Warm Dense Matter Regime Research Articles

Improved first-principles equation-of-state table of deuterium for high-energy-density applications

We present a first-principles equation-of-state (EOS) table of deuterium aimed at improving the previously established first-principles equation-of-state table (FPEOS) [S. X. Hu et al., Phys. Rev. B 84, 224109 (2011); S. X. Hu et al., Phys. Plasmas 22, 056304 (2015)]. The EOS table presented here, referred to as iFPEOS, introduces (1) a universal density functional theory (DFT) treatment of all density and temperature conditions, (2) a fully consistent treatment of exchange-correlation (XC) thermal effects across the entire range of temperatures covered, and (3) quantum treatment of ions. Based on ab initio molecular dynamics driven by thermal density functional theory, iFPEOS includes density points in the range $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1.6\ifmmode\times\else\texttimes\fi{}{10}^{3}$ $\mathrm{g}/{\mathrm{cm}}^{3}$ and temperature points in the range 800 K $\ensuremath{\le}T\ensuremath{\le}256$ MK, thus covering the challenging warm dense matter (WDM) regime. For an improved description of the electronic structure, iFPEOS employs an advanced free-energy XC density functional with explicit temperature dependence, which is at the metageneralized gradient approximation level of DFT. We use the latest orbital-free free-energy density functional for the high-temperature regime where it shows excellent agreement with standard Mermin-Kohn-Sham DFT. For quantum treatment of ions we use path-integral molecular dynamics in order to take into account nuclear quantum effects. Results are compared to other EOS models and most recent experimental measurements of deuterium properties such as the molecular-to-atomic fluid transition, the principal and reshock Hugoniot, and sound speed. We find that iFPEOS provides an improved agreement with experimental data compared to other first-principles EOS models in the WDM regime for pressures up to 200 GPa and temperatures up to $60\phantom{\rule{0.16em}{0ex}}000$ K. For higher pressures and temperatures, however, iFPEOS is in agreement with other models in predicting lower compressibility and higher sound speed along the Hugoniot, compared to experiment.

Density response of the warm dense electron gas beyond linear response theory: Excitation of harmonics

In a recent Letter, Dornheim et al. [PRL 125, 085001 (2020)] have investigated the nonlinear density response of the uniform electron gas in the warm dense matter regime. More specifically, they have studied the cubic response function at the first harmonic, which cannot be neglected in many situations of experimental relevance. In this work, we go one step further and study the full spectrum of excitations at the higher harmonics of the original perturbation based on extensive new ab initio path integral Monte Carlo (PIMC) simulations. We find that the dominant contribution to the density response beyond linear response theory is given by the quadratic response function at the second harmonic in the moderately nonlinear regime. Furthermore, we show that the nonlinear density response is highly sensitive to exchange-correlation effects, which makes it a potentially valuable new tool of diagnostics. To this end, we present a new theoretical description of the nonlinear electronic density response based on the recent effective static approximation to the local field correction [PRL 125, 235001 (2020)], which accurately reproduces our PIMC data with negligible computational cost.

Nonlinear electronic density response of the ferromagnetic uniform electron gas at warm dense matter conditions

AbstractIn a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 125, 085001 (2020)], we have presented the first ab initio results for the nonlinear density response of electrons in the warm dense matter (WDM) regime. In the present work, we extend these efforts by carrying out extensive new path integral Monte Carlo simulations of a ferromagnetic electron gas that is subject to an external harmonic perturbation. This allows us to unambiguously quantify the impact of spin‐effects on the nonlinear density response of the warm dense electron gas. In addition to their utility for the description of WDM in an external magnetic field, our results further advance our current understanding of the uniform electron gas as a fundamental model system, which is important in its own right.

Thermionic emission of electrons from metal surfaces in the warm dense matter regime

Thermionic emission of electrons is a process fundamental to our understanding of laser–matter interactions in the ultra-short pulse regime. Charge collected from an emission process, as well the secondary radiation generated by their collective motion, provides avenues for diagnosing and verifying existing laser–solid interaction models. Laser fluences (∼104 J/m2) are of particular interest as they heat the surface electrons to temperatures on the order of a few electron volts (eV), placing it in the warm dense matter regime where much underlying physics is yet to be fully understood. However, even at such moderate fluences the conventional Richardson–Dushman formula for the emission rate becomes invalid. We consider an additional barrier potential on the surface that appears due to space-charge effects, which then limits the thermionic emission. This provides feedback leading to a self-consistent solution with the emission rate. Unlike the work function, this barrier dynamically evolves during the emission process. Here, we present the first calculation of the barrier potential on the surface, along with analytical expression, from a one-dimensional electrostatic model. The result is a generalization of the Richardson–Dushman picture to moderate laser fluences. The potential barrier has been incorporated into a two-temperature model for thermionic emission from an Al target irradiated by a femtosecond laser. The collisional and transport data for Al have been obtained using an average atom model.

