Thermal equilibration in a shock wave

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'.

Assessment of the electron-proton energy relaxation rates extracted from molecular dynamics simulations in weakly-coupled hydrogen plasmas

Electron-proton energy relaxation rates are assessed using molecular dynamics (MD) simulations in weakly-coupled hydrogen plasmas. To this end, we use various approaches to extract the energy relaxation rate from MD-simulated temperatures, and we find that existing extracting approaches may yield results with a sizable discrepancy larger than the variance between analytical models, which is further verified by well-known case studies. Present results show that two of the extracting approaches can produce identical results, which is attributed to a proper treatment of relaxation evolution. To discriminate the use of various methods, an empirical criterion with respect to initial plasma temperatures is proposed, which can self-consistently explain the cases considered. In addition, for a transient electron-proton plasma, we show that it is possible to extrapolate the Coulomb logarithm from that derived by initial plasma parameters in a single MD calculation, which is reasonably consistent with previous MD data. Our results are helpful to obtain accurate MD-based energy relaxation rates.

Electron Kinetics Induced by Ultrafast Photoexcitation of Warm Dense Matter in a 30-nm-Thick Foil

We report on the study of electron kinetics induced by intense femtosecond (fs) laser excitation of electrons in the 5d band of Au. Changes in the electron system are observed from the temporal evolution of ac conductivity and conduction electron density. The results reveal an increase of electron thermalization time with excitation energy density, contrary to the Fermi-liquid behavior of the decrease of thermalization time associated with the heating of conduction electrons. This is attributed to the severe mitigation of photoexcitation by Auger decay. The study also uncovers the shortening of 5d hole lifetime with the increase of photoexcitation rates. These unique findings provide valuable insights for understanding electron kinetics under extreme nonequilibrium conditions.

Temperature Equilibration due to Charge State Fluctuations in Dense Plasmas

The charge states of ions in dense plasmas fluctuate due to collisional ionization and recombination. Here, we show how, by modifying the ion interaction potential, these fluctuations can mediate energy exchange between the plasma electrons and ions. Moreover, we develop a theory for this novel electron-ion energy transfer mechanism. Calculations using a random walk approach for the fluctuations suggest that the energy exchange rate from charge state fluctuations could be comparable to direct electron-ion collisions. This mechanism is, however, predicted to exhibit a complex dependence on the temperature and ionization state of the plasma, which could contribute to our understanding of significant variation in experimental measurements of equilibration times.

Theory of the electron-ion temperature relaxation rate spanning the hot solid metals and plasma phases

We present a theory for the rate of energy exchange between electrons and ions-also known as the electron-ion coupling factor-in physical systems ranging from hot solid metals to plasmas, including liquid metals and warm dense matter. The paper provides the theoretical foundations of a recent work [J. Simoni and J. Daligault, Phys. Rev. Lett. 122, 205001 (2019)PRLTAO0031-900710.1103/PhysRevLett.122.205001], where first-principles quantum molecular dynamics calculations based on this theory were presented for representative materials and conditions. We first derive a general expression for the electron-ion coupling factor that includes self-consistently the quantum mechanical and statistical nature of electrons, the thermal and disorder effects, and the correlations between particles. The electron-ion coupling is related to the friction coefficients felt by individual ions due to their nonadiabatic interactions with the electrons. Each coefficient satisfies a Kubo relation given by the time integral of the autocorrelation function of the interaction force of an ion with the electrons. Exact properties and different representations of the general expressions are discussed. We then show that our theory reduces to well-known models in limiting cases. In particular, we show that it simplifies to the standard electron-phonon coupling formula in the limit of hot solids with lattice and electronic temperatures much greater than the Debye temperature, and that it extends the electron-phonon coupling formula beyond the harmonic phonon approximation. For plasmas, we show that the theory readily reduces to the well-known Spitzer formula in the hot plasma limit, to the Fermi "golden rule" formula in the limit of weak electron-ion interactions, and to other models proposed to go beyond the latter approximation. We explain that the electron-ion coupling is particularly well adapted to average atom models, which offer an effective way to include nonideal interaction effects to the standard models and at a much reduced computational cost in comparison to first-principles quantum molecular dynamics simulations.

Anharmonic and Anomalous Trends in the High-Pressure Phase Diagram of Silicon

A multifaceted first-principles approach utilizing density functional theory, evolutionary algorithms, and lattice dynamics was used to construct the phase diagram of silicon up to 4 TPa and 26 000  K. These calculations predicted that (i) an anomalous sequence of face-centered cubic to body-centered cubic to simple cubic crystalline phase transitions occur at pressures of 2.87 and 3.89 TPa, respectively, along the cold curve, (ii) the orthorhombic phases of Imma and Cmce-16 appear on the phase diagram only when the anharmonic contribution to the Gibbs free energy is taken into account, and (iii) a substantial change in the slope of the principal Hugoniot is observed if the anharmonic free energy of the cubic diamond phase is considered.

