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An overview of Purgatorio, a new implementation of the Inferno [Liberman, Phys Rev B 1979;20:4981–9] equation of state model, is presented. The new algorithm emphasizes a novel subdivision scheme for automatically resolving the structure of the continuum density of states, circumventing limitations of the pseudo-R matrix algorithm previously utilized.

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... MEOS provides additional capabilities that are necessary for applications that demand higher accuracy. It is a multiphase framework that allows for the inclusion of phase-specific physics and adoption of more accurate free-energy models, such as the Cell model 24,25 for liquid and Purgatorio 26,27 for electronic ionization and excitation processes that become prominent at high temperatures and pressures. The implementation of Purgatorio and construction of phase boundaries Figure 1. ...

... Since a single analytical function is not sufficient for our purposes, we represent the cold curve as a series of cubic splines (specifically, the "bimond" scheme 28 based on cubic Hermite polynomials) that smoothly connect three domains in density ρ: (1) ideal-gas limit (e.g., ρ < 10 −3 g/cm 3 ); (2) conditions near ambient density up to moderate compression (this intermediate domain is often the one that is examined in DAC studies and is the one we pay most attention to in this study); (3) highly compressed states (up to 10 3 g/cm 3 ) for which we use the computed cold curve from the Purgator-io 26,27 averaged-atom-in jellium model that we describe later. Although the cold curve is ultimately represented by splines defined over a nonuniformly spaced set of control points in ρ, a piecewise-smoothing procedure (fitted by localized polynomials) over multiple regions is applied to reduce some of the numerical noise. ...

... Instead, we apply the same F electron model for all three phases. Specifically in the case of L42, we have employed LLNL's Purgatorio code 26,27 to compute DFT free energies under the averaged-atom-in-jellium framework 26,27,35 and then use these Purgatorio results to model F electron (T, V). This Purgatorio model is widely recognized as the best available global treatment for the F electron contribution of the liquid, which becomes significant at high temperatures. ...

We construct a family of beryllium (Be) multiphase equation of state (EOS) models that consists of a baseline ("optimal") EOS and variations on the baseline to account for physics-based uncertainties. The Be baseline EOS is constructed to reproduce a set of self-consistent data and theory including known phase boundaries, the principal Hugoniot, isobars, and isotherms from diamond-anvil cell experiments. Three phases are considered, including the known hexagonal closed-packed (hcp) phase, the liquid, and the theoretically predicted high-pressure body-centered cubic (bcc) phase. Since both the high-temperature liquid and high-pressure bcc phases lack any experimental data, we carry out ab initio density functional theory (DFT) calculations to obtain new information about the EOS properties for these two regions. At extremely high temperature conditions (>87 eV), DFT-based quantum molecular dynamics simulations are performed for multiple liquid densities using the state-of-the-art Spectral Quadrature methodology in order to validate our selected models for the ion- and electron-thermal free energies of the liquid. We have also performed DFT simulations of hcp and bcc with different exchange-correlation functionals to examine their impact on bcc compressibility, which bound the hcp-bcc transition pressure to within 4 ± 0.5 Mbar. Our baseline EOS predicts the first density maximum along the Hugoniot to be 4.4-fold in compression, while the hcp-bcc-liquid triple-point pressure is predicted to be at 2.25 Mbar. In addition to the baseline EOS, we have generated eight variations to accommodate multiple sources of potential uncertainties such as (1) the choice of free-energy models, (2) differences in theoretical treatments, (3) experimental uncertainties, and (4) lack of information. These variations are designed to provide a reasonable representation of nonstatistical uncertainties for the Be EOS and may be used to assess its sensitivity to different inertial-confinement fusion capsule designs.

... The measured sequence of data points along the shock Hugoniot is shown in Fig. 3 (red curve) with errors that correspond to uncertainty contours of 1σ (red shaded region). Also shown are previous shock Hugoniot measurements 19,[29][30][31] , theoretical calculations made with molecular dynamics based on Kohn-Sham density functional theory (KS-DFT) 32 (orange curve), and models using an average-atom (AA) single ion-in-jellium description with the electronic structure based on Kohn-Sham density functional theory (AA-DFT) [33][34][35][36] (solid black curve) and on Thomas-Fermi-Dirac theory (AA-TFD) 37 (dot-dashed black curve); see also Methods. These models are commonly used to generate EOS tables for, inertial confinement fusion experiments, for example. ...

... Calculations show insensitivity of the theoretical Hugoniot (AA-DFT [33][34][35] ) to fluorobenzene solvent (C 6 H 5 F) for concentrations up to 20%, corresponding to 1% atomic fraction of fluorine (green curve). Concentrations of 0.5% F (red curve) and 0% F (blue curve) are also shown, but not visibly distinguishable. ...

... Extracted shock-front compressions and pressures from radiation hydrodynamic simulations 38 of the experimental platform (red points). The theoretical shock Hugoniot [33][34][35] input to the simulations is also shown with ±2% deviation in compression from the input Hugoniot (black curves). ...

White dwarfs represent the final state of evolution for most stars1–3. Certain classes of white dwarfs pulsate4,5, leading to observable brightness variations, and analysis of these variations with theoretical stellar models probes their internal structure. Modelling of these pulsating stars provides stringent tests of white dwarf models and a detailed picture of the outcome of the late stages of stellar evolution6. However, the high-energy-density states that exist in white dwarfs are extremely difficult to reach and to measure in the laboratory, so theoretical predictions are largely untested at these conditions. Here we report measurements of the relationship between pressure and density along the principal shock Hugoniot (equations describing the state of the sample material before and after the passage of the shock derived from conservation laws) of hydrocarbon to within five per cent. The observed maximum compressibility is consistent with theoretical models that include detailed electronic structure. This is relevant for the equation of state of matter at pressures ranging from 100 million to 450 million atmospheres, where the understanding of white dwarf physics is sensitive to the equation of state and where models differ considerably. The measurements test these equation-of-state relations that are used in the modelling of white dwarfs and inertial confinement fusion experiments7,8, and we predict an increase in compressibility due to ionization of the inner-core orbitals of carbon. We also find that a detailed treatment of the electronic structure and the electron degeneracy pressure is required to capture the measured shape of the pressure–density evolution for hydrocarbon before peak compression. Our results illuminate the equation of state of the white dwarf envelope (the region surrounding the stellar core that contains partially ionized and partially degenerate non-ideal plasmas), which is a weak link in the constitutive physics informing the structure and evolution of white dwarf stars9. Researchers have measured the equation of state of hydrocarbon in a high-density regime, which is necessary for accurate modelling of the oscillations of white dwarf stars.

... where A is the set of all shells from which configurations are constructed. We note that, in general, the number of bound shells is a function of the atomic number, temperature and density, which determines the self consistent central potential -for example, a higher Z element has a larger number of bound shells, due to the higher nucleus charge and a higher density plasma may have a smaller number of shells due to an increased pressure ionization effect (bound states dissolving into the continuum [33,[37][38][39][40][41][42][43][44][45]). One can also define the number of configurations taking into account only charge states with probability larger than p: ...

... [28,29]. In this method, only the bound shells and chemical potential are needed -so that the estimate for the number of populated configurations can be obtained, for example, by solving the Dirac equation in a Thomas-Fermi potential [55] or in a more advanced average atom model potential [33,[37][38][39][40][41][42][43][44][45]. ...

... Detailed finite temperature density functional theory (DFT) calculations in the spherical average atom approximation [33,[37][38][39][40][41][42][43][44][45] followed by STA calculations using the atomic code STAR [28,29], were performed over a wide range of plasma temperatures: 100eV-10keV and densities: 10 −3 − 10 3 g/cm 3 , for the following low, mid and high Z elements -Silicone (Z=14), Iron (Z=26), Xenon (Z=54) and Gold (Z=79). The results are shown in Figs 5-8, and include the average ionization, the chem-ical potential, the number of bound shells, the combinatoric number of configurations over all ionization levels (eq. ...

