Atom-in-jellium equations of state in the high-energy-density regime

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.

Figure 1

Atom-in-jellium states calculated for Al. Dark blue: electron and ion calculations both completed. Light blue: ion calculation failed, indicating noninteracting atoms.

Figure 2

Debye temperature calculated for Al.

Figure 3

Variation of Debye temperature with compression for Al, calculated with atom-in-jellium perturbation theory, compared with experimental value [33].

Figure 4

Comparison of states calculated for Be along the 10-eV isotherm. AJ: atom-in-jellium (present work); QMD, PAMD, and average-atom Kohn-Sham (AA-KS) [17].

Figure 5

Comparison of states calculated for Be along the $10\mathrm{g}/{\mathrm{cm}}^3$ isochore. AJ: atom-in-jellium (present work); QMD, PAMD, and average-atom Kohn-Sham (AA-KS) [17].

Figure 6

Shock Hugoniot for Be, showing comparison between atom-in-jellium (AJ, present work) EOS constructed using models A and T (dashed, coincident) and B (solid), and TF EOS constructed with slightly different ion-thermal treatments [2, 40], with experimental measurements [35, 36, 37, 38, 39].

Figure 7

States calculated for Al along the $2.7\text{−}\mathrm{g}/{\mathrm{cm}}^3$ isochore. Atom-in-jellium (AJ, present work) is shown with and without the ion-thermal contribution, and compared with QMD, PAMD, and average-atom Kohn-Sham (AA-KS) [17].

Figure 8

Shock Hugoniot for Al, showing comparison between atom-in-jellium EOS (AJ, present work), an example TF-based EOS [46], and PAMD results [17], with experimental measurements [35, 36, 44].

Figure 9

States in Si along isochores at multiples of ${\rho }_0$ (lowest: $1\times {\rho }_0$; highest: $6\times {\rho }_0$). Atom-in-jellium calculations are shown for models A (dashed) and B (solid), compared with previous QMD calculations reported in [17].

Figure 10

Shock Hugoniot for Si, showing comparison between atom-in-jellium EOS (AJ, present work), two TF-based EOS [2, 48], PIMC and PAMD calculations [17], and experimental measurements [44, 50].

Figure 11

Shock Hugoniot for Fe, showing comparison between atom-in-jellium EOS (AJ, present work), a semiempirical EOS incorporating shock data and TF theory [2], Kerley's semiempirical multiphase EOS which blends into TF theory for the cold curve and previous atom-in-jellium calculations for the electron-thermal contribution [54], and experimental measurements [35, 36, 44].

Figure 12

Shock Hugoniot for Mo, showing comparison between atom-in-jellium EOS (AJ, present work), two semiempirical EOS using TF theory at high density and temperature [2, 57], and experimental measurements [35, 36, 44]. Results from a previous EOS using atom-in-jellium calculations for the electron-thermal EOS only [56] are also shown, demonstrating that the temperature range of this EOS is insufficient to show shell structure effects.