Extremely Low Electron-ion Temperature Relaxation Rates in Warm Dense Hydrogen: Interplay between Quantum Electrons and Coupled Ions

Theoretical and computational modeling of nonequilibrium processes in warm dense matter represents a significant challenge. The electron-ion relaxation process in warm dense hydrogen is investigated here by nonequilibrium molecular dynamics using the constrained electron force field (CEFF) method. CEFF evolves wave packets that incorporate dynamic quantum diffraction that obviates the Coulomb catastrophe. Predictions from this model reveal temperature relaxation times as much as three times longer than prior molecular dynamics results based on quantum statistical potentials. Through analyses of energy distributions and mean free paths, this result can be traced to delocalization. Finally, an improved GMS [Gericke, Murillo, and Schlanges, Phys. Rev. E 78, 025401 (2008)PRESCM1539-375510.1103/PhysRevE.78.025401] model is proposed, in which the Coulomb logarithms are in good agreement with CEFF results.

Quantum theory for the dynamic structure factor in correlated two-component systems in nonequilibrium: Application to x-ray scattering

We present a quantum theory for the dynamic structure factors in nonequilibrium, correlated, two-component systems such as plasmas or warm dense matter. The polarization function, which is needed as the input for the calculation of the structure factors, is calculated in nonequilibrium based on a perturbation expansion in the interaction strength. To make our theory applicable for x-ray scattering, a generalized Chihara decomposition for the total electron structure factor in nonequilibrium is derived. Examples are given and the influence of correlations and exchange on the structure and the x-ray-scattering spectrum are discussed for a model nonequilibrium distribution, as often encountered during laser heating of materials, as well as for two-temperature systems.

First-principles equation-of-state table of silicon and its effects on high-energy-density plasma simulations

Using density-functional theory-based molecular-dynamics simulations, we have investigated the equation of state for silicon in a wide range of plasma density and temperature conditions of ρ=0.001-500g/cm^{3} and T=2000-10^{8}K. With these calculations, we have established a first-principles equation-of-state (FPEOS) table of silicon for high-energy-density (HED) plasma simulations. When compared with the widely used SESAME-EOS model (Table 3810), we find that the FPEOS-predicted Hugoniot is ∼20% softer; for off-Hugoniot plasma conditions, the pressure and internal energy in FPEOS are lower than those of SESAME EOS for temperatures above T ≈ 1-10 eV (depending on density), while the former becomes higher in the low-T regime. The pressure difference between FPEOS and SESAME 3810 can reach to ∼50%, especially in the warm-dense-matter regime. Implementing the FPEOS table of silicon into our hydrocodes, we have studied its effects on Si-target implosions. When compared with the one-dimensional radiation-hydrodynamics simulation using the SESAME 3810 EOS model, the FPEOS simulation showed that (1) the shock speed in silicon is ∼10% slower; (2) the peak density of an in-flight Si shell during implosion is ∼20% higher than the SESAME 3810 simulation; (3) the maximum density reached in the FPEOS simulation is ∼40% higher at the peak compression; and (4) the final areal density and neutron yield are, respectively, ∼30% and ∼70% higher predicted by FPEOS versus the traditional simulation using SESAME 3810. All of these features can be attributed to the larger compressibility of silicon predicted by FPEOS. These results indicate that an accurate EOS table, like the FPEOS presented here, could be essential for the precise design of targets for HED experiments.

Molecular dynamics studies of electron-ion temperature equilibration in hydrogen plasmas within the coupled-mode regime

We use classical molecular dynamics (MD) to study electron-ion temperature equilibration in two-component plasmas in regimes for which the presence of coupled collective modes has been predicted to substantively reduce the equilibration rate. Guided by previous kinetic theory work, we examine hydrogen plasmas at a density of n=10^{26}cm^{-3}, T_{i}=10^{5}K, and 10^{7}K<T_{e}<10^{9}K. The nonequilibrium classical MD simulations are performed with interparticle interactions modeled by quantum statistical potentials (QSPs). Our MD results indicate (i) a large effect from time-varying potential energy, which we quantify by appealing to an adiabatic two-temperature equation of state, and (ii) a notable deviation in the energy equilibration rate when compared to calculations from classical Lenard-Balescu theory including the QSPs. In particular, it is shown that the energy equilibration rates from MD are more similar to those of the theory when coupled modes are neglected. We suggest possible reasons for this surprising result and propose directions of further research along these lines.

