Equation of state measurements of dense krypton up to the insulator-metal transition regime: Evaluating the exchange-correlation functionals

Motivated by the poor understanding of the applicability of new exchange-correlation (XC) functionals to warm dense matter (WDM), we designed and performed multiple-shock reverberation compression experiments on dense krypton to evaluate explicitly the implications of recently derived XC functionals. The equation of states of krypton up to 155 GPa and 45 000 K, which ranges from an initial dense gaseous state up to the insulator-metal transition regime, were determined accurately. It is found that the experimental data are better reproduced by the strongly constrained and appropriately normed (SCAN) XC functional compared to the conventional Perdew-Burke-Ernzerhof and Van der Waals (vdW) DF1 functionals, elucidating that the introduction of the kinetic energy density and the intermediate-range vdW interaction is decisive. However, the incorporation of long-range interactions into the SCAN ($\mathrm{SCAN}+\mathrm{rVV}10$ XC functional) results in a noticeably stiffer prediction due to an overestimation of the density and internal energy of the system at low densities and temperatures. Our evaluation of the Karasiev-Sjostrom-Dufty-Trickey free-energy functional experimentally validates the XC thermal effect in the WDM regime, verifies the previous predictions, and sheds light on a direction for future theoretical efforts. Finally, a phase diagram of krypton is given, which provides a clear picture for understanding the thermophysical behavior of krypton in a wider temperature-pressure range.

Figure 1

Experimental concept (not to scale). (a) Sketch of the experimental target used to obtain the multishock states of dense krypton. (b) The velocity history of the shock front monitored by the DPS. (c) The velocity profiles of sample/LiF interfaces measured by the DLHV. (d) The cross-check of the first-shock transmit time recorded by the MCOP and LPSBO system. (e) The time-resolved self-emission of the shocked krypton measured by MCOP at wavelength ranging from 404.5 to 702.7 nm, and the voltage signal amplitude $\left(h_i\right)$ shifting to these numbers in parentheses. The other detailed description and the arrangement of diagnostic probes can be seen in the SM [34].

Figure 2

The measured shock experimental $\rho$-P data for shots (a) GKr180604 and GKr180605, and (b) GKr180705 [for which the legend is the same as that in (a)]. The insets show the first-shock state (principal Hugoniot) of dense krypton. Our experimental data (solid ball and square) were compared with AIMD calculations obtained by using five different XC functionals (PBE [51], vdW-DF1 [52], SCAN [29], $\mathrm{SCAN}+\mathrm{rVV}10$ [53], and KSDT [31]) and the existing EOS models (SFVT [37], SESAME 5180, and SESAME 5181 [38]). Error bars represent standard deviations at the $1\sigma$ level and include random and systematic uncertainties.

Figure 3

The measured first-shock temperatures. Comparisons between the experimental data and different theoretical results for shots: (a) GKr180604 and GKr180605, and (b) GKr180705 [for which the legend is the same as that in (a)]. (c) Relative differences between the experimental data and the theoretical results for four shots. The experimental uncertainties at the $1\sigma$ level are shown as the central bright color region around 0. The experimental temperatures were extracted from MCOP signals using the Smith-Drude model [58] and the gray-body Planck function.

Figure 4

The distribution of our shock states in the T-P phase diagram. The first-shock temperatures were determined experimentally (solid ball) and the second- to fourth-shock temperatures (square) were estimated with the SFVT model. The conductivities were obtained from the DFT calculations, and the values are indicated by the color bar. The solid-fluid phase boundaries were derived from the melting curve of solid krypton [5, 20] using the laser heated diamond-anvil cell method. The predicted lower (${10}^4\mathrm{S}/\mathrm{m}$) and upper (${10}^5\mathrm{S}/\mathrm{m}$) insulator-metallic fluid boundaries are taken from Ref. [25]. The Hugoniot of liquid krypton [21] and the mantle geotherm of Earth's interior [5, 60] are also shown for comparison.