Session BO03: HED Warm Dense Matter; EOS - Theory

Two Temperature Thermodynamics of Classical-Map Hypernetted-Chain Theory for the Warm Dense Matter Regime

Warm dense plasmas are ubiquitous in astrophysical systems, but modeling this regime is complex due to the lack of simple expansion parameters. Properly modeling the partially degenerate electron and the strongly coupled ions traditionally requires expensive simulations via Kohn-Sham molecular dynamics or path integral monte carlo. These highly accurate methods allow the extraction of transport coefficients and equations of state for practical use in hydrodynamic models of real large-scale systems.

Faster, but more approximate methods such as hypernetted-chain theory are known to model strongly coupled systems with high accuracy. The issue with warm dense matter in this case is in the modeling of quantum electrons. We investigate the semi-classical picture wherein the quantum Hamiltonian is mapped onto a classical Hamiltonian, taking advantage of the speed and accuracy of classical integral equation theory. In particular, this mapping requires an effective classical electron temperature that differs from the physical temperature. This leads to subtleties regarding the thermodynamics of two temperature systems. We demonstrate an improved method for modeling warm dense matter in this framework and compare it with well-known models such as Dharma-Wardana's CHNC model.   

Conservative dielectric functions and electrical conductivities from multicomponent Bhatnagar-Gross-Krook

In high energy-density experimental campaigns, computational models are essential for designing, interpreting, and diagnosing experiments. Since experimental micro-physical data is often unavailable, macroscopic simulations rely on microscopic models to inform various properties. Important micro-physical properties include: energy loss functions, dielectric functions, dynamic structure factors, and electrical conductivities [Phys. Plasmas 25, 056306 (2018)]. Our objective is to examine these models using single species and multispecies BGK kinetic equations, particularly the recently published model by Haack et al. [J Stat. Phys., 168, 826-856 (2017)]. From the susceptibility, we utilize linear response theory to uncover the effects of conservation laws that are not readily apparent in dynamic structure function, dielectric function, and conductivity models. We establish connections between older literature, such as the classical BGK kinetic equation [Bhatnagar, Gross, and Krook, Phys. Rev. 94, 511]. Mermin's dielectric function [Phys. Rev. B 1, 2362 (1970)] and models for electrical conductivities (e.g., Drude and Drude-Smith [Smith Phys. Rev. B 64, 155106 (2001)]). Numerical calculations are performed to illustrate how microscopic models are effected by these conservative constraints. Finally, we present numerical results, which include applications to C impurities in inertial confinement fusion (ICF) targets composed of deuterium and tritium.   

Embedding Theory Approach to Average Atom Models for Warm Dense Matter

Density Functional Theory (DFT)-based Average Atom Models (AAM) provide useful physical insight for the Warm Dense Matter (WDM) regime by reducing the ionic many-body system to a spherical average over local environments or charge states. However, AAMs tend to fall short of providing an accurate picture of electron-electron and electron-ion interactions. Typical approaches account for interactions by imposing boundary conditions; other schemes use closure relations from plasma physics to model ion-ion correlations (two-component plasma TCP-AAM). We propose a DFT embedding theory approach to handle interactions in the AAM in which we consider the average atom embedded in a background plasma described by an ion pair correlation function, e.g. as calculated from TCP-AAM and full quantum mechanical treatments such as quantum molecular dynamics (QMD). The connection between the embedded atom and background plasma subsystems is made by the nonadditive kinetic potential V^NAD which can be calculated from Thomas-Fermi and von Weizsäcker or more advanced kinetic energy functionals. We apply this approach to modelling a dense hydrogen plasma. Thermodynamic properties generated from this scheme can provide a benchmark comparison against existing AAMs and QMD to validate our approach.   

Withdrawn

In the recent decades, there is an increasing interest in the properties of material under extreme conditions of temperature and density (T-rho). The physical description of stars and planets formation and evolution, controlled fusion experiments and laser-material interaction requires the knowledge of the equation of state, opacity, ionization degree and some other physical quantities in a wide range of T-rho for many atomic compounds. Since there are no experimentally measured databases for these quantities, in such wide ranges of T-rho, these physical quantities are calculated, theoretically, using computational simulations. A popular theoretical approach for calculating the required quantities for plasmas relies on spherical models, like the ion-sphere average atom (ISAA) model [1]. In this simplified model, the electron-electron (e-e) interaction is approximated by the LDA/GGA exchange-correlation potential, and the interaction with the surrounding plasma ions is poorly described through the boundary condition of the ISAA's eigenfunctions. I will present an improved ISAA model which uses approximate optimized effective potentials, like KLI-SIC and KLI-EXX [2], for the e-e potential and the hyper-netted chain (HNC) [3] approximation for realistic description of the ion-plasma interaction. Also, improved opacity calculations, which are based on the improved potentials, will be presented for astrophysical systems and laser-plasma experiments, using the CRSTA code [4].