In hot dense plasmas of intermediate or high-Z elements in the state of local thermodynamic equilibrium, the number of electronic configurations contributing to key macroscopic quantities such as the spectral opacity and equation of state, can be enormous. In this work we present systematic methods for the analysis of the number of relativistic electronic configurations in a plasma. While the combinatoric number of configurations can be huge even for mid-Z elements, the number of configurations which have non negligible population is much lower and depends strongly and non-trivially on temperature and density. We discuss two useful methods for the estimation of the number of populated configurations: (i) using an exact calculation of the total combinatoric number of configurations within superconfigurations in a converged super-transition-array (STA) calculation, and (ii) by using an estimate for the multidimensional width of the probability distribution for electronic population over bound shells, which is binomial if electron exchange and correlation effects are neglected. These methods are analyzed, and the mechanism which leads to the huge number of populated configurations is discussed in detail. Comprehensive average atom finite temperature density functional theory (DFT) calculations are performed in a wide range of temperature and density for several low, mid and high Z plasmas. The effects of temperature and density on the number of populated configurations are discussed and explained.

... Calculations have been done in carbon plasmas at gigabar pressures. Comparisons [8] with the PURGATORIO averageatom model [12,13] have shown that the QMD results are systematically about 0.5 higher than the average ionization predicted by the PURGATORIO code. ...

... The nonrelativistic quantum average-atom model in the muffin-tin approximation to describe the electronic structure in dense plasmas is well known [12][13][14]18,19]. We assume that the electrons are in LTE at T e and the ions at T i . ...

... To avoid this problem, we multiply σ (ω) by ω 2 /(γ 2 + ω 2 ). The free parameter γ is determined such that the sum rule [11] +∞ 0 dωσ (ω) = π e 2 2m eZ N i (12) is respected [14]. By definition [2], the average ionization Z = n(R W S ). ...

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.

... Purgatorio [52]. These models are intended to be valid for arbitrarily high densities where relativistic effects may become important, so they employ solutions to the Dirac equation rather than the Schrödinger equation. ...

... Purgatorio [52] is an AA model based on Liberman's Inferno [51], as discussed in section 2.2.3. ...

... This has implications for thermodynamic consistency of EOS tables calculated from numerical derivatives of neighboring density-temperature points [56]. Purgatorio uses an adaptive-energy gridding method based on Gaussian quadrature [52] and a phase-shift tracking method [56,85] to provide very robust numerics and thermodynamically-consistent quantities. ...

Accurate transport properties—such as opacity, and electrical & thermal conductivities—provide crucial input for the intricate physics models necessary to describe the dynamics of complex, high energy density (HED) systems. This includes stars, giant planets, and inertial confinement fusion plasmas. However, these theoretical transport models present challenges as the phase space often sits at the intersection of solid, liquid, gas, and plasma where many effects of comparable magnitude must be considered. Additionally, the transient nature of such high energy density materials complicates experimental measurement, and many theories remain sparsely benchmarked by data. In the laboratory, HED material must be created via some combination of material compression to very high densities or by adding large amounts of energy to the material in a very short time. This thesis focuses on experiments utilizing the second technique. X-ray free-electron lasers (tau < 100 fs) or short-pulse lasers (tau < 1 ps) are capable of heating materials from room temperature to tens or even many hundreds of eV while keeping densities at appreciable fractions of their ambient value. This allows for the probing of material properties before hydrodynamics phenomena become dominant. First, an experimental platform designed to constrain thermal conductivity models in warm dense matter is presented. Its basis relies on differentially heating multilayer targets (one high-Z layer and one low- to mid-Z layer) to generate a thermal gradient. This concept was first demonstrated using the Titan laser at the Jupiter Laser Facility, creating an intense proton beam to heat a gold/aluminum multilayer target. The temperature, reflectivity, and expansion of the rear surface were observed with time-resolved diagnostics as the thermal energy from the hot gold layer reached the coldest part of the aluminum layer. The data were compared with hydrodynamics models that xxiii self-consistently used the electrical and thermal conductivities to calculate observables. Measured temperatures were too low relative to predictions, possibly indicating the need to decrease tested conductivity models. This experiment was repeated using an X-ray free-electron laser at the Linac Coherent Light Source (LCLS) with gold/iron targets. Data are presented for this work along with calculations and a discussion of how the different drivers impact the experimental design and data quality. Finally, data from a platform designed to measure opacities using short-pulse lasers at the Orion Laser Facility are presented. Spectroscopic measurements of silicon’s K-shell that are both temporally and angularly resolved are benchmarked against the radiation transfer code Cretin. The validity of the commonly-used escape factor approximation is tested against the full solution of the radiation transfer equation and found to be in good agreement for presented experimental conditions. An analysis of the effects of radial gradients on spectroscopically inferred temperatures is found to lead to errors in the peak temperature as large as 50% as well as incorrect cooling rates. This emphasizes the importance of absolute emissivity calibrations and spatially resolved spot size measurements.

... More approximate methods include average atom models and their extensions to include ionic structure [21][22][23][24][25][26]. These are DFT models of one fictitious averaged atom * Electronic address: starrett@lanl.gov ...

... At 100 eV, the average atom model predicts two bound states in the energy range shown (figure 7), as well as a resonance structure in the continuum. The resonance structure in average atom models is well documented [22,23] and is a long lived quasi-stable state, caused by a potential minimum due to the sum of the effective potential and the centrifugal term l(l + 1)/r 2 in the radial Kohn-Sham equation. These bound and resonance states merge into one 'bump' in the KKR-GF result, and the continuum merges with the bound states. ...

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.

... This concept, which has clear computational advantages, has a long history in plasma physics and electronic structure theory. The earliest AA models [94][95][96][97][98] were based on the Thomas-Fermi (TF) approximation [99,100] and modifications thereof; subsequent models built on this premise by adopting a mixed KS-DFT and TF approach for the bound and continuum electrons respectively [101], then treating the full spectrum (discrete and continuous) via KS-DFT [102], and later incorporating effects from outside the central atom such as ionic correlations [103][104][105][106]. AA models continue to be extensively developed and used under a variety of different approaches and assumptions [107][108][109][110][111][112][113][114][115][116][117][118][119][120][121][122][123][124][125]; we also mention here the recent works in Refs. [126,127], which attempt to bridge the gap between AA models and full KS-DFT via novel approaches. ...

... We note that, in modern AA codes, the unbound electrons are usually treated in a more sophisticated manner, either semiclassically (TF) or (more typically) in a fully quantum manner [102,110,111], for example by expanding the continuum states in a discrete set of normalizable states [157,158]. The TF approximation for the unbound density is known to have certain limitations, for example systematically overestimating the chemical potential [119]; these limitations are likely to be exacerbated using the even simpler ideal approximation. ...

Finite-temperature Kohn-Sham density functional theory (KS-DFT) is a widely-used method in warm dense matter (WDM) simulations and diagnostics. Unfortunately, full KS-DFT-molecular dynamics models scale unfavourably with temperature and there remains uncertainty regarding the performance of existing approximate exchange-correlation (XC) functionals under WDM conditions. Of particular concern is the expected explicit dependence of the XC functional on temperature, which is absent from most approximations. Average-atom (AA) models, which significantly reduce the computational cost of KS-DFT calculations, have therefore become an integral part of WDM modeling. In this paper, we present a derivation of a first-principles AA model from the fully-interacting many-body Hamiltonian, carefully analyzing the assumptions made and terms neglected in this reduction. We explore the impact of different choices within this model—such as boundary conditions and XC functionals—on common properties in WDM, for example equation-of-state data, ionization degree and the behavior of the frontier energy levels. Furthermore, drawing upon insights from ground-state KS-DFT, we discuss the likely sources of error in KS-AA models and possible strategies for mitigating such errors.

... This concept, which has clear computational advantages, has a long history in plasma physics and electronic structure theory. The earliest AA models [94][95][96][97][98] were based on the Thomas-Fermi (TF) approximation [99,100] and modifications thereof; subsequent models built on this premise by adopting a mixed KS-DFT and TF approach for the bound and continuum electrons respectively [101], then treating the full spectrum (discrete and continuous) via KS-DFT [102], and later incorporating effects from outside the central atom such as ionic correlations [103][104][105][106]. AA models continue to be extensively developed and used under a variety of different approaches and assumptions [107][108][109][110][111][112][113][114][115][116][117][118][119][120][121][122][123][124][125]; we also mention here the recent works in Refs. 126 and 127 which attempt to bridge the gap between AA models and full KS-DFT via novel approaches. ...

