Equation of state for partially ionized carbon at high temperatures

Extended chemical picture of ionization balance to extremely dense plasmas

Understanding the ionization balance of extremely dense plasmas still remains a scientific challenge for both theory and experiment, which is very important in many research fields such as the equation of state, radiative opacity, and thermal and electrical conductivities. In a most recent experiment at the National Ignition Facility, the onset of pressure-driven K-shell delocalization was observed in hot dense Be plasmas [T. Döppner et al., Nature (London) 618, 270 (2023)0028-083610.1038/s41586-023-05996-8]. However, all the referenced widely used ionization models can only reproduce part of the experimental data on the ionization state. It is generally regarded that the normal Saha equation is difficult to be applied to the dense plasma regime of, for instance, above a mass density of 10g/cm^{3} when the interactions between the charged particles and the pressure ionization start to dominate the ionization balance. Herein, we show that the chemical picture of the ionization balance can be extended to an even denser regime up to a density of, for example, 100g/cm^{3} or higher when the nonideal effects due to the interactions between the electrons and ions and among the electrons themselves and the pressure-induced ionization can be properly considered in a modified Saha equation. An accurate prediction of the ionization potential depression is crucial to depict the transition of the pressure-induced ionization with increasing plasma density. Comparison of our calculated average degree of ionization with the above-mentioned experiment shows good agreement for all the experimental data before and after the K-shell delocalization transition.

Nonideal effects on ionization potential depression and ionization balance in dense Al and Au plasmas

For low-density plasmas, the ionization balance can be properly described by the normal Saha equation in the chemical picture. For dense plasmas, however, nonideal effects due to the interactions between the electrons and ions and among the electrons themselves affect the ionization potential depression and the ionization balance. With the increasing of plasma density, the pressure ionization starts to play a more obvious role and competes with the thermal ionization. Based on a local-density temperature-dependent ion-sphere model, we develop a unified and self-consistent theoretical formalism to simultaneously investigate the ionization potential depression, the ionization balance, and the charge states distributions of the dense plasmas. In this work, we choose Al and Au plasmas as examples as Al is a prototype light element and Au is an important heavy element in many research fields such as in the inertial confinement fusion. The nonideal effect of the free electrons in the plasmas is considered by the single-electron effective potential contributed by both the bound electrons of different charge states and the free electrons in the plasmas. For the Al plasmas, we can reconcile the results of two experiments on measuring the ionization potential depression, in which one experiment can be better explained by the Stewart-Pyatt model while the other fits better with the Ecker-Kröll model. For dense Au plasmas, the results show that the double peak structure of the charge state distribution appears to be a common phenomenon. In particular, the calculated ionization balance shows that the two- and three-peak structures can appear simultaneously for denser Au plasmas above ∼30g/cm^{3}.

Plasma ionization balance in chemical-picture and average-atom models

We propose an approximate method to calculate ion partition functions in the context of the chemical-picture representation of plasmas as an interacting mixture of various ions and free electrons under the local-thermodynamic-equilibrium conditions. The method uses the superconfiguration approach and implies that the first-order corrections to the energies of excited electron configurations due to the electron-electron interaction may be replaced by a similar first-order correction to the energy of the basic configuration of an ion with the same number of bound electrons. The method enables one to significantly speed up the calculations and generally provides quite accurate results. Using the method proposed, plasma ionization balance and average ion charges calculated on the base of the chemical-picture representation show a good agreement with the relevant average-atom data. For the case of weak electron-ion nonideality, we provide approximate relations between the chemical-picture and average-atom values of the average ion charge, chemical potential, and plasma-density depression of ionization potential.

Nonideal effect of free electrons on ionization equilibrium and radiative property in dense plasmas

The thermodynamic as well as optical properties of strongly coupled plasmas depend crucially on the average degree of ionization and the ionic state composition, which, however, cannot be determined by using the normal Saha equation usually used for the ideal plasmas. Hence, an adequate treatment of the ionization balance and the charge state distribution of strongly coupled plasmas is still a challenge for theory due to the interactions between the electrons and ions and among the electrons themselves. Based on a local density temperature-dependent ion-sphere model, the Saha equation approach is extended to the regime of strongly coupled plasmas by taking into account the free-electron-ion interaction, the free-free-electron interaction, the nonuniform free-electron space distribution, and the free-electron quantum partial degeneracy. All the quantities, including the bound orbitals with ionization potential depression, free-electron distribution, and bound and free-electron partition function contributions, are calculated self-consistently in the theoretical formalism. This study shows that the ionization equilibrium is evidently modified by considering the above nonideal characteristics of the free electrons. Our theoretical formalism is validated by the explanation of a recent experimental measurement of the opacity of dense hydrocarbon.

