Pressure in warm and hot dense matter using the average-atom model

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.

Figure 1

Comparison of pressure between experiment (Expt.), the ion perfect gas (PG), QMD simulations (QMD), and the average-atom model (AAM) for boron at $0.094\mathrm{g}/{\mathrm{cm}}^3$, aluminum at $0.3\mathrm{g}/{\mathrm{cm}}^3$, silicon at $0.21\mathrm{g}/{\mathrm{cm}}^3$, and titanium at $0.2\mathrm{g}/{\mathrm{cm}}^3$.

Figure 2

Comparison of pressure between experiment (Expt.), the ion perfect gas (PG), QMD simulations (QMD), and the average-atom model (AAM) for nickel at $0.1\mathrm{g}/{\mathrm{cm}}^3$, copper at $0.5\mathrm{g}/{\mathrm{cm}}^3$, silver at $0.43\mathrm{g}/{\mathrm{cm}}^3$, and gold at $0.5\mathrm{g}/{\mathrm{cm}}^3$.

Figure 3

Pressure calculated from the average-atom model (AAM) compared to three expressions using an effective average ionization for gold at $0.5\mathrm{g}/{\mathrm{cm}}^3$. ${\overline{Z}}_B$ is the average ionization obtained from the the number of bound electrons, ${\overline{Z}}_R$ from the electron density at the surface of the average-atom sphere, and ${\overline{Z}}_{\text{CB}}$ from the continuum background. We give the QMD pressure (QMD) for the eyes.

Figure 4

Relative error of pressure calculated using ${\overline{Z}}_R$ and ${\overline{Z}}_{\text{CB}}$ with respect to the average-atom model (AAM) for gold at $0.5\mathrm{g}/{\mathrm{cm}}^3$. ${\overline{Z}}_R$ is the average ionization obtained from the electron density at the surface of the average-atom sphere and ${\overline{Z}}_{\text{CB}}$ from the continuum background.

Figure 5

Nonrelativistic (NR) and the relativistic (Rel) pressures calculated using the average-atom model for gold at $0.5\mathrm{g}/{\mathrm{cm}}^3$.

Figure 6

Comparison of pressure between DFT-MD and PIMC simulations and the average-atom model (AAM) for aluminum at five times the solid density.

Figure 7

Comparison of pressure between DFT-MD simulations (DFT-MD) and the average-atom model (AAM) for aluminum at ${10}^5$ K as a function of compression.