... We note that, in modern AA codes, the unbound electrons are usually treated in a more sophisticated manner, either semi-classically (TF) or (more typically) in a fully quantum manner [102,110,111], for example by expanding the continuum states in a discrete set of normalizable states [156,157]. The TF approximation for the unbound density is known to have certain limitations, for example systematically over-estimating the chemical potential [119]; these limitations are likely to be exacerbated using the even simpler ideal approximation. ...

Finite-temperature Kohn–Sham density-functional theory (KS-DFT) is a widely-used method in warm dense matter (WDM) simulations and diagnostics. Unfortunately, full KS-DFT-molecular dynamics models scale unfavourably with temperature and there remains uncertainty regarding the performance of existing approximate exchange-correlation (XC) functionals under WDM conditions. Of particular concern is the expected explicit dependence of the XC functional on temperature, which is absent from most approximations. Average-atom (AA) models, which significantly reduce the computational cost of KS-DFT calculations, have therefore become an integral part of WDM modelling. In this paper, we present a derivation of a first-principles AA model from the fully-interacting many-body Hamiltonian, carefully analysing the assumptions made and terms neglected in this reduction. We explore the impact of different choices within this model — such as boundary conditions and XC functionals — on common properties in WDM, for example equation-of-state data, ionization degree and the behaviour of the frontier energy levels. Furthermore, drawing upon insights from ground-state KS-DFT, we discuss the likely sources of error in KS-AA models and possible strategies for mitigating such errors.

... In comparison, average atom (AA) models are computationally far less demanding because they place only a single nucleus at the center of a cloud of electrons and determine the resulting states. Many variations of the AA approach exist [25,[31][32][33]. However, when these methods are applied to guide or interpret experiments at extreme densities or highly degenerate plasma conditions, the results appear to deviate from predictions of ab initio methods [26,[34][35][36], which provided the initial motivation for this study. ...

Average atom (AA) models allow one to efficiently compute electronic and optical properties of materials over a wide range of conditions and are often employed to interpret experimental data. However, at high pressure, predictions from AA models have been shown to disagree with results from ab initio computer simulations. Here we reconcile these deviations by developing an innovative type of AA model, AVION, that computes the electronic eigenstates with novel boundary conditions within the ion sphere. Bound and free states are derived consistently. We drop the common AA image that the free-particle spectrum starts at the potential threshold, which we found to be incompatible with ab initio calculations. We perform ab initio simulations of crystalline and liquid carbon and aluminum over a wide range of densities and show that the computed band structure is in very good agreement with predictions from AVION.

... This breaks the consistency between free and bound electrons so important for thermodynamic consistency of equation of state (EOS) models [31], but it is of less importance for photon spectra due to the (generally) large number of configurations [25]. In average atom based EOS models a large amount of effort has been devoted to accurate treatment of the free orbitals [28,[31][32][33]. However, for spectral resolution, an important physical effect completely missing in these single center models for dense plasmas is multiple scattering, which destroys the atomic eigenstate character of loosely bound orbitals [34]. ...

Charge state distributions in hot, dense plasmas are a key ingredient in the calculation of spectral quantities like the opacity. However, they are challenging to calculate, as models like Saha-Boltzmann become unreliable for dense, quantum plasmas. Here we present a new variational model for the charge state distribution, along with a simple model for the energy of the configurations that includes the orbital relaxation effect. Comparison with other methods reveals generally good agreement with average atom based calculations, the breakdown of the Saha-Boltzmann method, and mixed agreement with a chemical model. We conclude that the new model gives a relatively inexpensive, but reasonably high fidelity method of calculating the charge state distribution in hot dense plasmas, in local thermodynamic equilibrium.

... The first AA model that was developed is the well-known semi-classical Thomas-Fermi cell model [1][2][3]. A significant advance was the quantum mechanical (Kohn-Sham) and relativistic AA model INFERNO [4] and its modern implementations PURGATORIO [5] and TARTARUS [6]. The most formally rigorous AA model published to date is the VAAQP [7] and numerous variations of AA models have been developed. ...

We have developed an efficient and versatile model to describe warm and hot dense matter that couples an average atom model with the integral fluid equations for the ion correlations. This model provides all bound and free electronic states and wave functions, the interaction potentials and all correlation functions without adjustable parameters. The electrons can be described quantum mechanically (Schrödinger equation) or semi-classically with the Thomas-Fermi model. The ion-ion pair potential can be used in a classical molecular dynamics simulation to yield equation of state and dynamic properties of dense plasmas, including mixtures. This "pseudo-atom molecular dynamics" model, or PAMD, is more approximate than ab initio methods but presents distinct computational advantages. Extensions of the model allow the calculation of diffusion coefficients, viscosity, X-ray Thomson Scattering spectra, DC conductivities and opacities. LA-UR-19-27814

... (unlike the ARES model) accounts for the effects of electron shell structure (Hansen et al. 2006;Wilson et al. 2006;Sterne et al. 2007). The PURGATORIO results are expected to be of higher fidelity than those from the ARES model, and the former are treated here as benchmark data. ...

This paper describes a computational investigation of multimode instability growth and multimaterial mixing induced by multiple shock waves in a high-energy-density (HED) environment, where pressures exceed 1 Mbar. The simulations are based on a series of experiments performed at the National Ignition Facility (NIF) and designed as an HED analogue of non-HED shock-tube studies of the Richtmyer-Meshkov instability and turbulent mixing. A three-dimensional computational modelling framework is presented. It treats many complications absent from canonical non-HED shock-tube flows, including distinct ion and free-electron internal energies, non-ideal equations of state, radiation transport and plasma-state mass diffusivities, viscosities and thermal conductivities. The simulations are tuned to the available NIF data, and traditional statistical quantities of turbulence are analysed. Integrated measures of turbulent kinetic energy and enstrophy both increase by over an order of magnitude due to reshock. Large contributions to enstrophy production during reshock are seen from both the baroclinic source and enstrophy-dilatation terms, highlighting the significance of fluid compressibility in the HED regime. Dimensional analysis reveals that Reynolds numbers and diffusive Péclet numbers in the HED flow are similar to those in a canonical non-HED analogue, but conductive Péclet numbers are much smaller in the HED flow due to efficient thermal conduction by free electrons. It is shown that the mechanism of electron thermal conduction significantly softens local spanwise gradients of both temperature and density, which causes a minor but non-negligible decrease in enstrophy production and small-scale mixing relative to a flow without this mechanism. © The Author(s), 2021. Published by Cambridge University Press.

... To address these problems, a revised program, PURGATORIO, was written [19]. PURGATORIO did not however include the ion-thermal calculation. ...

Recent path-integral Monte Carlo and quantum molecular dynamics simulations have shown that computationally efficient average-atom models can predict thermodynamic states in warm dense matter to within a few percent. One such atom-in-jellium model has typically been used to predict the electron-thermal behavior only, although it was previously developed to predict the entire equation of state (EOS). We report completely atom-in-jellium EOS calculations for Be, Al, Si, Fe, and Mo, as elements representative of a range of atomic number and low-pressure electronic structure. Comparing the more recent method of pseudoatom molecular dynamics, atom-in-jellium results were similar: sometimes less accurate, sometimes more. All these techniques exhibited pronounced effects of electronic shell structure in the shock Hugoniot which are not captured by Thomas-Fermi based EOS. These results demonstrate the value of a hierarchical approach to EOS construction, using average-atom techniques with shell structure to populate a wide-range EOS surface efficiently, complemented by more rigorous three-dimensional multiatom calculations to validate and adjust the EOS.

... Zimmerman's model [110] is an example of the first approach; however, it does not give any opinion on what to use forZ. One option is to use values from More's fit to the electron density at the ion-sphere radius of a Thomas-Fermi atom [43] or from a more sophisticated equation-of-state code like Purgatorio [114,115,83], which outputs the total number of electrons with positive energy per atom, but the best practice is to use ā Z designed for the observable of interest. The one approximation used in warm dense matter for stopping power is the local density approximation (LDA): ...

