Equation of state of dense plasmas with pseudoatom molecular dynamics

We present an approximation for calculating the equation of state (EOS) of warm and hot dense matter that is built on the previously published pseudoatom molecular dynamics (PAMD) model of dense plasmas [Starrett et al., Phys. Rev. E 91, 013104 (2015)]. While the EOS calculation with PAMD was previously limited to orbital-free density functional theory (DFT), the new approximation presented here allows a Kohn-Sham DFT treatment of the electrons. The resulting EOS thus includes a quantum mechanical treatment of the electrons with a self-consistent model of the ionic structure, while remaining tractable at high temperatures. The method is validated by comparisons with pressures from ab initio simulations of Be, Al, Si, and Fe. The EOS in the Thomas-Fermi approximation shows remarkable thermodynamic consistency over a wide range of temperatures for aluminum. We calculate the principal Hugoniots of aluminum and silicon up to 500 eV. We find that the ionic structure of the plasma has a modest effect that peaks at temperatures of a few eV and that the features arising from the electronic structure agree well with ab initio simulations.

Figure 1

Percentage difference in the virial and thermodynamic pressures (see text) for aluminum. A positive difference indicates that $P^{\mathrm{th}}>P^{\mathrm{vir}}$. This is a stringent test of the accuracy of the superposition approximation [Eq. (2)]. In the limit of perfect physical and numerical accuracy the difference would be zero.

Figure 2

Pressures from PAMD-KS compared to QMD [26] for beryllium. Also shown are pressures from an average atom model (AA-KS). PAMD-KS agrees significantly better with the QMD calculations.

Figure 3

Pressures from PAMD-KS compared to QMD [27] for aluminum. Also shown are pressures from an average atom model (AA-KS). Both PAMD-KS and AA-KS agree well with the QMD calculations.

Figure 4

Isochores of the total pressure for silicon from PAMD-KS compared to the QMD and PIMC results of Militzer and Driver [8]. The isochores shown are in multiples of the normal solid density of ${\rho }_0=2.33\mathrm{g}/{\mathrm{cm}}^3$, from ${\rho }_0$ (bottom) to $6{\rho }_0$ (top). The differences remain below 9% everywhere and a generally below 5%.

Figure 5

Principal Hugoniot of aluminum from the PAMD-KS model for different choices of the initial internal energy $U_0$ (indicated in parentheses) in Hartree. The curve labeled “AA” uses the initial internal energy calculated using the KS average atom model. The experimental data are from Refs. [29, 30, 31]. The Hugoniot is shown as a function of the compression ratio $\rho /{\rho }_0$ where ${\rho }_0=2.7{\mathrm{g}/\mathrm{cm}}^3$ is the initial density. All PAMD curves cover a temperature range of 1 to 500 eV. The structure seen at high compression ratios is caused by the electronic shell structure of the bound states of Al.

Figure 6

Principal Hugoniot of aluminum from several models. The experimental data are from Refs. [29, 30, 31]. The range of temperatures for the calculated Hugoniots is 1–500 eV. The QMD Hugoniot is from Ref. [32]. The top panel shows the full range of compression ration $\rho /{\rho }_0$, while the lower panel focuses on the smaller compression region only. Results from the VAAQP [22] model are also shown. The initial density is ${\rho }_0=2.7\mathrm{g}/{\mathrm{cm}}^3$.

Figure 7

Percentage difference in the total pressure of aluminum calculated with the PAMD-TF and AA-TF models. The solid white line is the temperature-density path of the PAMD-TF Hugoniot. The dashed white line marks ${\mathrm{\Gamma }}_{\mathrm{eff}}=30$. The reference density is ${\rho }_0=2.7\mathrm{g}/{\mathrm{cm}}^3$.

Figure 8

Percentage difference in total pressure of aluminum calculated with the PAMD-KS and AA-KS models. The solid white line is the temperature-density path of the PAMD-KS Hugoniot. The dashed white line marks ${\mathrm{\Gamma }}_{\mathrm{eff}}=30$. The reference density, ${\rho }_0=2.7\mathrm{g}/{\mathrm{cm}}^3$, is the density of solid Al at room temperature.

Figure 9

Percentage difference in total pressure of aluminum between the PAMD-TF and PAMD-KS. Here ${\rho }_0=2.7\mathrm{g}/{\mathrm{cm}}^3$ is the density of solid Al at room temperature.

Figure 10

Principal Hugoniot of silicon from PAMD-KS compared to the DFT-MD and PIMC results of Militzer and Driver [8]. The initial density is ${\rho }_0$ = 2.33 g/${\mathrm{cm}}^3$. Ionization of, first, the $2s^{2}2p^6$ and, second, the $1s^2$ electronic shells cause the structure seen at compressions $\rho /{\rho }_0>4.5$.