 

 

[1]      Balazs F. Rozsnyai, Phys. Rev. A 5, 1137 (1972); D.A. Liberman, Journal of Quantitative Spectroscopy and Radiative Transfer, 27, 335 (1982).

 [2] J. B. Krieger, Yan Li, and G. J. Iafrate, Phys. Rev. A 45, 101 (1992); J. B. Krieger, Yan Li, and G. J. Iafrate, Phys. Rev. A 46, 5453 (1992).

[3] S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982).

[4] G. Hazak and Y. Kurzweil, High Energy Density Phys. 8, 290 (2012); Y. Kurzweil and N. Polack-Schupper, Phys. Plasmas 29, 012704 (2022).   

Atomic structure considerations for the low-temperature opacity of Xenon

We have investigated the opacity of Xe at low temperatures (<50 eV). The emissivity and opacity of Xe is a vital factor in determining the utility of Xe in EUV lithography, with numerous industrial applications. To this end, we have explored the accuracy of some approximations used in opacity models for the pertinent ion stages of Xe (6 times ionized through 20 times ionized). Due to the complexity of Xe atomic structure, one needs to use full configuration-interaction to appropriately describe the strong mixing in the various n=4 sub-shells that give rise to the Δn=0 and Δn=1 transitions that dominate the opacity spectrum at low temperatures. Calculations that consist of full configuration-interaction for large numbers of configurations quickly become computationally prohibitive, thus we have explored hybrid calculations, in which full configuration-interaction is retained for the most important transitions, while intermediate coupling is employed for all other transitions. After verifying atomic structure properties, local-thermodynamic-equilibrium (LTE) opacities are generated using the ATOMIC code at selected temperatures and densities and compared to several experimental results.   

An Equation of State Model for Low-temperature Hydrogen and Its Isotopes

The low-temperature equations of state (EOS) of hydrogen and its isotopes are important to the design of high-energy-density (HED) experiments and improving the models of gas giants. In this presentation, I will discuss the development of a quantum statistical approach to model the EOS of solid hydrogen in various structures and across a broad range of pressures. This model explicitly considers the cold, vibration, and nuclear spin (coupled with molecular rotation in certain phases) contributions and combines first-principles calculations and quantum statistics to derive thermodynamically consistent EOS, including internal energy, Helmholtz free energy, entropy, and pressure as functions of temperature and density. The derived heat capacity, Hugoniot, and isentropes show good agreement with expectations based on previous low-pressure experiments. Finally, I will discuss the dependence of the results on nuclear spin states and masses, and connections to experiments aiming to reach the quantum metallic regime.   

Calculations of static and dynamic properties of hydrogen at warm dense matter conditions using quantum-trajectory molecular dynamics

The wave-packet approximation for solving many-body quantum dynamics allows for the time propagation of electronic degrees of freedom over ionic time scales and the investigation of non-adiabatic coupling between the electronic and ionic subsystems. We recently presented an extension of the formulation -- appropriate for quantum plasmas -- where the electron wave packet can be elongated in arbitrary directions [1]. The implementation of the model in the molecular dynamics framework LAMMPS is demonstrated to scale close to linear in particle number which in combination with MPI parallelisation allows for the treatment of a large number of particles needed for the computation of transport properties. Focusing on hydrogen and deuterium plasmas in the warm dense matter regime, the structural properties of the plasma within the model are compared with variational density matrix methods, path integral Monte Carlo and DFT-based methods, however, transport properties such as diffusion and viscosity derived from time-resolved quantities are also presented. The explicit treatment of the electron dynamics allows for a straightforward description of electron transport coefficients, such as electrical conductivity, without further approximation. Experimentally relevant quantities such as the dynamic structure factor can be computed directly from the time-resolved electron density without the need for the Chihara Decomposition where separate models are employed for the description of the adiabatic electron envelope around ions and the free electron motion.

[1] P. Svensson et al., Philos. Trans. R. Soc. A, 381:20220325 (2023)   

Investigating the Impact of Non-adiabatic Effects in the Warm Dense Matter Regime

The investigation of warm dense matter (WDM) involves exploring the significance of electron-ion interactions, also known as non-adiabatic effects, on ion dynamics. In our recent publication, we investigated the impact of non-adiabatic effects on ion self-diffusion in warm dense hydrogen, using a novel method to characterize these effects [1]. Non-adiabatic effects were included using the LAMMPS implementation of the electron force field (eFF). Our approach is a straightforward yet effective way to examine dynamic properties in WDM since it includes more physics than classical molecular dynamics while remaining computationally efficient compared to time-dependent density functional theory. Our analysis using eFF also led us to examine the impact of electron dynamics on other dynamic properties of WDM systems, such as viscosity and the dynamic structure factor, further emphasizing the importance of developing higher fidelity non-adiabatic simulation methods. Ultimately, this will enable us to better understand WDM and its properties.