Development of Path Integral Monte Carlo Simulations with Localized Nodal Surfaces for Second-Row Elements

We extend the applicability range of fermionic path integral Monte Carlo simulations to heavier elements and lower temperatures by introducing various localized nodal surfaces. Hartree-Fock nodes yield the most accurate prediction for pressure and internal energy, which we combine with the results from density functional molecular dynamics simulations to obtain a consistent equation of state for hot, dense silicon under plasma conditions and in the regime of warm dense matter (2.3-18.6  g cm(-3), 5.0×10(5)-1.3×10(8)  K). The shock Hugoniot curve is derived and the structure of the fluid is characterized with various pair correlation functions.

Electron-ion equilibration in ultrafast heated graphite

We have employed fast electrons produced by intense laser illumination to isochorically heat thermal electrons in solid density carbon to temperatures of ∼10,000  K. Using time-resolved x-ray diffraction, the temperature evolution of the lattice ions is obtained through the Debye-Waller effect, and this directly relates to the electron-ion equilibration rate. This is shown to be considerably lower than predicted from ideal plasma models. We attribute this to strong ion coupling screening the electron-ion interaction.

Evolution of ac conductivity in nonequilibrium warm dense gold

Using a chirped pulse probe technique, we have obtained single-shot measurements of temporal evolution of ac conductivity at 1.55 eV (800 nm) during electron energy relaxation in nonequilibrium warm dense gold with energy densities up to 4.1  MJ/kg (8×10(10)  J/m3). The results uncover important changes that have been masked in an earlier experiment. Equally significant, they provide valuable tests of an ab initio model for the calculation of electron heat capacity, electron-ion coupling, and ac conductivity in a single, first principles framework. While measurements of the real part of ac conductivity corroborate our theoretical temperature-dependent electron heat capacity, they point to an electron-ion coupling factor of ∼2.2×10(16)  W/m3 K, significantly below that predicted by theory. In addition, measurements of the imaginary part of ac conductivity reveal the need to improve theoretical treatment of intraband contributions at very low photon energy.

Observation of inhibited electron-ion coupling in strongly heated graphite

Creating non-equilibrium states of matter with highly unequal electron and lattice temperatures (T(ele)≠T(ion)) allows unsurpassed insight into the dynamic coupling between electrons and ions through time-resolved energy relaxation measurements. Recent studies on low-temperature laser-heated graphite suggest a complex energy exchange when compared to other materials. To avoid problems related to surface preparation, crystal quality and poor understanding of the energy deposition and transport mechanisms, we apply a different energy deposition mechanism, via laser-accelerated protons, to isochorically and non-radiatively heat macroscopic graphite samples up to temperatures close to the melting threshold. Using time-resolved x ray diffraction, we show clear evidence of a very small electron-ion energy transfer, yielding approximately three times longer relaxation times than previously reported. This is indicative of the existence of an energy transfer bottleneck in non-equilibrium warm dense matter.

Energy relaxation in dense, strongly coupled two-temperature plasmas

A quantum kinetic approach for the energy relaxation in strongly coupled plasmas with different electron and ion temperatures is presented. Based on the density operator formalism, we derive a balance equation for the energies of electrons and ions connecting kinetic, correlation, and exchange energies with a quite general expression for the electron-ion energy-transfer rate. The latter is given in terms of the correlation function of density fluctuations which allows for a derivation of increasingly realistic approximation schemes including a coupled-mode expression. The equilibration of the contributions of the total energy including the species temperatures in dense hydrogen and beryllium relevant for inertial confinement fusion is investigated as an example.

Electron-ion thermal equilibration after spherical shock collapse

A comprehensive set of dual nuclear product observations provides a snapshot of imploding inertial confinement fusion capsules at the time of shock collapse, shortly before the final stages of compression. The collapse of strong convergent shocks at the center of spherical capsules filled with D(2) and (3)He gases induces D-D and D-(3)He nuclear production. Temporal and spectral diagnostics of products from both reactions are used to measure shock timing, temperature, and capsule areal density. The density and temperature inferred from these measurements are used to estimate the electron-ion thermal coupling and demonstrate a lower electron-ion relaxation rate for capsules with lower initial gas density.

Molecular dynamics simulations of electron-ion temperature equilibration in an SF6 plasma

We use classical molecular dynamics to investigate electron-ion temperature equilibration in a two-temperature SF6 plasma. We choose a density of 1.0 x 10;{19}SF_{6} molecules per cm;{3} and initial temperatures of T_{e} = 100 eV and T_{S} = T_{F} = 15 eV, in accordance with experiments currently underway at Los Alamos National Laboratory. Our computed relaxation time lies between two oft-used variants of the Landau-Spitzer relaxation formula which invoke static screening. Discrepancies are also found when comparing to the predictions made by more recent theoretical approaches. These differences should be large enough to be measured in the upcoming experiments.