Properties of carbon up to 10 million kelvin from Kohn-Sham density functional theory molecular dynamics

Accurately modeling dense plasmas over wide-ranging conditions of pressure and temperature is a grand challenge critically important to our understanding of stellar and planetary physics as well as inertial confinement fusion. In this work, we employ Kohn-Sham density functional theory (DFT) molecular dynamics (MD) to compute the properties of carbon at warm and hot dense matter conditions in the vicinity of the principal Hugoniot. In particular, we calculate the equation of state (EOS), Hugoniot, pair distribution functions, and diffusion coefficients for carbon at densities spanning 8 g/cm^{3} to 16 g/cm^{3} and temperatures ranging from 100 kK to 10 MK using the Spectral Quadrature method. We find that the computed EOS and Hugoniot are in good agreement with path integral Monte Carlo results and the sesame database. Additionally, we calculate the ion-ion structure factor and viscosity for selected points. All results presented are at the level of full Kohn-Sham DFT-MD, free of empirical parameters, average-atom, and orbital-free approximations employed previously at such conditions.

Multi-Configuration Calculation of Ionization Potential Depression

The modelling of ionization potential depression in warm and hot dense plasmas constitutes a real theoretical challenge due to ionic coupling and electron degeneracy effects. In this work, we present a quantum statistical model based on a multi-configuration description of the electronic structure in the framework of Density Functional Theory. We discuss different conceptual issues inherent to the definition of ionization potential depression and compare our results with the famous and widely-used Ecker-Kröll and Stewart-Pyatt models.

Carbon ionization from a quantum average-atom model up to gigabar pressures

We use a nonrelativistic average-atom model to calculate carbon ionization at megabar and gigabar pressures. The pressure is calculated using the stress-tensor method. The electronic electrical conductivity is also considered using the Kubo-Greenwood approach. Comparisons are made with quantum molecular dynamic simulations. A good agreement is obtained for the pressure between the average-atom model and the quantum molecular dynamic simulations in the regime of gigabar pressures. However, the discrepancy already seen with the PURGATORIO code for the average ionization deduced from the quantum molecular dynamic simulations is also observed here with the present average-atom model. Excellent agreement with the PURGATORIO code is found for the average ionization.

Influence of dust particles on ionization equilibrium in partially ionized plasmas

A chemical model is proposed for a dusty plasma consisting of electrons, ions, neutrals, and positively charged dust particles all being at thermodynamic equilibrium. An expression is derived for the Helmholtz free energy, which comprises the ideal part, taking into account the charge of dust particles, and the excess part, evaluated in the framework of the self-consistent chemical model [Phys. Rev. E 83, 016405 (2011)PLEEE81539-375510.1103/PhysRevE.83.016405]. The ionization potential depression for a dust-free partially ionized hydrogen is analytically evaluated for weakly and strongly ionized states to consistently account for the presence of charged and neutral components. An ad hoc interpolation of the ionization potential depression, valid across the whole ionization region, is put forward and subsequent solution of the generalized Saha equation is found to be in a perfect agreement with exact calculations. Minimization of the Helmholtz free energy of dusty plasmas provides the number densities of free electrons, free ions, neutrals, and the dust electric charge as well. Based on consideration of weakly and strongly ionized states, a straightforward comparison is made of the ionization equilibrium in a partially ionized plasma with and without dust particles to demonstrate that at thermal equilibrium positively charged dusts are responsible for an increase in the number density of free electrons and a decrease in the number density of free ions. It is analytically proved that nonideality effects result in a growth of the number densities of free electrons and ions by obtaining the so-called electron and proton ionization potential depressions. Electric charge of dust particles is systematically studied as a full plasma component rather than considering a detailed balance of the electron and ion fluxes on the surface of a solitary dust grain.

Ionization potential depression and Pauli blocking in degenerate plasmas at extreme densities

New facilities explore warm dense matter (WDM) at conditions with extreme densities (exceeding ten times condensed matter densities) so that electrons are degenerate even at temperatures of 10-100 eV. Whereas in the nondegenerate region correlation effects such as Debye screening are relevant for the ionization potential depression (IPD), new effects have to be considered in degenerate plasmas. In addition to the Fock shift of the self-energies, the bound-state Pauli blocking becomes important with increasing density. Standard approaches to IPD such as Stewart-Pyatt and widely used opacity tables (e.g., OPAL) do not contain Pauli blocking effects for bound states. The consideration of degeneracy effects leads to a reduction of the ionization potential and to a higher degree of ionization. As an example, we present calculations for the ionization degree of carbon plasmas at T = 100 eV and extreme densities up to 40 g/cm^{3}, which are relevant to experiments that are currently scheduled at the National Ignition Facility.

Density effects on electronic configurations in dense plasmas

We present a quantum mechanical model to describe the density effects on electronic configurations inside a plasma environment. Two different approaches are given by starting from a quantum average-atom model. Illustrations are shown for an aluminum plasma in local thermodynamic equilibrium at solid density and at a temperature of 100 eV and in the thermodynamic conditions of a recent experiment designed to characterize the effects of the ionization potential depression treatment. Our approach compares well with experiment and is consistent in that case with the approach of Stewart and Pyatt to describe the ionization potential depression rather than with the method of Ecker and Kröll.