We present the results of the first Charged-Particle Transport Coefficient Code Comparison Workshop, which was held in Albuquerque, NM October 4-6, 2016. In this first workshop, scientists from eight institutions and four countries gathered to compare calculations of transport coefficients including thermal and electrical conduction, electron-ion coupling, inter-ion diffusion, ion viscosity, and charged particle stopping powers. Here, we give general background on Coulomb coupling and computational expense, review where some transport coefficients appear in hydrodynamic equations, and present the submitted data. Large variations are found when either the relevant Coulomb coupling parameter is large or computational expense causes difficulties. Understanding the general accuracy and uncertainty associated with such transport coefficients is important for quantifying errors in hydrodynamic simulations of inertial confinement fusion and high-energy density experiments.

... Ab initio molecular dynamics (AIMD) simulations, when the classical molecular dynamics for the ionic degrees of freedom is combined with the DFT treatment for the electrons, take into account quantum effects and treat all electronic thermally occupied and empty states on the same basis. All of this is important to accurately predict transport coefficients via Kubo-Greenwood formalism [14,15], especially in the partially degenerate, strongly coupled regime when the chemical-and plasma-based models [16,17] become inaccurate [4]. ...

Exchange-correlation (XC) thermal effects for transport and optical properties of deuterium along the principal Hugoniot are investigated. The study is performed using ab initio molecular dynamics simulations within the Mermin-Kohn-Sham density functional theory. XC thermal effects are taken into account via the temperature-dependent Karasiev-Dufty-Trickey generalized gradient approximation functional [V. V. Karasiev et al., Phys. Rev. Lett. 120, 076401 (2018)]. We find that XC thermal effects account for the softening of the Hugoniot at pressures P>250 GPa and improve agreement with recent experimental measurements. Also, XC thermal effects lead to the reflectivity increase by about 2% for shock speeds above 20 km/s. The calculated reflectivity for shock speeds up to 50 km/s is in excellent agreement with recent experimental measurements on the Omega Laser System. The dc conductivity is increased by about 4% due to XC thermal effects. The system evolution along the Hugoniot crosses the so-called warm-dense-matter regime, and XC thermal effects must be taken into account to accurately predict the thermophysical properties across warm-dense conditions.

... For example, many widely used EOS tables are based on so-called average atom models. These attempt to capture the properties of one averaged atom that is representative of the system [10][11][12][13][14][15], but do not include the effect of ionic disorder. As a result, ionic disorder has to be included via a separate model [16][17][18]. ...

The effect of ionic disorder on the principal Hugoniot is investigated using Multiple Scattering Theory to very high pressure (Gbar). Calculations using molecular dynamics to simulate ionic disorder, are compared to those with a fixed crystal lattice, for both carbon and aluminum. For the range of conditions considered here, we find that ionic disorder is most important at the onset of shell ionization and that at higher pressures, the subtle effect of the ionic environment is overwhelmed by the larger number of ionized electrons with higher thermal energies.

... In comparison, average atom (AA) models are computationally far less demanding because they place only a single nucleus at the center of a cloud of electrons and determine the resulting states. Many variations of the AA approach exist [25,[31][32][33]. However, when these methods are applied to guide or interpret experiments at extreme densities or highly degenerate plasma conditions, the results appear to deviate from predictions of ab initio methods [26,[34][35][36], which provided the initial motivation for this study. ...

... Such a modern multiphase EOS generated at LANL for aluminum is shown in Fig. 2. Other modern EOS models include that of the Lawrence Livermore National Laboratory PURGATORIO, a novel implementation of the INFERNO EOS physical model. 50 Examining the compression of materials at zero temperature, DFT calculations are leveraged with diamond anvil cell (DAC) measurements and then extended to high compression by matching TFD calculations. DAC is a measurement of static compression, typically measured at room temperature. ...

The hydrodynamic response of materials under extreme conditions of pressure, temperature, and strain is dependent on the equation of state of the matter in all its states of existence. The Trinity plutonium implosion device development required the Los Alamos physics and engineering research community to advance the understanding of equations of state further and faster than ever before. The unpredicted high yield from the Trinity fission device explosion and the push to design the “Super” thermonuclear device initiated 75 years of unprecedented research and technological progress in equation of state development. This paper describes the progress made on equation of state development during and since the Manhattan Project at Los Alamos.

... For example, many widely used EOS tables are based on so-called average atom models. These attempt to capture the properties of one averaged atom that is representative of the system [10][11][12][13][14][15], but do not include the effect of ionic disorder. As a result, ionic disorder has to be included via a separate model [16][17][18]. ...

The effect of ionic disorder on the principal Hugoniot is investigated using multiple scattering theory to very high pressure (Gbar). Calculations using molecular dynamics to simulate ionic disorder are compared to those with a fixed crystal lattice, for both carbon and aluminum. For the range of conditions considered here we find that ionic disorder has a relatively minor influence. It is most important at the onset of shell ionization and we find that, at higher pressures, the subtle effect of the ionic environment is overwhelmed by the larger number of ionized electrons with higher thermal energies.

... The methodology applied here is the product of a long and continuing evolution in NIF postshot modeling. [19][20][21][22][23][24] As described previously, the simulations presented below are run with the radiation hydrodynamics code HYDRA 26 and include multigroup diffusive radiation transport, thermonuclear burn, Monte Carlo burn product transport, and tabular equation of state (EOS) [27][28][29] and opacity data. 30 In these capsule-only simulations, the hohlraum is treated as a spherical boundary surface where a prescribed inward x-ray flux is applied. ...

The goal of an inertially confined, igniting plasma on the National Ignition Facility (NIF) [M. L. Spaeth, Fusion Sci. Technol. 69, 25 (2016)] remains elusive. However, there is a growing understanding of the factors that appear to be limiting current implosion performance. And with this understanding, the question naturally arises: What conditions will ultimately be required to achieve ignition, either by continuing to improve the quality of current implosions, or by hydrodynamically scaling those implosions to larger driver energies on some future facility? Given the complexity of NIF implosions, answering this question must rely heavily on sophisticated numerical simulations. In particular, those simulations must respect the three-dimensionality of real NIF implosions and also resolve the wide range of scales for the many perturbation sources that degrade them. This prospectus article reviews the current state of detailed modeling of NIF implosions, the scaling to ignition from recent experiments that that modeling implies, and areas for future improvements in modeling technique that could increase understanding and further enhance predictive capabilities. Given the uncertainties inherent in any extrapolation, particularly for a process as nonlinear as ignition, there will be no definitive answer on the requirements for ignition until it is actually demonstrated experimentally. However, with continuing improvements in modeling technique and a growing experience base from NIF, the requirements for ignition are becoming clearer.

The equation of state (EOS) of materials at warm dense conditions poses significant challenges to both theory and experiment. We report a combined computational, modeling, and experimental investigation leveraging new theoretical and experimental capabilities to investigate warm-dense boron nitride (BN). The simulation methodologies include path integral Monte Carlo (PIMC), several density functional theory (DFT) molecular dynamics methods [plane-wave pseudopotential, Fermi operator expansion (FOE), and spectral quadrature (SQ)], activity expansion (actex), and all-electron Green's function Korringa-Kohn-Rostoker (mecca), and compute the pressure and internal energy of BN over a broad range of densities and temperatures. Our experiments were conducted at the Omega laser facility and the Hugoniot response of BN to unprecedented pressures (1200–2650 GPa). The EOSs computed using different methods cross validate one another in the warm-dense matter regime, and the experimental Hugoniot data are in good agreement with our theoretical predictions. By comparing the EOS results from different methods, we assess that the largest discrepancies between theoretical predictions are ≲4% in pressure and ≲3% in energy and occur at 106K, slightly below the peak compression that corresponds to the K-shell ionization regime. At these conditions, we find remarkable consistency between the EOS from DFT calculations performed on different platforms and using different exchange-correlation functionals and those from PIMC using free-particle nodes. This provides strong evidence for the accuracy of both PIMC and DFT in the high-pressure, high-temperature regime. Moreover, the recently developed SQ and FOE methods produce EOS data that have significantly smaller statistical error bars than PIMC, and so represent significant advances for efficient computation at high temperatures. The shock Hugoniot predicted by PIMC, actex, and mecca shows a maximum compression ratio of 4.55±0.05 for an initial density of 2.26g/cm3, higher than the Thomas-Fermi predictions by about 5%. In addition, we construct tabular EOS models that are consistent with the first-principles simulations and the experimental data. Our findings clarify the ionic and electronic structure of BN over a broad range of temperatures and densities and quantify their roles in the EOS and properties of this material. The tabular models may be utilized for future simulations of laser-driven experiments that include BN as a candidate ablator material.

