High-temperature ion-thermal behavior from average-atom calculations

Atom-in-jellium calculations of the Einstein frequency were used to calculate the mean displacement of an ion over a wide range of compression and temperature. Expressed as a fraction of the Wigner-Seitz radius, the displacement is a measure of the asymptotic freedom of the ion at high temperature, and thus of the change in heat capacity from six to three quadratic degrees of freedom per atom. A functional form for free energy was proposed based on the Maxwell-Boltzmann distribution as a correction to the Debye free energy, with a single free parameter representing the effective density of potential modes to be saturated. This parameter was investigated using molecular dynamics simulations, and found to be $\sim 0.2$ per atom. In this way, the ion-thermal contribution can be calculated for a wide-range equation of state (EOS) without requiring a large number of molecular dynamics simulations. Example calculations were performed for carbon, including the sensitivity of key EOS loci to ionic freedom.

Figure 1

Root-mean-square fractional displacement $u_{\text{rms}}/r_{\text{WS}}$ calculated for carbon using atom-in-jellium theory. Contours shown are from 0.1 to 1.0 at intervals of 0.1, and then 2.0 to 4.0 at intervals of 1.0. A fractional displacement of 1 would correspond to ionic freedom, ignoring the velocity distribution of the ions.

Figure 2

Characteristic binding temperature $T_b=T{r}_{\text{WS}}^2/\overline{u^2}$ calculated for carbon. Contours shown are $\{1,2,5\}\times {10}^{\{4,5,6,7\}}\mathrm{K}$. Where contours of $T_b$ are not vertical, its temperature dependence is significant.

Figure 3

Variation of heat capacities along sample isochores, deduced from QMD simulations. Fully activated vibrational modes give $3k_{\text{B}}/\mathrm{atom}$, and unbound kinetic modes $\frac{3}{2}{k}_{\text{B}}/\mathrm{atom}$.

Figure 4

Variation of ion-thermal heat capacity along the $10\mathrm{g}/{\mathrm{cm}}^3$ isochore, from QMD (dashed) and atom-in-jellium with different values of the parameter $N$ in Eq. (11) (solid).

Figure 5

Variation of ion-thermal heat capacity along sample isochores, from QMD (dashed) and atom-in-jellium with $N=0.2$ (solid).

Figure 6

Principal shock Hugoniot for carbon calculated using previous EOS models [9, 32], atom-in-jellium (AJ) predictions using different values of the parameter $N$ in Eq. (11), and compared with experimental data [35]. Where distinguishable, the solid curves are $N=0.1$ (lower), 0.2 (middle), and 0.5 (upper).

Figure 7

Ambient isochore ($3.51\mathrm{g}/{\mathrm{cm}}^3$) for carbon from previous EOS models [9, 32], compared with atom-in-jellium (AJ) predictions using different values of the parameter $N$ in Eq. (11). Where distinguishable, the solid curves are $N=0.1$ (lower), 0.2 (middle), and 0.5 (upper).