Ion-ion dynamic structure factor of warm dense mixtures

The ion-ion dynamic structure factor of warm dense matter is determined using the recently developed pseudoatom molecular dynamics method [Starrett et al., Phys. Rev. E 91, 013104 (2015)]. The method uses density functional theory to determine ion-ion pair interaction potentials that have no free parameters. These potentials are used in classical molecular dynamics simulations. This constitutes a computationally efficient and realistic model of dense plasmas. Comparison with recently published simulations of the ion-ion dynamic structure factor and sound speed of warm dense aluminum finds good to reasonable agreement. Using this method, we make predictions of the ion-ion dynamical structure factor and sound speed of a warm dense mixture—equimolar carbon-hydrogen. This material is commonly used as an ablator in inertial confinement fusion capsules, and our results are amenable to direct experimental measurement.

Figure 1

$S(k,\omega )$ for aluminum at $5.2\mathrm{g}/{\mathrm{cm}}^3$ and $3.5\mathrm{eV}$. Solid lines are PAMD calculations and the numbers beside each curve indicate the $k$ value. The dashed lines are from QMD simulations [10] and the legend indicates their $k$ values. Despite the very different simulation methods involved, the agreement is good. The differences result from a combination of physical approximations and computational limitations (see text).

Figure 2

Dispersion relation ${\omega }_s$ (top panel) and adiabatic sound speed $c_s={\omega }_s/k$ (bottom panel) for aluminum at $5.2\mathrm{g}/{\mathrm{cm}}^3$ and $3.5\mathrm{eV}$. Solid lines with triangles are PAMD calculations using 10 000 particles. The blue dots are QMD results from [10], where 256 particles were used. For comparison, we also plot the results of a PAMD simulation using 256 particles (solid lines with squares).

Figure 3

Static structure factor (top panel) and $S(k,\omega )$ (bottom panel) for aluminum at $2.7 {\mathrm{g}/\mathrm{cm}}^3$ and $5\mathrm{eV}$ compared to TF-DFT-MD simulations [9]. We show PAMD results using both Thomas-Fermi (TF) and Kohn-Sham (KS) functionals. ${\omega }_p$ is the ion plasma frequency. For KS-PAMD $\hslash {\omega }_p=0.123$ eV; for TF-PAMD $\hslash {\omega }_p=0.125$ eV.

Figure 4

Dispersion relation ${\omega }_s$ (top panel) and adiabatic sound speed $c_s={\omega }_s/k$ (bottom panel) for aluminum at $2.7\mathrm{g}/{\mathrm{cm}}^3$ and $5\mathrm{eV}$. We show PAMD results using both Thomas-Fermi (TF) and Kohn-Sham (KS) functionals.

Figure 5

Static structure factor (top panel) for aluminum at $5.2\mathrm{g}/{\mathrm{cm}}^3$ and $3.5\mathrm{eV}$. The bottom two panels show the calculated dynamical structure factor and the hydrodynamical fit [Eq. (1)]. The four fit parameters are determined by a least squares fit of the calculated data for $k=0.143{\text{Å}}^{-1}$. On increasing $k$ (values are marked beside the corresponding lines) this hydrodynamical fit begins to deviate from the calculated curves, as is expected.

Figure 6

Static structure factor from PAMD (top panel) for a carbon-hydrogen mixture at $10\mathrm{g}/{\mathrm{cm}}^3$ and $10\mathrm{eV}$. The bottom three panels show the dynamic structure factors for C-C, H-H, and C-H at various fixed values of $k$.

Figure 7

Elastic part of the electron-electron dynamic structure factor for CH at 10 eV and $10 {\mathrm{g}/\mathrm{cm}}^3$. The $k$ values and line labels in this figure and inset are the same as in Fig. 6.

Figure 8

Top panel shows the dispersion relations extracted from the PAMD dynamic structure factors $S_{\mathrm{CC}}(k,\omega ),S_{\mathrm{HH}}(k,\omega )$, and $S_{ee}^{el}(k,\omega )$ as well as the dispersion relations for a pure hydrogen and a pure carbon plasma at the same temperature and density as the mixture (10 eV and $10 {\mathrm{g}/\mathrm{cm}}^3$). In the bottom panel the corresponding adiabatic sound speeds $\left(c_s={\omega }_s/k\right)$ are shown. For clarity, the sound speed for the pure hydrogen case has been shifted by $-35\mathrm{km}{\mathrm{s}}^{-1}$.