Energy relaxation rates in dense hydrogen plasmas

A comprehensive study is made of the energy relaxation rates between ions and electrons in a dense hydrogen plasma. Results of classical molecular dynamics (MD) simulations are compared with quantal calculations using the Fermi golden rule and using dimensional continuation. The rates from the molecular dynamics simulations employing a screened potential are found to be in reasonable agreement with the Landau-Spitzer relaxation rates, and are around 30% higher than the Fermi golden rule rates. By inverting the classical MD relaxation rate vs the quantal result, a semiclassical value for the screening length is suggested. We present energy relaxation rates relevant for radiation-hydrodynamic simulations of inertial confinement fusion devices.

Molecular dynamics simulations of temperature equilibration in dense hydrogen

The temperature equilibration rate between electrons and protons in dense hydrogen has been calculated with molecular dynamics simulations for temperatures between 10 and 600eV and densities between 10;{20}cm;{-3}to10;{24}cm;{-3} . Careful attention has been devoted to convergence of the simulations, including the role of semiclassical potentials. We find that for Coulomb logarithms L greater, similar1 , a model by Gericke-Murillo-Schlanges (GMS) [D. O. Gericke, Phys. Rev. E 65, 036418 (2002)] based on a T -matrix method and the approach by Brown-Preston-Singleton [L. S. Brown, Phys. Rep. 410, 237 (2005)] agrees with the simulation data to within the error bars of the simulation. For smaller Coulomb logarithms, the GMS model is consistent with the simulation results. Landau-Spitzer models are consistent with the simulation data for L>4 .

Quantum corrections and bound-state effects in the energy relaxation of hot dense hydrogen

Simple analytic formulas for energy relaxation (ER) in electron-ion systems, with quantum corrections, ion dynamics, and RPA-type screening are presented. ER in the presence of bound electrons is examined in view of recent simulations for ER in hydrogen in the range 10{20}-10{24} electrons/cc.

Temperature relaxation in hot dense hydrogen

Temperature equilibration of hydrogen is studied for conditions relevant to inertial confinement fusion. New molecular-dynamics simulations and results from quantum many-body theory are compared with Landau-Spitzer predictions for temperatures T with 50<T<4000 eV and densities with Wigner-Seitz radii rs=1.0 and 0.5. The relaxation is slower than the Landau-Spitzer result, even for T in the kilo-electron-volt range, but converge to agreement in the high-T limit.

Ultrafast dynamics of strongly coupled plasmas

The ultrafast dynamics of a strongly coupled plasma following an energy landscape shift is studied theoretically and with simulation. To lowest order in time, the inertial dynamics on the new landscape can be characterized by the plasma microfield, which, for the randomly ordered case of an ultracold neutral plasma, is dominated by nearest neighbor interactions. Formation of the pair correlation function arises after ballistic overshoot, which leads to oscillations in the effective temperature. Warm dense matter systems are also considered in this context.

Nonequilibrium and non-steady-state evolution of a shock state

Using numerical simulations, we study the evolution of silicon in the passage of a constant-pressure shock wave launched from an adjacent pusher. We examine also its optical characteristics of reflectivity and emission. Our finding points to the study of shocked interfaces as a novel means to explore nonequilibrium, non-steady-state behaviors of shock states and an alternative approach to assess electron-ion equilibration rate in a shock wave. It also reveals important structures in such a shock wave in contrast to its usual notion as a propagating discontinuity. This offers some possibilities for reconciling the different findings on the compressibility of deuterium.

Dense plasma temperature equilibration in the binary collision approximation

Temperature equilibration in dense, strongly coupled plasmas has been investigated without most of the usual simplifying assumptions. A quantum kinetic approach is used that accounts for strong electron-ion collisions through an exact T-matrix treatment of the scattering cross section using a screened interaction. Our results reveal the accuracy of the usual Spitzer formula for Coulomb logarithms larger than about three. Moreover, a simple model based on hyperbolic orbits yields surprisingly accurate results. We also have included equation of state effects to describe realistic plasmas.

Results on the energy-relaxation rates of dense two-temperature aluminum, carbon, and silicon plasmas close to liquid-metal conditions

We present results for the electron-ion energy relaxation coupling constants g(ei)(T(e),T(i),kappa) for aluminum, carbon, and silicon plasmas at several electron and ion temperatures T(i), T(e) of experimental interest. The calculations use the Fermi golden rule and the Landau-Spitzer model valid at weak electron-ion coupling, as well as the coupled-mode approach suitable for strong coupling. A physically motivated simple derivation of the coupled-mode energy relaxation formula for two-component charged fluids is presented. While the commonly used weak-coupling theories predict relaxation constants relatively independent of the ion temperature, the strong-coupling theory predicts energy relaxation constants that become smaller by an order of magnitude as the ion temperature is lowered.