We assess the relative importance of ionic structure on the opacity of dense plasmas by using the potential of mean force as a scattering potential within the Kubo-Greenwood formalism. We compare results from the potential of mean force, which includes realistic ionic structure, to results using an average atom potential, which includes only a crude treatment of ionic structure. Comparisons with less approximate but more expensive DFT-MD simulations for aluminum plasma reveal that the mean force generally improves agreement for DC conductivity. We also see improvement when applying the mean force to free-free transitions, whereas for bound-bound and bound-free transitions the mean force leads to poorer agreement on transition energies. Further, we assess the impact of accounting for correlations within the plasma at the temperature and density conditions relevant to iron opacity measurements at Sandia's Z machine facility [Bailey et al., Nature 517:56-59, 2015] and find that these correlations do not account for the discrepancy between the measurements and leading opacity calculations.

We are reporting the observation of the breakdown of electrons’ degeneracy and emergence of classical statistics in the simplest element: metallic deuterium. We have studied the optical reflectance, shock velocity, and temperature of dynamically compressed liquid deuterium up to its Fermi temperature TF. Above the insulator-metal transition, the optical reflectance shows the distinctive temperature-independent resistivity saturation, which is prescribed by Mott’s minimum metallic limit, in agreement with previous experiments. At T>0.4 TF, however, the reflectance of metallic deuterium starts to rise with a temperature-dependent slope, consistent with the breakdown of the Fermi surface. The experimentally inferred electron-ion collisional time in this region exhibits the characteristic temperature dependence expected for a classical Landau-Spitzer plasma. Our observation of electron degeneracy lifting extends studies of degeneracy to new fermionic species—electron Fermi systems—and offers an invaluable benchmark for quantum statistical models of Coulomb systems over a wide range of temperatures relevant to dense astrophysical objects and ignition physics.

The electron-ion relaxation dynamics in warm dense iron were investigated by time-resolved x-ray absorption near-edge spectroscopy (XANES). A novel analysis, combining ab initio density functional theory (DFT) and two temperature model (TTM) simulations, was developed to calculate the x-ray absorption spectra as a function of delay time. Here we present experimental evidence of changes at the XAS L3 edge of iron that are consistent with the reduction of the electron-phonon coupling under warm dense matter conditions. The experimental results are in agreement with the model that takes both the electron (Te) and the ion temperature (Ti) dependence of the thermophysical properties into consideration, while models where either constant electron-phonon coupling factor (G) or only Te-dependent G are used do not agree with the observed relaxation dynamics of iron.

We compare two formulas obtained from first principles to calculate the electron-ion coupling factor for temperature relaxation in dense plasmas. The quantum average-atom model is used to calculate this electron-ion coupling factor. It is shown that if the two formulas agree at sufficiently high temperature so that the potential energy is of limited importance, i.e., when the plasma is said to be kinetic, and are consistent with the Landau-Spitzer formula, then they strongly differ in the warm-dense-matter regime. Only one of the two is shown to be consistent with quantum molecular dynamics approach. We use this point to determine which formula is valid to describe temperature relaxation between electrons and ions in warm and hot dense plasmas.

As a simple metal, aluminum's valence band is usually described as a free-electron gas with three electrons per atom. The discrepancies between the experimental electronic Grüneisen parameter and heat capacity and their free-electron-gas counterparts are usually attributed to electron-phonon coupling. We recently calculated thermal electronic contributions to aluminum's internal energies with our average-atom code paradisio and obtained results that contradict this point of view. Our code also pointed out the overlap of the sp valence band by the 3d one, resulting in an sp to d electron transfer. Applying Sommerfeld's temperature expansion method to the electron-electron Coulomb part of the internal energy, we relate the electronic Grüneisen parameter, T=0 K isotherm, and thermal contributions to the internal energy to a parameter α describing the fraction of d electrons resulting from p to d transfer. Finally, we find a unique value of this parameter that provides a consistent explanation for the experimental Grüneisen parameter, for T=0 K energy deduced from experimental shock Hugoniot data, as well as for our average-atom thermal contributions to internal energy.

We report measurements of K-shell fluorescence lines induced by fast electrons in ramp-compressed Co targets. The fluorescence emission was stimulated by fast electrons generated through short-pulse laser-solid interaction with an Al target layer. Compression up to 2.1× solid density was achieved while maintaining temperatures well below the Fermi energy, effectively removing the thermal effects from consideration. We observed small but unambiguous redshifts in the Kβ fluorescence line relative to unshifted Cu Kα. Redshifts up to 2.6 eV were found to increase with compression and to be consistent with predictions from self-consistent models based on density-functional theory.

A notorious challenge in high-pressure science is to develop an equation of state (EOS) that explicitly treats chemical reactions. For instance, many materials tend to dissociate at high pressures and temperatures where the chemical bonds that hold them together break down. We present an EOS for carbon dioxide (CO2) that allows for dissociation and captures the key material behavior in a wide range of pressure-temperature conditions. Carbon dioxide is an ideal prototype for the development of a wide-ranging EOS that allows for chemical-dissociation equilibria since it is one of the simplest polyatomic systems and because it is of great interest in planetary science and in the study of detonations. Here, we show that taking dissociation into account significantly improves the accuracy of the resulting EOS compared to other EOSs that either neglect chemistry completely or treat CO2 dissociation in a more rudimentary way.

We present a relativistic quantum average-atom model where bound and free electrons are treated self-consistently. We propose a relativistic exchange potential that generalizes the nonrelativistic Kohn-Sham exchange potential. The relativistic exchange potential is considered in detail. We interpolate smoothly between the nonrelativistic and ultrarelativistic limits. We neglect the correlation effect. Numerical examples are given.

We present calculations of electrical resistivity in dense plasmas using the average-atom model. The Born approximation is proposed to improve the computations especially in the hot domain of the density and temperature plane. Both the nonrelativistic and relativistic regimes are considered. Numerical examples are given.

This chapter describes the theoretical equations of state (EOSs) for porous and granular materials. In principle the EOS for a porous/granular material is identical to the EOS for the equivalent non-porous material; however, the requirement that the EOS must provide a realistic model of the material in its porous/granular state presents additional challenges. Broadly these challenges are first that the regions of thermodynamic phase space of interest are poorly described by standard wide-ranging EOS models and second, accurate measurement of materials properties that are routinely used to constrain an EOS can be more difficult to obtain. This chapter describes in detail the challenges and methods for EOS generation of porous materials.

Expressions of pressure in warm and hot dense matter using the average-atom model are presented. They are based on the stress-tensor approach. Nonrelativistic and relativistic cases are considered. The obtained formulas are simple and can be easily implemented in an average-atom model code. Comparisons with experimental data and quantum molecular dynamics and path integral Monte Carlo simulations are shown. The present formalism agrees well with experimental results for a large variety of elements in the warm dense matter regime and with ab initio simulations in the warm and hot dense matter regime for aluminum.

Argon is the most abundant noble gas on Earth and its noble, atomic fluid nature makes it an excellent candidate for comparison of experiment and theory at extreme conditions. We performed a combined computational and experimental study on shock compressed cryogenic liquid argon. Using Sandia's Z machine, we shock compressed liquid argon to 600 GPa and reshock states up to 950 GPa. Laser shock experiments at the Omega Laser facility extend the principal Hugoniot to 1000 GPa and provided temperature data along the principal Hugoniot. The plate impact experiments and laser shock experiments used well-characterized impedance matching standards and demonstrate consistent results between the two platforms over a common range. Density functional theory based molecular dynamics simulations provided additional data on the Hugoniot to 600 GPa. The combined experimental data and simulation results provide constraints on the development of new equation of state models at extreme conditions.

We present the results of the first Charged-Particle Transport Coefficient Code Comparison Workshop, which was held in Albuquerque, NM October 4–6, 2016. In this first workshop, scientists from eight institutions and four countries gathered to compare calculations of transport coefficients including thermal and electrical conduction, electron–ion coupling, inter-ion diffusion, ion viscosity, and charged particle stopping powers. Here, we give general background on Coulomb coupling and computational expense, review where some transport coefficients appear in hydrodynamic equations, and present the submitted data. Large variations are found when either the relevant Coulomb coupling parameter is large or computational expense causes difficulties. Understanding the general accuracy and uncertainty associated with such transport coefficients is important for quantifying errors in hydrodynamic simulations of inertial confinement fusion and high-energy density experiments.