Real-space Green's function approach for x-ray spectra at high temperature

There has been considerable interest in properties of condensed matter at finite temperature, including non-equilibrium behavior and extreme conditions up to the warm dense matter regime. Such behavior is encountered, e.g., in experimental time resolved x-ray absorption spectroscopy (XAS) in the presence of intense laser fields. In an effort to simulate such behavior, we present an approach for calculations of finite-temperature x-ray absorption spectra in arbitrary materials, using a generalization of the real-space Green's function formalism. The method is incorporated as an option in the core-level x-ray spectroscopy code FEFF10. To illustrate the approach, we present calculations for several materials together with comparisons to experiment and with other methods.

Interplay of high-precision shock wave experiments with first-principles theory to explore molecular systems at extreme conditions: A perspective

Conventional methods for probing molecular changes in condensed matter systems, such as electronic and vibrational spectroscopy, are difficult to implement at the extreme conditions associated with dynamic compression experiments. This is particularly true for experiments in the multimegabar regime; to achieve the requisite energy density to produce such pressures, sample sizes are necessarily quite small and experimental timescales are, therefore, extremely short. Furthermore, these extreme pressure conditions also result in high temperatures and, therefore, significant thermal emission even in the visible to infrared regime and in some cases render the sample opaque or reflective, thereby precluding bulk spectroscopy techniques, such as Raman scattering. These experimental challenges require a different approach to evaluating shock-induced changes at the molecular or atomic level in the multimegabar or the so-called warm dense matter regime. The past few decades have seen significant advances in the use of first-principles methods to investigate materials under extreme conditions, enabling these methods to become a powerful tool for exploring molecular systems at extreme conditions. Here, we discuss the construct of combining high-precision shock wave experiments with first-principles theory to explore molecular systems at extreme conditions. The results from high-fidelity dynamic compression experiments are used to evaluate first-principles theoretical frameworks and identify the framework that best reproduces experimental results in the regime of interest. That validated framework is then used to perform detailed simulations of the system of interest, providing unique insight into the response of the system at the molecular level.

Ab initio path integral Monte Carlo approach to the momentum distribution of the uniform electron gas at finite temperature without fixed nodes

We present extensive new \textit{ab intio} path integral Monte Carlo results for the momentum distribution function $n(\mathbf{k})$ of the uniform electron gas (UEG) in the warm dense matter (WDM) regime over a broad range of densities and temperatures. This allows us to study the nontrivial exchange--correlation induced increase of low-momentum states around the Fermi temperature, and to investigate its connection to the related lowering of the kinetic energy compared to the ideal Fermi gas. In addition, we investigate the impact of quantum statistics on both $n(\mathbf{k})$ and the off-diagonal density matrix in coordinate space, and find that it cannot be neglected even in the strongly coupled electron liquid regime. Our results were derived without any nodal constraints, and thus constitute a benchmark for other methods and approximations.

Analytical representation of the local field correction of the uniform electron gas within the effective static approximation

The description of electronic exchange--correlation effects is of paramount importance for many applications in physics, chemistry, and beyond. In a recent Letter, Dornheim \textit{et al.} [\textit{Phys. Rev. Lett.}~\textbf{125}, 235001 (2020)] have presented the \emph{effective static approximation} (ESA) to the local field correction (LFC), which allows for the highly accurate estimation of electronic properties such as the interaction energy and the static structure factor. In the present work, we give an analytical parametrization of the LFC within ESA that is valid for any wave number, and available for the entire range of densities ($0.7\leq r_s \leq20$) and temperatures ($0\leq \theta\leq 4$) that are relevant for applications both in the ground state and in the warm dense matter regime. A short implementation in Python is provided, which can easily be incorporated into existing codes. In addition, we present an extensive analysis of the performance of ESA regarding the estimation of various quantities like the dynamic structure factor, static dielectric function, the electronically screened ion-potential, and also stopping power in electronic medium. In summary, we find that the ESA gives an excellent description of all these quantities in the warm dense matter regime, and only becomes inaccurate when the electrons start to form a strongly correlated electron liquid ($r_s\sim20$). Moreover, we note that the exact incorporation of exact asymptotic limits often leads to a superior accuracy compared to the neural-net representation of the static LFC [\textit{J.~Chem.~Phys.}~\textbf{151}, 194104 (2019)].