First-principles equation of state and shock compression predictions of warm dense hydrocarbons

We use path integral Monte Carlo and density functional molecular dynamics to construct a coherent set of equations of state (EOS) for a series of hydrocarbon materials with various C:H ratios (2:1, 1:1, 2:3, 1:2, and 1:4) over the range of 0.07-22.4gcm^{-3} and 6.7×10^{3}-1.29×10^{8}K. The shock Hugoniot curve derived for each material displays a single compression maximum corresponding to K-shell ionization. For C:H = 1:1, the compression maximum occurs at 4.7-fold of the initial density and we show radiation effects significantly increase the shock compression ratio above 2 Gbar, surpassing relativistic effects. The single-peaked structure of the Hugoniot curves contrasts with previous work on higher-Z plasmas, which exhibit a two-peak structure corresponding to both K- and L-shell ionization. Analysis of the electronic density of states reveals that the change in Hugoniot structure is due to merging of the L-shell eigenstates in carbon, while they remain distinct for higher-Z elements. Finally, we show that the isobaric-isothermal linear mixing rule for carbon and hydrogen EOS is a reasonable approximation with errors better than 1% for stellar-core conditions.

All-electron path integral Monte Carlo simulations of warm dense matter: application to water and carbon plasmas

We develop an all-electron path integral Monte Carlo method with free-particle nodes for warm dense matter and apply it to water and carbon plasmas. We thereby extend path integral Monte Carlo studies beyond hydrogen and helium to elements with core electrons. Path integral Monte Carlo results for pressures, internal energies, and pair-correlation functions compare well with density functional theory molecular dynamics calculations at temperatures of (2.5-7.5)×10(5) K, and both methods together form a coherent equation of state over a density-temperature range of 3-12 g/cm(3) and 10(4)-10(9) K.

Equation of state for partially ionized carbon and oxygen mixtures at high temperatures

The equation of state (EOS) for partially ionized carbon, oxygen, and carbon-oxygen mixtures at temperatures 3×10(5)K is less than or approximately equal to T is less than or approximately equal to 3×10(6) K is calculated over a wide range of densities, using the method of free energy minimization in the framework of the chemical picture of plasmas. The free energy model is an improved extension of our model previously developed for pure carbon [Potekhin, Massacrier, and Chabrier, Phys. Rev. E 72, 046402 (2005)]. The internal partition functions of bound species are calculated by a self-consistent treatment of each ionization stage in the plasma environment taking into account pressure ionization. The long-range Coulomb interactions between ions and screening of the ions by free electrons are included using our previously published analytical model, recently improved, in particular for the case of mixtures. We also propose a simple but accurate method of calculation of the EOS of partially ionized binary mixtures based on detailed ionization balance calculations for pure substances.

Self-consistent chemical model of partially ionized plasmas

A simple renormalization theory of plasma particle interactions is proposed. It primarily stems from generic properties of equilibrium distribution functions and allows one to obtain the so-called generalized Poisson-Boltzmann equation for an effective interaction potential of two chosen particles in the presence of a third one. The same equation is then strictly derived from the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for equilibrium distribution functions in the pair correlation approximation. This enables one to construct a self-consistent chemical model of partially ionized plasmas, correctly accounting for the close interrelation of charged and neutral components thereof. Minimization of the system free energy provides ionization equilibrium and, thus, permits one to study the plasma composition in a wide range of its parameters. Unlike standard chemical models, the proposed one allows one to study the system correlation functions and thereby to obtain an equation of state which agrees well with exact results of quantum-mechanical activity expansions. It is shown that the plasma and neutral components are strongly interrelated, which results in the short-range order formation in the corresponding subsystem. The mathematical form of the results obtained enables one to both firmly establish this fact and to determine a characteristic length of the structure formation. Since the cornerstone of the proposed self-consistent chemical model of partially ionized plasmas is an effective pairwise interaction potential, it immediately provides quite an efficient calculation scheme not only for thermodynamical functions but for transport coefficients as well.

Equation of state for a partially ionized gas. III

The derivation of equations of state for fluid phases of a partially ionized gas or plasma is addressed from a fundamental point of view. The results of the Thomas-Fermi model always yield pressures which are less than or equal to that of an ideal Fermi gas. On the other hand, the spherical cellular model shows significant "overpressure" relative to the ideal Fermi gas in certain regions of low density and low temperature. This effect is studied in considerable detail. A nonthermodynamic region, more or less overlapping the regions of overpressure, is found. It is characterized by a negative specific heat at constant volume. An independent electron model within a Z -electron cell is employed. The inadequacy of the wave function in the low-density, low-temperature nonthermodynamic region is shown to be the cause of this overpressure. Numerical examples of the theory for several elements (Li, N, Al, K, and Er) are reported. These results reduce in various limits of temperature and density to the expected behavior, except in the aforementioned region.