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.

Boron carbide (B4C) is of both fundamental scientific and practical interest due to its structural complexity and how it changes upon compression, as well as its many industrial uses and potential for use in inertial confinement fusion (ICF) and high-energy density physics experiments. We report the results of a comprehensive computational study of the equation of state (EOS) of B4C in the liquid, warm dense matter, and plasma phases. Our calculations are cross-validated by comparisons with Hugoniot measurements up to 61 megabar from planar shock experiments performed at the National Ignition Facility (NIF). Our computational methods include path integral Monte Carlo, activity expansion, as well as all-electron Green's function Korringa-Kohn-Rostoker and molecular dynamics that are both based on density functional theory. We calculate the pressure-internal energy EOS of B4C over a broad range of temperatures (∼6×103–5×108 K) and densities (0.025–50 g/cm3). We assess that the largest discrepancies between theoretical predictions are ≲5% near the compression maximum at 1–2×106 K. This is the warm-dense state in which the K shell significantly ionizes and has posed grand challenges to theory and experiment. By comparing with different EOS models, we find a Purgatorio model (LEOS 2122) that agrees with our calculations. The maximum discrepancies in pressure between our first-principles predictions and LEOS 2122 are ∼18% and occur at temperatures between 6×103–2×105 K, which we believe originate from differences in the ion thermal term and the cold curve that are modeled in LEOS 2122 in comparison with our first-principles calculations. To account for potential differences in the ion thermal term, we have developed three new equation-of-state models that are consistent with theoretical calculations and experiment. We apply these new models to 1D hydrodynamic simulations of a polar direct-drive NIF implosion, demonstrating that these new models are now available for future ICF design studies.

The detailed radiative properties of plasmas in non-local thermodynamic equilibrium (NLTE) are important for determining experimental plasma states. However, a complete detailed-level-accounting approach calculation is impractical for mid- and high-Z elements. Herein, we propose a hybrid method for obtaining the detailed radiative properties of mid-Z NLTE plasmas. First, a large-scale rate equation within the framework of a detailed-configuration-accounting method is established using atomic data in a configuration–configuration formalism. Second, we assume that the population distributions in fine-structure levels belonging to a particular configuration are in equilibrium. Thus, the populations at fine-structure levels are obtained through the populations in the corresponding configurations. Finally, detailed radiative properties are calculated using the populations in fine-structure levels and radiative data in level–level formalism. Such a method can balance computation costs and accuracy. Examples utilizing Ge plasmas demonstrate that the proposed method can better predict detailed structures in emission spectra than the detailed-configuration-accounting method.

SpK is part of the numerical codebase at Imperial College London used to model high energy density physics (HEDP) experiments. SpK is an efficient atomic and microphysics code used to perform detailed configuration accounting calculations of electronic and ionic stage populations, opacities and emissivities for use in post-processing and radiation hydrodynamics simulations. This is done using screened hydrogenic atomic data supplemented by the NIST energy level database. An extended Saha model solves for chemical equilibrium with extensions for non-ideal physics, such as ionisation potential depression, and non thermal equilibrium corrections. A tree-heap (treap) data structure is used to store spectral data, such as opacity, which is dynamic thus allowing easy insertion of points around spectral lines without a-priori knowledge of the ion stage populations. Results from SpK are compared to other codes and descriptions of radiation transport solutions which use SpK data are given. The treap data structure and SpK's computational efficiency allows inline post-processing of 3D hydrodynamics simulations with a dynamically evolving spectrum stored in a treap.

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.

High-pressure equation of state and isentropic sound speed data for fluid silicon to pressures of 2100 GPa (21 Mbar) are reported. Principal Hugoniot measurements were performed using impedance matching techniques with α-quartz as the reference. Sound speeds were determined by time correlating imposed shock-velocity perturbations in both the sample (Si) and reference material (α-quartz). A change in shock velocity versus particle velocity (us–up) slope on the fluid silicon principal Hugoniot is observed at 200 GPa. Density functional theory based quantum molecular dynamics simulations suggest that both an increase in ionic coordination and a 50% increase in average ionization are coincident with this experimentally observed change in slope.

In hot dense plasmas of intermediate or high-Z elements in the state of local thermodynamic equilibrium, the number of electronic configurations contributing to key macroscopic quantities such as the spectral opacity and equation of state can be enormous. In this work we present systematic methods for the analysis of the number of relativistic electronic configurations in a plasma. While the combinatoric number of configurations can be huge even for mid-Z elements, the number of configurations which have non-negligible population is much lower and depends strongly and nontrivially on temperature and density. We discuss two useful methods for the estimation of the number of populated configurations: (i) using an exact calculation of the total combinatoric number of configurations within superconfigurations in a converged super-transition-array (STA) calculation, and (ii) by using an estimate for the multidimensional width of the probability distribution for electronic population over bound shells, which is binomial if electron exchange and correlation effects are neglected. These methods are analyzed, and the mechanism which leads to the huge number of populated configurations is discussed in detail. Comprehensive average-atom finite-temperature density functional theory (DFT) calculations are performed in a wide range of temperature and density for several low-, mid-, and high-Z plasmas. The effects of temperature and density on the number of populated configurations are discussed and explained.

We study the interaction of extremely short and high-intensity x-ray pulses with a 1.0μm thick Al foil. Four pulse lengths – 100 fs, 200 fs, 300 fs, and 400 fs – are considered. The photon energy is 1830 eV and the pulse intensity is 1017W/cm2. The interaction dynamics are calculated via a radiation hydrodynamic code. The x-ray laser pulse heats the target isochorically. It generates a homogeneous hot dense matter; electrons are hotter than ions. The simulation of the interaction of pump and probe pulses with a delay time in the fs scale provides that the probe pulse heats the target significantly. A Monte-Carlo method is used to provide a microscopic description; the electron distribution function shows a two-temperature system. The electron distribution has spikes at the energy difference between the k-edges of Al ions and the energy of incident photons. The energies of these spikes depend on the considered ionization depression model. The Chihara formula and the non-equilibrium random phase approximation are utilized to calculate the x-ray Thomson scattering spectrum (XRTS). For collective scattering, the plasmon peaks are a function of the pulse lengths and the electron distribution function. Therefore, when XRTS is fitted to a measured spectrum may give the target density, the target temperature, and the microscopic electron distribution function.

We study the notion of Friedel sum rule at finite temperature in hot dense plasmas. Using the average-atom model, we establish expressions for the Friedel sum rule at zero and finite temperature using non-relativistic or relativistic approaches. Formulas are also given using the Born approximation for the phase shifts. Numerical examples are provided. The Friedel sum rule is a stringent test of the internal consistency of a quantum average-atom model. The question of normalization of free wavefunctions is also discussed.

Using a modified version of the pseudoatom molecular-dynamics approach, the silicon and oxygen equations of state were generated and then employed to construct the equation of state of silicon dioxide. The results are supported by the close agreement with ab initio simulations of the silicon pressure and experimental shock Hugoniot of silicon dioxide. Ion thermal contributions to thermodynamic functions provided by the PAMD simulations are compared to their counterparts obtained with the one-component plasma and charged-hard-sphere approximations.

We show with a recently devised extended first-principles molecular dynamics method that calculated Hugoniots of poly-α-methylstyrene agree well with precision experimental results of Kritcher et al. [Nature (London) 584, 51 (2020)] and Döppner et al. [Phys. Rev. Lett. 121, 025001 (2018)]. The deviation is smaller than 0.8%. This agreement does not sensitively rely on the approximations in the employed first-principles methods as long as underlying physics are well described, as illustrated in the calculation of equation of state for polystyrene covering the warm dense regime. These results may stimulate a broad range of quantitative investigations on warm dense matter that were not thought possible before, and may thus afford a new prospect to the field of inertial confinement fusion and high-energy-density physics.