Lattice stability of ordered Au-Cu alloys in the warm dense matter regime

In the warm dense regime, where the electron temperature is increased to the same order of the Fermi temperature, the dynamical stability of elemental metals depends on its electronic band structure as well as its crystal structure. It has been known that phonon hardening occurs due to an enhanced internal pressure caused by electron excitations as in close-packed simple metals, whereas phonon softening occurs at a specific point in the Brillouin zone as in body-centered cubic metals. Here, we investigate the dynamical stability of binary ordered alloys (Au and Cu) in the L1$_0$ and L1$_2$ structures. By performing first principles calculations on phonon dispersions, we demonstrate that warm dense Au-Cu systems show phonon hardening behavior except the lowest frequency phonon mode at point R of AuCu in the L1$_0$ structure. We show that when a phonon mode is stabilized by long-range interatomic interactions at ambient condition, such a phonon will be destabilized by the short-range nature of the warm dense matters.

Modeling of short-pulse laser-metal interactions in the warm dense matter regime using the two-temperature model.

A numerical model for laser-matter interactions in the warm dense matter regime is presented with broad applications, e.g., ablation, thermionic emission, and radiation. A unique approach is adopted, in which a complete set of collisional and transport data is calculated using a quantum model and incorporated into the classical two-temperature model for the electron and lattice-ion temperatures. The data set was produced by the average atom model that combines speed, conceptual simplicity, and straightforward numerical development. Such data are suitable for use in the warm dense matter regime, where most of the laser-matter interactions at moderate intensities occur, thus eliminating deficiencies of previous models, e.g., interpolation between solid and ideal plasma regimes. In contrast to other works, we use a more rigorous definition of solid and plasma states of the metal, based on the physical condition of the lattice, crystalline (ordered) versus melted (disordered), rather than a definition based on electron temperature. The synergy between the two-temperature and average atom models has been demonstrated on a problem involving heating and melting of the interior of Al by a short-pulse laser with duration 0.1-1 ps and laser fluences 1×10^{3}-3×10^{4}J/m^{2}(0.1-3J/cm^{2}). The melting line, which separates the solid and plasma regimes, has been tracked in time and space. The maximum melting depth has been determined as a function of laser fluence: l_{melt}(μm)≅4×10^{3}F(J/m^{2}).

Optical and physical properties of hydrocarbons with metal impurities in the warm dense matter regime

The hydrocarbon (CH) polymer is often chosen as the converter material with potential applications to Z-pinch driven dynamic hohlraum implosion experiments. Its physical and optical properties in the warm dense matter regime are important for dynamic hohlraum platform designs. Using the quantum molecular dynamics (QMD) method, we have obtained the equation of state, absorption coefficient, and reflectivity of hydrocarbon and Al–CH mixtures with the temperature and density ranging from 104–106 K and 0.1–0.9 g/cm3, respectively. The QMD-predicted principal Hugoniot data are compared with experiments as well as the theoretical calculations, and both show good agreement. The optical reflectivity from the corresponding dielectric functions is calculated using the corrected refraction index of the ambient (n0 = 1.59). Besides, we have further analyzed the atomic structure and bond dissociation process of polystyrene and Al–CH mixture systems using a bond tracking method with the temperature ranging from 1000 K to 10 000 K. The Al impurities have a slightly promoting effect on the initial stage of polystyrene pyrolysis. The calculation results can be helpful for future theoretical and experimental studies in high energy density physics research.

Equation of state measurements of dense krypton up to the insulator-metal transition regime: Evaluating the exchange-correlation functionals

Motivated by the poor understanding of the applicability of new exchange-correlation (XC) functionals to warm dense matter (WDM), we designed and performed multiple-shock reverberation compression experiments on dense krypton to evaluate explicitly the implications of recently derived XC functionals. The equation of states of krypton up to 155 GPa and 45 000 K, which ranges from an initial dense gaseous state up to the insulator-metal transition regime, were determined accurately. It is found that the experimental data are better reproduced by the strongly constrained and appropriately normed (SCAN) XC functional compared to the conventional Perdew-Burke-Ernzerhof and Van der Waals (vdW) DF1 functionals, elucidating that the introduction of the kinetic energy density and the intermediate-range vdW interaction is decisive. However, the incorporation of long-range interactions into the SCAN ($\mathrm{SCAN}+\mathrm{rVV}10$ XC functional) results in a noticeably stiffer prediction due to an overestimation of the density and internal energy of the system at low densities and temperatures. Our evaluation of the Karasiev-Sjostrom-Dufty-Trickey free-energy functional experimentally validates the XC thermal effect in the WDM regime, verifies the previous predictions, and sheds light on a direction for future theoretical efforts. Finally, a phase diagram of krypton is given, which provides a clear picture for understanding the thermophysical behavior of krypton in a wider temperature-pressure range.

