DFT calculations have reproduced the recently well-characterized stopping power experiment in warm dense beryllium. For α-particle stopping in warm and solid-density DT plasmas, the ab initio TD-OF-DFT simulations show a lower stopping power up to ∼25% in comparison with three stopping-power models often used in the high-energy-density physics community.

Experimental observation of the stratified electrothermal instability on aluminum with thickness greater than a skin depth

A direct observation of the stratified electrothermal instability on the surface of thick metal is reported. Aluminum rods coated with 70μm Parylene-N were driven to 1 MA in 100ns, with the metal thicker than the skin depth. The dielectric coating suppressed plasma formation, enabling persistent observation of discrete azimuthally correlated stratified thermal perturbations perpendicular to the current whose wave numbers, k, grew exponentially with rate γ(k)=0.06ns^{-1}-(0.4ns^{-1}μm^{2}rad^{-2})k^{2} in ∼1g/cm^{3}, ∼7000K aluminum.

Thermal conductivity measurements of proton-heated warm dense aluminum

Thermal conductivity is one of the most crucial physical properties of matter when it comes to understanding heat transport, hydrodynamic evolution, and energy balance in systems ranging from astrophysical objects to fusion plasmas. In the warm dense matter regime, experimental data are very scarce so that many theoretical models remain untested. Here we present the first thermal conductivity measurements of aluminum at 0.5–2.7 g/cc and 2–10 eV, using a recently developed platform of differential heating. A temperature gradient is induced in a Au/Al dual-layer target by proton heating, and subsequent heat flow from the hotter Au to the Al rear surface is detected by two simultaneous time-resolved diagnostics. A systematic data set allows for constraining both thermal conductivity and equation-of-state models. Simulations using Purgatorio model or Sesame S27314 for Al thermal conductivity and LEOS for Au/Al release equation-of-state show good agreement with data after 15 ps. Discrepancy still exists at early time 0–15 ps, likely due to non-equilibrium conditions.

Multidimensional Chebyshev interpolation for warm and hot dense matter

We propose a scheme based on a multidimensional Chebyshev interpolation to approximate smooth functions that depend on more than one variable. The present method generalizes the one dimensional Chebyshev approximation. The multidimensional approach can be used for generating databases like equation of state in the warm and hot dense matter. It is well suited to the present advance of massively parallel supercomputers.

Density-functional calculations of transport properties in the nondegenerate limit and the role of electron-electron scattering

We compute electrical and thermal conductivities of hydrogen plasmas in the nondegenerate regime using Kohn-Sham density functional theory (DFT) and an application of the Kubo-Greenwood response formula, and demonstrate that for thermal conductivity, the mean-field treatment of the electron-electron (e-e) interaction therein is insufficient to reproduce the weak-coupling limit obtained by plasma kinetic theories. An explicit e-e scattering correction to the DFT is posited by appealing to Matthiessen's Rule and the results of our computations of conductivities with the quantum Lenard-Balescu (QLB) equation. Further motivation of our correction is provided by an argument arising from the Zubarev quantum kinetic theory approach. Significant emphasis is placed on our efforts to produce properly converged results for plasma transport using Kohn-Sham DFT, so that an accurate assessment of the importance and efficacy of our e-e scattering corrections to the thermal conductivity can be made.

Equation of state of dense plasmas with pseudoatom molecular dynamics

We present an approximation for calculating the equation of state (EOS) of warm and hot dense matter that is built on the previously published pseudoatom molecular dynamics (PAMD) model of dense plasmas [Starrett et al., Phys. Rev. E 91, 013104 (2015)PLEEE81539-375510.1103/PhysRevE.91.013104]. While the EOS calculation with PAMD was previously limited to orbital-free density functional theory (DFT), the new approximation presented here allows a Kohn-Sham DFT treatment of the electrons. The resulting EOS thus includes a quantum mechanical treatment of the electrons with a self-consistent model of the ionic structure, while remaining tractable at high temperatures. The method is validated by comparisons with pressures from ab initio simulations of Be, Al, Si, and Fe. The EOS in the Thomas-Fermi approximation shows remarkable thermodynamic consistency over a wide range of temperatures for aluminum. We calculate the principal Hugoniots of aluminum and silicon up to 500 eV. We find that the ionic structure of the plasma has a modest effect that peaks at temperatures of a few eV and that the features arising from the electronic structure agree well with ab initio simulations.

X-ray Thomson Scattering in Warm Dense Matter without the Chihara Decomposition

X-ray Thomson scattering is an important experimental technique used to measure the temperature, ionization state, structure, and density of warm dense matter (WDM). The fundamental property probed in these experiments is the electronic dynamic structure factor. In most models, this is decomposed into three terms [J. Chihara, J. Phys. F 17, 295 (1987)] representing the response of tightly bound, loosely bound, and free electrons. Accompanying this decomposition is the classification of electrons as either bound or free, which is useful for gapped and cold systems but becomes increasingly questionable as temperatures and pressures increase into the WDM regime. In this work we provide unambiguous first principles calculations of the dynamic structure factor of warm dense beryllium, independent of the Chihara form, by treating bound and free states under a single formalism. The computational approach is real-time finite-temperature time-dependent density functional theory (TDDFT) being applied here for the first time to WDM. We compare results from TDDFT to Chihara-based calculations for experimentally relevant conditions in shock-compressed beryllium.

