Path integral Monte Carlo simulations of warm dense aluminum

We perform first-principles path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) calculations to explore warm dense matter states of aluminum. Our equation of state (EOS) simulations cover a wide density-temperature range of $0.1–32.4\mathrm{g}{\mathrm{cm}}^{-3}$ and ${10}^4–{10}^8$ K. Since PIMC and DFT-MD accurately treat effects of the atomic shell structure, we find two compression maxima along the principal Hugoniot curve attributed to K-shell and L-shell ionization. The results provide a benchmark for widely used EOS tables, such as SESAME, QEOS, and models based on Thomas-Fermi and average-atom techniques. A subsequent multishock analysis provides a quantitative assessment for how much heating occurs relative to an isentrope in multishock experiments. Finally, we compute heat capacity, pair-correlation functions, the electronic density of states, and $\langle Z\rangle$ to reveal the evolution of the plasma structure and ionization behavior.

Figure 1

(a) Comparison of the combined PIMC and DFT-MD principal shock Hugoniot curve ($P-\rho /{\rho }_0$ space, ${\rho }_0=2.70$ g ${\mathrm{cm}}^{-3}$) with experiments [11, 13, 15, 53] and various models, including the QEOS [50, 73] and SESAME-1100 [49] tables, Purgatorio [64], ACTEX [56], Pseudoatom MD [68], and extended-plane-wave DFT [70]. Methods that treat the quantum-mechanical shell structure of the ions reveal two compression maxima due to K-shell and L-shell ionization. The QEOS and SESAME tables are derived from models that do not explicitly treat shell effects and, thus, only predict an average ionization behavior. (b) A magnification of the compression maxima.

Figure 2

Temperature-pressure conditions for the PIMC and DFT-MD calculations along isochores corresponding to the densities of 0.1-fold (0.27 g ${\mathrm{cm}}^{-3}$) to 12-fold (32.38 $\mathrm{g}{\mathrm{cm}}^{-3}$). The blue, dash-dotted line shows the principal Hugoniot curve for an initial density of ${\rho }_0=2.70\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 3

Aluminum excess pressure, relative to the ideal Fermi gas, computed with PIMC, DFT-MD, and the Debye-Hückel plasma model. The results are plotted at densities corresponding to 4-fold, 4.5-fold, and 5-fold compression as a function of temperature. DFT-MD results using a more robust pseudopotential in ABINIT validate the VASP DFT-MD calculations. The 4- and 4.5-fold curves were shifted vertically for visibility by $-3$ and $-1.5$, respectively.

Figure 4

Aluminum excess internal energy, relative to the ideal Fermi gas, computed with PIMC, DFT-MD, and the Debye-Hückel plasma model. The results are plotted for densities corresponding to 4-fold, 4.5-fold, and 5-fold compression as a function of temperature. DFT-MD results using a more robust pseudopotential in ABINIT validate the VASP DFT-MD calculations. The 4- and 4.5-fold curves were shifted vertically for visibility by $-5.5$ and $-3$, respectively.

Figure 5

The heat capacity of aluminum, spanning the warm dense matter and plasma regimes, derived from DFT-MD and PIMC calculations.

Figure 6

(a) Comparison of the combined PIMC and DFT-MD shock Hugoniot curve with SESAME-1100 [49], in $T-\rho /{\rho }_0$ space. As in Fig. 1, PIMC and DFT-MD predict shell structure effects along the Hugoniot, while SESAME predicts the average ionization behavior, without distinct shell effects.

Figure 7

Comparison of an isentrope and various multishock Hugoniot curves.

Figure 8

Pair-correlation functions of Al nuclei computed with PIMC and DFT-MD simulations over a wide range of temperatures in 8-atom simulations cells. Functions are compared for densities of (a) 5.40 (2-fold compression) and (b) 32.38 g ${\mathrm{cm}}^{-3}$ (12-fold compression).

Figure 9

Number of electrons contained in a sphere of radius, $r$, around an Al ion. PIMC data at two densities of (a) 5.40 (2-fold compression) and (b) 32.38 g ${\mathrm{cm}}^{-3}$ (12-fold compression) and four temperatures is compared with the doubly occupied Al 1s core ground state (eight-atom simulation cells).

Figure 10

The electron-electron pair-correlation functions for electrons with opposite spins in PIMC calculations of Al plasma. Results are compared for densities of (a) 5.40 (2-fold compression) and (b) 32.38 g ${\mathrm{cm}}^{-3}$ (12-fold compression).

Figure 11

The electron-electron pair-correlation functions for electrons with parallel spins in PIMC calculations of Al plasma. Results are compared for densities of (a) 5.40 (2-fold compression) and (b) 32.38 g ${\mathrm{cm}}^{-3}$ (12-fold compression).

Figure 12

(a) Density-driven upshifting of the occupied DOS and Fermi energy at a fixed temperature of $5\times {10}^5$ K. (b) Temperature-driven downshifting of the occupied DOS and Fermi energy at a fixed, ambient density.

Figure 13

The computed (a) total (all) and (b) occupied (occ.) electronic DOS at rho-T conditions along the principal Hugoniot curve from DFT-MD. The Fermi energies are marked by vertical lines with colors and line-styles corresponding to their respective DOS curve.

Figure 14

The valence band gap, computed with DFT-MD as a function of compression ratio at fixed temperature 8.6 eV. Results are compared with 2p-ionization energies from REODP and Stewart-Pyatt models for the isolated atom at a temperature of 12.5 eV [124].

Figure 15

An example DOS showing that $\langle Z\rangle$ is computed by integrating occupied conducting states above the valence band gap, which is shaded red for clarity.

Figure 16

$\langle Z\rangle$ as a function of temperature, computed from our DFT-MD and PIMC simulations, for densities corresponding to ambient (2.70 g ${\mathrm{cm}}^{-3}$) and 12-fold (32.38 g ${\mathrm{cm}}^{-3}$) compression. The results are compared with a variety of other models at ambient density conditions. The inset shows a zoom-in of the comparisons.

Figure 17

$\langle Z\rangle$ as a function of compression, computed from our DFT-MD and PIMC simulations, for temperatures of 12.5 and 174 eV. The results are compared with a variety of other models evaluated a temperature of 12.5 eV.