Influence of different charge-state ion distribution on elastic X-ray scattering in warm dense matter

The study of warm dense matter is very important for the evolution of celestial bodies and inertial confinement fusion, which often contains a mixture of multiple elements and different charge-state ions. The ionic structure and distribution of different charge-states directly affect the diagnosis and physical properties of warm dense matter. At the same time, the influence of high-temperature dense plasma on the ionic structure should be considered when we study the physical properties from the first-principle calculation of electron structure. In the present work, the radial distribution functions of multiple charge-state ions (gold, carbon-hydrogen mixture, and aluminum) are developed in the hypernetted-chain approximation, and elastic x-ray scattering of different charge-state ions are calculated in the warm dense matter regime. Firstly, the electron structure of different charge-state ions is self-consistently computed in the ionic sphere, in which the ion-sphere radii are determined by the plasma density and their charges. And then the ionic fraction is obtained by solving the modified Saha equation, with the interactions among different charge-state ions taken into account, and ion-ion pair potentials are obtained by Yukawa model. Finally, the ion features of x-ray elastic scattering for Al are calculated on the basis of electronic distribution around the nuclei and ionic radial distribution function. By comparing the results of different charge-sate ions with the result of mean charge-sate ion, it is shown that different statistical methods can affect the physical properties which are dependent on the electronic and ionic structure.

Nonideal mixing effects in warm dense matter studied with first-principles computer simulations.

We study nonideal mixing effects in the regime of warm dense matter (WDM) by computing the shock Hugoniot curves of BN, MgO, and MgSiO3. First, we derive these curves from the equations of state (EOS) of the fully interacting systems, which were obtained using a combination of path integral Monte Carlo calculations at high temperature and density functional molecular dynamics simulations at lower temperatures. We then use the ideal mixing approximation at constant pressure and temperature to rederive these Hugoniot curves from the EOS tables of the individual elements. We find that the linear mixing approximation works remarkably well at temperatures above ∼2 × 105 K, where the shock compression ratio exceeds ∼3.2. The shape of the Hugoniot curve of each compound is well reproduced. Regions of increased shock compression, which emerge because of the ionization of L and K shell electrons, are well represented, and the maximum compression ratio of the Hugoniot curves is reproduced with high precision. Some deviations are seen near the onset of the L shell ionization regime, where ionization equilibrium in the fully interacting system cannot be well reproduced by the ideal mixing approximation. This approximation also breaks down at lower temperatures, where chemical bonds play an increasingly important role. However, the results imply that the equilibrium properties of binary and ternary mixtures in the regime of WDM can be derived from the EOS tables of the individual elements. This significantly simplifies the characterization of binary and ternary mixtures in the WDM and plasma phases, which otherwise requires large numbers of more computationally expensive first-principles computer simulations.

Ion modes in dense ionized plasmas through nonadiabatic molecular dynamics

We perform nonadiabatic simulations of warm dense aluminum based on the electron-force field (EFF) variant of wave-packet molecular dynamics. Comparison of the static ion-ion structure factor with density functional theory (DFT) is used to validate the technique across a range of temperatures and densities spanning the warm dense matter regime. Focusing on a specific temperature and density (3.5 eV, 5.2 g/cm3), we report on differences in the dynamic structure factor and dispersion relation across a variety of adiabatic and nonadiabatic techniques. We find the dispersion relation produced with EFF is in close agreement with the more robust and adiabatic Kohn-Sham DFT.

Evaluating diffusion and the thermodynamic factor for binary ionic mixtures

Molecular dynamics (MD) simulations are a powerful tool for the calculation of transport properties in mixtures. Not only are MD simulations capable of treating multicomponent systems, they are also applicable over a wide range of temperatures and densities. In plasma physics, this is particularly important for applications such as inertial confinement fusion. While many studies have focused on the effect of plasma coupling on transport properties, here we focus on the effects of mixing. We compute the thermodynamic factor, a measure of ideal/non-ideal mixing, for three binary ionic mixtures. We consider mixtures of hydrogen and carbon, hydrogen and argon, and argon and carbon, each at 500 randomly generated state points in the warm dense matter and plasma regimes. The calculated thermodynamic factors indicate different mixing behavior across phase space, which can significantly affect the corresponding mutual diffusion coefficients. As MD simulations are still computationally expensive, we apply modern data science tools to predict the thermodynamic factor over a large phase space. Further, we propose a more accurate approximation to the mutual diffusion coefficient than the commonly applied Darken relation.