Time-resolved radiography can be used to obtain absolute shock Hugoniot states by simultaneously measuring at least two mechanical parameters of the shock, and this technique is particularly suitable for one-dimensional converging shocks where a single experiment probes a range of pressures as the converging shock strengthens. However, at sufficiently high pressures, the shocked material becomes hot enough that the x-ray opacity falls significantly. If the system includes a Lagrangian marker such that the mass within the marker is known, this additional information can be used to constrain the opacity as well as the Hugoniot state. In the limit that the opacity changes only on shock heating, and not significantly on subsequent isentropic compression, the opacity of the shocked material can be determined uniquely. More generally, it is necessary to assume the form of the variation of opacity with isentropic compression or to introduce multiple marker layers. Alternatively, assuming either the equation of state or the opacity, the presence of a marker layer in such experiments enables the non-assumed property to be deduced more accurately than from the radiographic density reconstruction alone. An example analysis is shown for measurements of a converging shock wave in polystyrene at the National Ignition Facility.

The discrepancies between theoretical and experimental opacities reported by experiments performed at the Sandia National Laboratory Z-pinch relevant to the solar interior remain unexplained. The suggestion that two-photon ionization could help resolve the discrepancies was recently examined and found not to account for the higher than predicted measured opacities. That test, however, was limited in scope and is now extended to include excited configurations and different charge states of several elements. Comparisons of one- and two-photon ionization cross-sections show that the latter fail to resolve the aforementioned discrepancies.

Charge state distributions in hot, dense plasmas are a key ingredient in the calculation of spectral quantities like the opacity. However, they are challenging to calculate, as models like Saha–Boltzmann become unreliable for dense, quantum plasmas. Here, we present a new variational model for the charge state distribution, along with a simple model for the energy of the configurations that includes the orbital relaxation effect. Comparison with other methods reveals generally good agreement with average atom-based calculations, the breakdown of the Saha–Boltzmann method, and mixed agreement with a chemical model. We conclude that the new model gives a relatively inexpensive, but reasonably high fidelity method of calculating the charge state distribution in hot dense plasmas, in local thermodynamic equilibrium.

We are reporting the observation of the breakdown of electrons degeneracy and emergence of classical statistics in the simplest element: metallic deuterium. We have studied the optical reflectance, shock velocity and temperature of dynamically compressed liquid deuterium up to its Fermi temperature, TF. Above the insulator-metal transition, the optical reflectance shows the distinctive temperature-independent resistivity saturation, which is prescribed by Mott minimum metallic limit, in agreement with previous experiments. At T > 0.4 TF, however, the reflectance of metallic deuterium starts to rise with a temperature-dependent slope, consistent with the breakdown of the Fermi surface. The experimentally inferred collisional time in this region exhibits the characteristic temperature dependence expected for classical Landau-Spitzer plasma. Our observation of electron degeneracy lifting extends studies of degeneracy to new fermionic species, electron Fermi systems and offers an invaluable benchmark for quantum statistical models of Coulomb systems over a wide range of temperatures relevant to dense astrophysical objects and ignition physics.

A method for discretizing the continuum by using a transformed harmonic oscillator basis has recently been presented [Phys. Rev. A 63, 052111 (2001)]. In the present paper, we propose a generalization of that formalism which does not rely on the harmonic oscillator for the inclusion of the continuum in the study of weakly bound systems. In particular, we construct wave functions that represent the continuum by making use of families of orthogonal polynomials whose weight function is the square of the ground state wave function, expressed in terms of a suitably scaled variable. As an illustration, the formalism is applied to one-dimensional Morse, Pöschl-Teller, and square well potentials. We show how the method can deal with potentials having several bound states, and for the square well case we present a comparison of the discretized and exact continuum wave functions.

First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.

Electrical resistivity, pressure, and internal energy variation of warm dense correlated titanium (density 0.2 g/cm(3)) and aluminum (density 0.1 g/cm(3)) plasmas are measured using a homogeneous and thermally equilibrated media produced inside an isochoric closed-vessel plasma. These data are compared to detailed calculations based on the density functional theory. In the studied temperature range (15,000-30,000 K), it appears that both exchange-correlation and ion-ion interaction treatments are of great importance to calculate accurate theoretical values.

The quotidian equation of state (QEOS) is a general‐purpose equation of state model for use in hydrodynamic simulation of high‐pressure phenomena. Electronic properties are obtained from a modified Thomas–Fermi statistical model, while ion thermal motion is described by a multiphase equation of state combining Debye, Grüneisen, Lindemann, and fluid‐scaling laws. The theory gives smooth and usable predictions for ionization state, pressure, energy, entropy, and Helmholtz free energy. When necessary, the results may be modified by a temperature‐dependent pressure multiplier which greatly extends the class of materials that can be treated with reasonable accuracy. In this paper a comprehensive evaluation of the resulting thermodynamic data is given including comparison with other theories and shock‐wave data.

This is the first of a series of papers concerning the electrical and thermal transport properties of dense plasmas. Temperatures and densities considered range from zero to 2(13) eV and 2(-13) to 2(13) times compressed. Theoretical calculations of electrical conductivities using the t-matrix version of the Ziman theory with various self-consistent ionic potential models are described. The theoretical basis is described and illustrative results are given.

When calculating radial wavefunctions for use in such problems as electron scattering, the step size of a step-by-step method is severely limited by the oscillatory nature of the solution. However, the radial Schrödinger equation may be solved by computing the phase and amplitude of the oscillations; these two functions are slowly varying and permit the use of a much larger step size. In this paper it is shown how the method may be adapted to the solution of the relativistic radial Dirac equations. This method also enables oscillatory solutions over the whole radial distance to be calculated on a logarithmic grid.

Two codes to evaluate the real zeros (jv.s) of the Bessel functions of the first kind Jv(x) for real orders v are presented. The codes are based on a Newton-Raphson iteration over the monotonic function ƒv(x) = x2v−1Jv(x)/Jv−1(x).The code ELF is a remarkably short program for finding, given any starting value x0 > 0 and any real order, the zero of Jv(x) in the neighborhood of x0 (x0 and the zero in the same branch of ƒv(x)). GNOME is a modification of ELF for finding the zeros of Jv(x) inside a given interval [xmin, xmax; for simplicity, we restrict the code GNOME to work for v > −1, which is the region of greatest practical use, where all the zeros of Jv(x) are real.The method is especially efficient for moderate values of v and for small zeros, where asymptotic expansions tend to fail and, besides, contrary to existing algorithms, enables the search of the real zeros for real orders, including negative orders.

We present a density functional theory for hot dense plasmas, applicable to high Z atom,where electronic structure of atom as microscopic aspect and ionic distribution as macroscopic aspect are determined self-consistently. In this theory, the plasma is regarded as an ensemble of the virtual \lqneutral atom' or, in other words, pseudo neutral atom. We apply our theory to Fe plasmas at temperatures T=100 eV and 1 keV in the density range 0.1rho solid˜200rho solid (rho solid=8.465×1022 cm-3), neglecting the polarization of continuum electrons, and consequently obtain the result that the interatomic effective potential is almost the same as the bare-Coulomb type potential in one-component plasmas (OCP). From the obtained profile of the potential veff( r) which determines bound electron states, we suggest a possibility of resonance states. The obtained internal energy for ion subsystem is lower than the Monte Carlo result for OCP, but the difference is within a few percent.

A model for condensed matter is described in which the ions surrounding a particular atom are replaced by a positive charge distribution which is constant outside of a sphere containing the atom and zero inside. The orbital functions, both bound and free, are obtained as solutions of the Dirac equation and are used to self-consistently determine the potential function. In order to obtain the desired equation-of-state data from the calculations, three different and somewhat arbitrary prescriptions are used to separate quantities pertaining to the atom from those of the electron gas in which it is imbedded. Results are shown for 14 elements, including the 5d transition metals, in the neighborhood of normal solid density. Equation-of-state data for nickel, copper, and zinc are also given and are compared with experiment.

We present a method for evaluating resonances and bound states present in any one-dimensional potential. Semiclassical theory (WKB) provides a fast estimate of resonance positions, but it can be off in calculating resonance lifetimes. Conversely, there are a multitude of numerical techniques, solving the one-dimensional Schrödinger equation, that give precise resonance energies and widths, but they are much more difficult to apply. The phase-amplitude method presented here yields accurate resonance energies and widths and is as easy to apply as semiclassical theory. Long-lived vibrational states of He22+ and CO2+ molecular ions are computed for demonstration.

