Theoretical and experimental investigation of the equation of state of boron plasmas

We report a theoretical equation of state (EOS) table for boron across a wide range of temperatures ($5.1\times {10}^4$–$5.2\times {10}^8$ K) and densities (0.25–49 g/${\mathrm{cm}}^3$) and experimental shock Hugoniot data at unprecedented high pressures ($5608\pm 118$ GPa). The calculations are performed with first-principles methods combining path-integral Monte Carlo (PIMC) at high temperatures and density-functional-theory molecular-dynamics (DFT-MD) methods at lower temperatures. PIMC and DFT-MD cross-validate each other by providing coherent EOS (difference $<1.5$ Hartree/boron in energy and $<5\%$ in pressure) at $5.1\times {10}^5$ K. The Hugoniot measurement is conducted at the National Ignition Facility using a planar shock platform. The pressure-density relation found in our shock experiment is on top of the shock Hugoniot profile predicted with our first-principles EOS and a semiempirical EOS table (LEOS 50). We investigate the self-diffusivity and the effect of thermal and pressure-driven ionization on the EOS and shock compression behavior in high-pressure and -temperature conditions. We also study the sensitivity of a polar direct-drive exploding pusher platform to pressure variations based on applying pressure multipliers to LEOS 50 and by utilizing a new EOS model based on our ab initio simulations via one-dimensional radiation-hydrodynamic calculations. The results are valuable for future theoretical and experimental studies and engineering design in high-energy density research.

Figure 1

Temperature-density conditions in our PIMC (red squares) and DFT-MD (blue diamonds) calculations are shown. The black curves depict the computed principal Hugoniot with (dashed) and without (solid) radiation correction [96] to the EOS. The dashed lines in green represent the conditions with different values of the degeneracy parameter, $\mathrm{\Theta }$, and the dotted lines in cyan denote the effective ionic coupling parameter, $\mathrm{\Gamma }$. The Hugoniot curve is constructed by choosing the initial density to be the same as ${\rho }_0$ ($\sim 2.46$ g/${\mathrm{cm}}^3$).

Figure 2

Target design for the impedance-matching experiment at the NIF.

Figure 3

(a) Energy- and (b) pressure-temperature EOS plots along isochores for boron from our PIMC and DFT-MD simulations. For comparison, the ideal Fermi-gas theory, Debye-Hückel model, and LEOS 50 are also shown. In (a), the LEOS 50 data have been aligned with DFT by setting their energies to be equal at 2.46 g/${\mathrm{cm}}^3$ and 0 K. Subplots (c) and (d) are the comparison in internal energy and pressure between PIMC and DFT-MD along four isotherms as functions of density. In subplots (a) and (b), results at different isochores have been shifted apart for clarity.

Figure 4

Boron EOS and shock Hugoniot curves shown in (a) $P\text{−}\rho /{\rho }_0$ and (b) $T\text{−}P$ plots. The Hugoniot results from LEOS 50, ACTEX, and SESAME 2330 are coplotted for comparison. Gray curves in panels (a) and (b) denote isotherms and isochores, respectively. The Hugoniot curves are constructed by choosing the initial density to be the same as ${\rho }_0$ ($\sim 2.46$ g/${\mathrm{cm}}^3$). The difference between ACTEX and others in (a) at $P>{10}^7$ GPa is because of the inclusion of electronic relativistic effects in ACTEX.

Figure 5

Comparison of the experimental boron shock Hugoniot result with predictions from our first-principles EOS data and LEOS 50, SESAME 2330, X52 models, and ACTEX. The initial density for the PIMC–DFT-MD, LEOS 50, SESAME 2330, and X52 curves are 2.31 g/${\mathrm{cm}}^3$, i.e., the same as that of the experimental sample. The initial density for the ACTEX curve is 2.46 g/${\mathrm{cm}}^3$.

Figure 6

Nuclear pair correlation function $g\left(r\right)$ of boron at three densities and three temperatures. The $g\left(r\right)$ curves analyzed based on MD simulations of 120-atom cells at 6736.47 K are coplotted for comparison. Curves at different densities have been off set for clarity. The consistency between $g\left(r\right)$ of 30- and 120-atom cells show negligible finite-size effect in describing the ionic structures. The numbers at the inset of each panel show the values of self diffusivity at the corresponding density. Numbers in parentheses denote the standard error of the corresponding data. Red, blue, and dark texts correspond to temperatures of $6.7\times {10}^4$, $1.3\times {10}^5$, and $5.1\times {10}^5$ K, respectively.

Figure 7

The average number of electrons around each nucleus at different densities and a series of temperatures; ${\rho }_0\approx 2.46$ g/${\mathrm{cm}}^3$. The long dashed curve denotes the corresponding profile of the isolated ${\mathrm{B}}^{3+}$ ion. The profile was derived by integrating the doubly occupied $1s$ orbitals that we computed with the GAMESS quantum chemistry code [112].

Figure 8

Comparison of the $1s$ binding energy $E_b^{1s}$ (solid curves) with the chemical potential $E_{\text{CP}}$ (thin dashed curves) as functions of temperature at four densities. The data are obtained using the Purgatorio method. The dash-dotted curves represent $E_{\text{CP}}$–$5k_{B}T$. The diamonds indicate the points at which the $1s$ level starts to be ionized (by 0.67%).

Figure 9

Total heat capacity $C_V^{\text{tot}}=\left(\partial E/\partial T\right){|}_V$ of boron obtained from our DFT-MD and PIMC data along several isochores. All curves converge to the ideal-gas limit of $9k_B$/atom at high temperature.