Equation of state of hot, dense magnesium derived with first-principles computer simulations

Using two first-principles computer simulation techniques, path integral Monte Carlo and density functional theory molecular dynamics, we derive the equation of state of magnesium in the regime of warm dense matter, with densities ranging from 0.43 to 86.11 g cm−3 and temperatures from 20 000 K to 5×108 K. These conditions are relevant for the interiors of giant planets and stars as well as for shock compression measurements and inertial confinement fusion experiments. We study ionization mechanisms and the electronic structure of magnesium as a function of density and temperature. We show that the L shell electrons, 2s and 2p energy bands, merge at high densities. This results in gradual ionization of the L-shell with increasing density and temperature. In this regard, Mg differs from MgO, which is also reflected in the shape of its principal shock Hugoniot curve. For Mg, we predict a single broad pressure-temperature region, where the shock compression ratio is approximately 4.9. Mg thus differs from Si and Al plasmas that exhibit two well-separated compression maxima on the Hugoniot curve for L and K shell ionizations. Finally, we study multiple shocks and effects of preheat and precompression.

Fast and Universal Kohn-Sham Density Functional Theory Algorithm for Warm Dense Matter to Hot Dense Plasma.

Understanding many processes, e.g., fusion experiments, planetary interiors, and dwarf stars, depends strongly on microscopic physics modeling of warm dense matter and hot dense plasma. This complex state of matter consists of a transient mixture of degenerate and nearly free electrons, molecules, and ions. This regime challenges both experiment and analytical modeling, necessitating predictive abinitio atomistic computation, typically based on quantum mechanical Kohn-Sham density functional theory (KS-DFT). However, cubic computational scaling with temperature and system size prohibits the use of DFT through much of the warm dense matter regime. A recently developed stochastic approach to KS-DFT can be used at high temperatures, with the exact same accuracy as the deterministic approach, but the stochastic error can converge slowly and it remains expensive for intermediate temperatures (<50 eV). We have developed a universal mixed stochastic-deterministic algorithm for DFT at any temperature. This approach leverages the physics of KS-DFT to seamlessly integrate the best aspects of these different approaches. We demonstrate that this method significantly accelerated self-consistent field calculations for temperatures from 3 to 50eV, while producing stable molecular dynamics and accurate diffusion coefficients.

Influence of finite temperature exchange-correlation effects in hydrogen

We use density functional molecular dynamics (DFT-MD) to study the effect of finite temperature exchange-correlation (xc) in Hydrogen. Using the Kohn-Sham approach, the xc energy of the system, $E_{xc}(r_{s})$ is replaced by the xc free energy $f_{xc}(r_{s},\Theta)$ within the local density approximation (LDA) based on parametrized path integral Monte Carlo (PIMC) data for the uniform electron gas (UEG) at warm dense matter (WDM) conditions. We observe insignificant changes in the equation of state (EOS) at the region of metal-insulator transition compared to the regular LDA form, whereas significant changes are observed for T$>$10000 K, i.e., in the important WDM regime. Thus, our results further corroborate the need for temperature-dependent xc functionals for DFT simulations of WDM systems. Moreover, we present the first finite-temperature DFT results for the EOS of Hydrogen in the electron liquid regime up to $r_{s}=14$ and find a drastic impact (about $20\%$) of thermal xc effects, which manifests at lower temperatures compared to WDM. We expect our results to be important for many applications beyond DFT, like quantum hydrodynamics and astrophysical models.

Ab initio study of structure and electrical conductivity of warm dense oxygen

Using quantum molecular dynamics simulations, the equation of state and electrical conductivity of warm dense oxygen is calculated in the density and temperature ranges of 2.0–4.3 g/cm3 and 103–105 K, respectively. The simulations show that the dissociation of oxygen molecules at about 2000 K and 2.6 g/cm3 reaches 89%, and complex clusters form with increasing temperature and density. The dissociation of oxygen molecules significantly affects the electrical conductivity. The electrical conductivity of warm dense oxygen is greater than 105 S/m at pressures above 20 GPa and the oxygen is metallic, and then the electrical conductivity weakly dependent on pressure, up to 200 GPa. The density of states of liquid oxygen indicates conduction-state electron behavior in the warm dense matter regime.