Ionic and electronic transport properties in dense plasmas by orbital-free density functional theory

We validate the application of our recent orbital-free density functional theory (DFT) approach [Phys. Rev. Lett. 113, 155006 (2014);] for the calculation of ionic and electronic transport properties of dense plasmas. To this end, we calculate the self-diffusion coefficient, the viscosity coefficient, the electrical and thermal conductivities, and the reflectivity coefficient of hydrogen and aluminum plasmas. Very good agreement is found with orbital-based Kohn-Sham DFT calculations at lower temperatures. Because the computational costs of the method do not increase with temperature, we can produce results at much higher temperatures than is accessible by the Kohn-Sham method. Our results for warm dense aluminum at solid density are inconsistent with the recent experimental results reported by Sperling et al. [Phys. Rev. Lett. 115, 115001 (2015)].

First-principles equation of state of polystyrene and its effect on inertial confinement fusion implosions

Obtaining an accurate equation of state (EOS) of polystyrene (CH) is crucial to reliably design inertial confinement fusion (ICF) capsules using CH/CH-based ablators. With first-principles calculations, we have investigated the extended EOS of CH over a wide range of plasma conditions (ρ=0.1to100g/cm(3) and T=1000 to 4,000,000 K). When compared with the widely used SESAME-EOS table, the first-principles equation of state (FPEOS) of CH has shown significant differences in the low-temperature regime, in which strong coupling and electron degeneracy play an essential role in determining plasma properties. Hydrodynamic simulations of cryogenic target implosions on OMEGA using the FPEOS table of CH have predicted ∼30% decrease in neutron yield in comparison with the usual SESAME simulations. This is attributed to the ∼5% reduction in implosion velocity that is caused by the ∼10% lower mass ablation rate of CH predicted by FPEOS. Simulations using CH-FPEOS show better agreement with measurements of Hugoniot temperature and scattered light from ICF implosions.

Conductivity of warm dense matter including electron-electron collisions

We present an approach that can resolve the controversy with respect to the role of electron-electron collisions in calculating the dynamic conductivity of dense plasmas. In particular, the dc conductivity is analyzed in the low-density, nondegenerate limit where the Spitzer theory is valid and electron-electron collisions lead to the well-known reduction in comparison to the result considering only electron-ion collisions (Lorentz model). With increasing degeneracy, the contribution of electron-electron collisions to the dc conductivity is decreasing and can be neglected for the liquid metal domain where the Ziman theory is applicable. We give expressions for the effect of electron-electron collisions in calculating the conductivity in the warm dense matter region, i.e., for strongly coupled Coulomb systems at arbitrary degeneracy.

First-principles opacity table of warm dense deuterium for inertial-confinement-fusion applications

Accurate knowledge of the optical properties of a warm dense deuterium-tritium (DT) mixture is important for reliable design of inertial confinement fusion (ICF) implosions using radiation-hydrodynamics simulations. The opacity of a warm dense DT shell essentially determines how much radiation from hot coronal plasmas can be deposited in the DT fuel of an imploding capsule. Even for the simplest species of hydrogen, the accurate calculation of their opacities remains a challenge in the warm-dense matter regime because strong-coupling and quantum effects play an important role in such plasmas. With quantum-molecular-dynamics (QMD) simulations, we have derived a first-principles opacity table (FPOT) of deuterium (and the DT mixture by mass scaling) for a wide range of densities from ρ(D)=0.5 to 673.518g/cm(3) and temperatures from T=5000K up to the Fermi temperature T(F) for each density. Compared with results from the astrophysics opacity table (AOT) currently used in our hydrocodes, the FPOT of deuterium from our QMD calculations has shown a significant increase in opacity for strongly coupled and degenerate plasma conditions by a factor of 3-100 in the ICF-relevant photon-energy range. As conditions approach those of classical plasma, the opacity from the FPOT converges to the corresponding values of the AOT. By implementing the FPOT of deuterium and the DT mixture into our hydrocodes, we have performed radiation-hydrodynamics simulations for low-adiabat cryogenic DT implosions on the OMEGA laser and for direct-drive-ignition designs for the National Ignition Facility. The simulation results using the FPOT show that the target performance (in terms of neutron yield and energy gain) could vary from ∼10% up to a factor of ∼2 depending on the adiabat of the imploding DT capsule; the lower the adiabat, the more variation is seen in the prediction of target performance when compared to the AOT modeling.

Properties of warm dense polystyrene plasmas along the principal Hugoniot

Polystyrene (CH) is often chosen as the ablator material for inertial confinement fusion (ICF) targets. Its static, dynamical, and optical properties in warm, dense conditions (due to shock compression) are important for ICF designs. Using the first-principles quantum molecular dynamics (QMD) method, we have investigated the equation of state (EOS) and optical reflectivity of shock-compressed CH up to an unprecedentedly high pressure of 62 Mbar along the principal Hugoniot. The QMD results are compared with existing experimental measurements as well as the SESAME EOS model. Although the Hugoniot pressure and/or temperature from QMD calculations agrees with experiments and the SESAME EOS model at low pressures below 10 Mbar, we have identified for the first time a stiffer behavior of shocked CH at higher pressures (>10 Mbar). Such a stiffer behavior of warm, dense CH can affect the ablation pressure (shock strength), shock coalescence dynamics, and nonuniformity growth in ICF implosions. In addition, we corrected the mistake made in literature for calculating the reflectivity of shocked CH and obtained good agreements with experimental measurements, which should lend credence to future opacity calculations in a first-principles fashion.