We consider the single particle level density g(ε) of a realistic finite depth potential well, concentrating on the continuum (ε>0) region. We carry out quantum-mechanical calculations of the partial level density gl(ε), associated with a well-defined orbital angular momentum l<~40, using the phase-shift derivative method and the Green’s-function method and compare the results with those obtained using the Thomas-Fermi approximation. We also numerically calculate g(ε) as a l sum of gl(ε) up to a certain value of lmax<~40 and determine the corresponding smooth level densities using the Strutinsky smoothing procedure. We demonstrate, in accordance with Levinson’s theorem, that the partial contribution gl(ε) to the single particle level density from continuum states has positive and negative values. However, g(ε) is nonnegative. We also point out that this is not the case for an energy-dependent potential well.

The composition of dense metal plasmas is calculated considering higher ionization stages of the atoms. A system of coupled mass action laws is solved self-consistently taking into account medium corrections which lead to pressure ionization at high densities. The electrical conductivity is calculated within linear response theory. The interactions between the various species are treated on T matrix level. The numerical results for the electrical conductivity are in reasonable agreement with new experimental data for nonideal Al and Cu plasmas. Comparison with other theories is performed.

Calculation of the physical properties of reacting plasmas depends on knowing the state of ionization and/or the state occupation numbers. Simple methods have often been used to estimate ionization balance in plasmas, but they are not adequate for understanding a variety of new experimental and observational measurements. Theoretical methods to determine the ionization state of partially ionized plasmas must confront the effects of density on bound states and strong ion coupling. These methods can be separated into two categories. Chemical picture methods consider the system to be composed of distinct chemical species. Consequently, it is necessary to assert the effect of the plasma environment on internal states of these species. On the other hand, physical picture methods view the plasma in terms of its fundamental constituents; i.e., electrons and nuclei, so that plasma effects on bound states are a basic component of the theory. A discussion of some work representative of both of these philosophies will be given. Some comparisons between theories and with recent helioseismic observations and shock experiments will also be given. © 2000 American Institute of Physics.

We propose a new thermodynamic approach to ions in dense plasmas taking into account the screening by free electrons. The ions in fundamental and excited states are represented by a group of superconfigurations. Each superconfiguration containing an integer number of bound electrons is totally screened by free electrons in a Wigner–Seitz (WS) sphere. The minimisation of the plasma free energy with respect to the WS radii of the ions, under the condition that the specific mass of the plasma is preserved, leads to the equality of the electronic pressure for all ions. This pressure, as well as the WS radii and the distribution of ionic probabilities, are calculated by iteration. In this way, our approach not only gives the charge neutrality of the plasma, but also assures that the plasma environment is the same for each ion. In principle, the proposed approach allows one to apply the superconfiguration method to calculate radiative properties of plasmas for higher densities than those encountered in transmission experiments. Numerical examples for nickel, aluminium and samarium plasmas will be given. Comparisons with results from the previous statistical approach using the same WS radius for each ion charge state will also be discussed.

We present a new, fast and accurate phase-amplitude algorithm for the calculation of atomic continuum orbitals needed for cross sections computations of various atomic processes in plasmas. A coarse, energy independent, mesh is sufficient to achieve high accuracy. A straightforward application of a predictor-corrector procedure to the non-linear differential equations would fail, in particular for high energy free electrons in any atomic potential. The present algorithm overcomes this problem. In addition, we describe a novel method for calculating the radial integrals by integration over the phases instead of r. With the use of Gaussian trigonometric formulas over half periods, the integrals are expressed as alternating series. Levin's transform for convergence acceleration then provides the sum of the series with a few terms only. These methods are applicable in a relativistic framework as well as non-relativistic.

An algorithm for the evaluation with arbitrary accuracy of the Fermi-Dirac integral for any real value of j and x is presented. A new rapidly convergent series representation for x ⩾ 0 is derived. It involves confluent hypergeometric functions, for which several efficient implementations are available. Application of Euler transformation is proposed to improve the convergence of the two classical series expansions for x ⩽ 0 and |xvnb < π. The incomplete Fermi-Dirac integral is defined, and series expansions are provided for its evaluation. Finally, the computation of the inverse functions of and is discussed, and a comparison between three iterative methods is performed.

A simple method is given to evaluate the equation of state of a weakly relativistic, partially degenerate electron gas.

Construction and study of resonance wave functions corresponding to poles of the Green's function for several illustrative models of theoretical interest. Resonance wave functions obtained from the Siegert and Kapur-Peierls definitions of the resonance energies are compared. The comparison especially clarifies the meaning of the normalization constant of the resonance wave functions. It is shown that the wave functions may be considered renormalized in a sense analogous to that of quantum field theory. However, this renormalization is entirely automatic, and the theory has neither ad hoc procedures nor infinite quantities.

Diverse semianalytical constructions of wave functions, approximate or exact, are combined through the introduction of novel concepts. These include representing wave functions by a mosaic of analytic functions of a phase variable phi, each of them adapted to the problem's features over a restricted range of a coordinate x, and focusing the numerical effort on the metric relation between phi and x. This shift of attention from the wave function proper to metric relations is viewed as the essence of the Milne approach and of its WKB approximation. Illustrative examples are worked out and discussed.

A simplified version of the density-functional theory of Dharma-wardana
and Perrot is introduced in which the Kohn-Sham-Schrödinger
equation for the electrons is replaced by the Thomas-Fermi (TF)
statistical approximation, while interionic correlations are still
treated classically in the hypernetted-chain (HNC) approximation. The
nonlinear TF and HNC equations are coupled by a defining relation for
the effective ion-ion interaction potential. The resulting TF HNC theory
smoothly interpolates between the Debye-Hückel and confined-atom TF
theories. It is easily solved in the strong-coupling regime, while for
weak and intermediate couplings it agrees well with solutions of the
more elaborate density-functional theories for both ion distributions
and effective potentials. We have found that the computer code solving
our model does not converge in certain areas of ρ-T space, but at
present we can only speculate as to the significance, if any, of this
fact.

The electronic structure of many of the elemental solids is fairly well represented by a single atom embedded in a degenerate electron gas. This is particularly true for bulk properties such as the equation of state of highly compressed matter. Because the atom-in-jellium model is spherically symmetric, it is simpler than band-structure models. We study what happens when the nucleus is moved off center in the atomic cell to form a nonspherical system. This forms the basis of an Einstein model of atomic vibrations. The model is used to calculate Einstein temperatures and Grüneisen constants of simple solids by self-consistent-field electronic-structure methods.

Thermoelectric transport coefficients of metal plasmas are calculated within the linear response theory applied previously to determine the electrical conductivity of Al and Cu plasmas [R. Redmer, Phys. Rev. E 59, 1073 (1999)]. We consider temperatures of 1-3 eV and densities of 0.001-1 g/cm(3) as relevant in rapid wire evaporation experiments. The plasma composition is calculated considering higher ionization stages of atoms up to 5+, and solving the respective system of coupled mass action laws. Interactions between charged particles are treated on T matrix level. Results for the electrical conductivity of various metal plasmas are in reasonable agreement with experimental data. Thermal conductivity and thermopower are also given. In addition, we compare with experimental data for temperatures up to 25 eV and liquidlike densities.

The generalized Fermi-Dirac functions and their derivatives are important in evaluating the thermodynamic quantities of partially degenerate electrons in hot dense stellar plasmas. New recursion relations of the generalized Fermi-Dirac functions have been found. An effective numerical method to evaluate the derivatives of the generalized Fermi-Dirac functions up to third order with respect to both degeneracy and temperature is then proposed, following Aparicio. A Fortran program based on this method, together with a sample test case, is provided. Accuracy and domain of reliability of some other, popularly used analytic approximations of the generalized Fermi-Dirac functions for extreme conditions are investigated and compared with our results. Comment: accepted for publication in Comp. Phys. Comm

Numerical recipes, the art of scientific computing

- Press
- Wh

Press WH, et al. Numerical recipes, the art of scientific computing. 2nd ed. Cambridge: 1992. p. 727–35.

Semiclassical theory of atoms. Lecture notes in physics 300

- B G Englert

Englert BG. Semiclassical theory of atoms. Lecture notes in physics 300. Berlin: Springer, 1988.

Atomic physics: a numerical approach

- W R Johnson

Free-particle continuum density

- W R Johnson