Variational-average-atom-in-quantum-plasmas (VAAQP) code and virial theorem: Equation-of-state and shock-Hugoniot calculations for warm dense Al, Fe, Cu, and Pb

The numerical code VAAQP (variational average atom in quantum plasmas), which is based on a fully variational model of equilibrium dense plasmas, is applied to equation-of-state calculations for aluminum, iron, copper, and lead in the warm-dense-matter regime. VAAQP does not impose the neutrality of the Wigner-Seitz ion sphere; it provides the average-atom structure and the mean ionization self-consistently from the solution of the variational equations. The formula used for the electronic pressure is simple and does not require any numerical differentiation. In this paper, the virial theorem is derived in both nonrelativistic and relativistic versions of the model. This theorem allows one to express the electron pressure as a combination of the electron kinetic and interaction energies. It is shown that the model fulfills automatically the virial theorem in the case of local-density approximations to the exchange-correlation free-energy. Applications of the model to the equation-of-state and Hugoniot shock adiabat of aluminum, iron, copper, and lead in the warm-dense-matter regime are presented. Comparisons with other approaches, including the inferno model, and with available experimental data are given. This work allows one to understand the thermodynamic consistency issues in the existing average-atom models. Starting from the case of aluminum, a comparative study of the thermodynamic consistency of the models is proposed. A preliminary study of the validity domain of the inferno model is also included.

Figure 1

Thermodynamic consistency of the different models along the aluminum 2-eV isotherm. Relative difference between thermodynamic and virial electron pressures is plotted vs the compression ratio (we use ${\rho }_0=2.7\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$).

Figure 2

Aluminum $2.7\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$ (a) and $10.8\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$ (b) isochoric curves (one time and four times solid density, respectively) for the three options of VAAQP.

Figure 3

Value of $2b_0{R}_{\mathrm{WS}}$ found with VAAQP for aluminum as a function of compression ratio and temperature (we use ${\rho }_0=2.7\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$). The $2b_0{R}_{\mathrm{WS}}=4$ isoline is displayed in black on the figure as well as some $\Gamma$ isolines.

Figure 4

Thermodynamic consistency of the inferno option. The relative difference between thermodynamic and virial electron pressures is displayed as a function of compression ratio and temperature in the case of aluminum (we use ${\rho }_0=2.7\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$). The black line is the contour of the $>1\%$ region. The two dashed curves correspond to the Hugoniot shock adiabats obtained using, respectively, the thermodynamic (for the lower curve) and the virial (for the upper curve) definition of the pressure with the inferno option.

Figure 5

Aluminum mean ionization obtained with VAAQP, inferno option, and TFD.

Figure 6

Iron mean ionization obtained with VAAQP, inferno option, and TFD.

Figure 7

Copper mean ionization obtained with VAAQP, inferno option, and TFD.

Figure 8

Lead mean ionization obtained with VAAQP, inferno option, and TFD.

Figure 9

Aluminum principal Hugoniot shock adiabat obtained using VAAQP, inferno option, TFD, and the SESAME3713 table. We use ${\rho }_0=2.7\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$.

Figure 10

Iron principal Hugoniot shock adiabat obtained using VAAQP, inferno option, TFD, and the SESAME2140 table. We use ${\rho }_0=7.874\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$.

Figure 11

Copper principal Hugoniot shock adiabat obtained using VAAQP, inferno option, TFD, and the SESAME3336 table. We use ${\rho }_0=8.92\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$.

Figure 12

Lead principal Hugoniot shock adiabat obtained using VAAQP, inferno option, TFD, and the SESAME3200 table. We use ${\rho }_0=11.34\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$.

Figure 13

Aluminum principal Hugoniot shock adiabat obtained using VAAQP with the SESAME3713-306 cold-curve contribution. We use ${\rho }_0=2.7\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$.

Figure 14

Effect of a relativistic treatment along the ytterbium 30-eV isotherm. Relatives differences between electron pressures and mean ionizations stemming from a nonrelativistic calculation and from a relativistic calculation are plotted vs the compression ratio (we use ${\rho }_0=7.019\hspace{0.5em}\mathrm{g}\mathrm{\hspace{0.5em}}{\mathrm{cm}}^{-3}$).