1. W. A. Angermeier, B. S. Scheiner, N. R. Shaffer, and T. G. White, Philos. Trans. R. Soc. A. 381, (2023).   

Study of the H2/H-He mixtures at extreme conditions: demixing, insulator-metal transition and miscibility boundaries

Accurate knowledge of the H-He demixing conditions and location of the insulator-metal-transition (IMT) boundary at Mbar pressures is important for planetary models. The most successful approach, to date, is the evaluation of the Gibbs free-energy of mixing based on the non-ideal entropy of mixing. Here, we explore a different approach. Using an NPT ensemble, large-scale ab-initio molecular dynamics (AIMD) simulations are performed along a set of selected isobars. At low-T the system spontaneously separates resulting in two regions, one that is He-rich and one that is molecular hydrogen-rich. Calculations of the conductivity indicate this demixed system is an insulator. Admixture with a small fraction of He makes the H2 bonds stronger leading to a H2 dissociation and related IMT that occurs at temperatures significantly higher as compared to pure H2. With further increase of temperature the H-He mixing will occur. Analysis of the first peak of the H-He radial distribution function provides a sharp signature of the transition from the demixed to the perfectly mixed phase. The use of a thermal exchange-correlation (XC) functional in our AIMD simulations shifts the miscibility boundary by about 10% up in temperature as compared to the standard ground-state XC functional.   

Improving efficient electronic stopping power models with first-principles benchmarks

Hydrodynamic simulations of fusion experiments rely on tabulated electronic stopping powers, which govern self-heating on the way to ignition. We use real-time time-dependent density functional theory (TDDFT) to benchmark and constrain efficient stopping models suitable for tabulation in the warm dense regime. First, we consider different treatments of electron-ion collisions in the Mermin dielectric formulation for the free-electron contribution to proton stopping powers and find that T-matrix or Kubo-Greenwood approaches improve agreement with TDDFT over the Born approximation [1]. Then, we examine deviations from Z² scaling between proton and alpha stopping powers predicted by TDDFT near and below the Bragg peak, and we evaluate effective charge models for capturing this behavior. Finally, we leverage recent methodological advances [2] to study the validity of additivity laws for mixtures. Throughout, we survey a range of warm dense systems, including hydrogen, deuterium, carbon, and aluminum. These efforts will enable more accurate stopping power models used to design and interpret fusion experiments.

1. Hentschel et al., Physics of Plasmas 30, 2023

2. Kononov et al., arXiv:2307.03213, 2023   

Electron-phonon thermalization of warm dense beryllium from first-principles

Understanding the thermalization processes in warm dense matter (WDM) are essential to improving the modeling, design, and understanding of high energy density physics (HEDP) experiments. In this work, we study electron-phonon (e-ph) thermalization in isochorically heated solid-density beryllium using first-principles methods. Density functional theory (DFT) and density functional perturbation theory (DFPT) are used to obtain e-ph matrix elements that are then used as inputs to solve the electron Boltzmann transport equation (BTE). We solve the BTE with the software PERTURBO, which allows us to determine how out of equilibrium electrons and phonons populations equilibrate over time, with sub-femtosecond resolution. In our study, we consider how electrons with initial electronic temperatures from ~0.1-1 eV thermalize to the phonon temperature of 400 K due to e-ph scattering. While the initial and final distributions obey Fermi-Dirac statistics by construction, we observe that the intermediate distrubutions are non-thermal. We use the results from these simulations to determine the time scales of e-ph equilibration, model effective electronic temperatures for intermediate steps, and determine energy transfer rates. These results will benchmark and reveal the limitations of the two-temperature models that are often used in HEDP simulations.   

First-principles electron-electron scattering simulations in warm dense matter

Obtaining accurate descriptions of the scattering processes in the warm dense matter (WDM) regime will result in better predictions of electrical and thermal conductivities, which are an essential part in the modeling and design of inertial confinement fusion (ICF) experiments. In our approach, we use first-principles calculations to study isochorically heated solid-density beryllium and hydrogen which is closer to plasma conditions, both materials are commonly found in ICF experiments. We go beyond the commonly used Kubo-Greenwood formalism, and instead we utilize the GW approximation to study electron-electron (e-e) scattering through the relationship between the electron self-energy and the e-e scattering rate. We compare e-e scattering rates using two methods: one in which we fit the full-frequency GW self-energies to the Landau theory of the Fermi liquid to approximate energy-dependent lifetimes. Since this model is calculated within the zero-temperature formalism, we model a temperature-dependence by averaging the energy-dependent lifetimes over Fermi occupations and the density of states to obtain an average e-e- scattering rate. The second approach utilizing a low-scaling GW calculation which uses a compressed Matsubara frequency grid to determine state-dependent lifetimes which have an explicit temperature dependence. Our results will lead to the improvement of models used to generate data for simulations of ICF experiments.