Sound velocity in shock compressed molybdenum obtained by ab initio molecular dynamics

The sound velocity of Mo along the Hugoniot adiabat is calculated from first principles using density-functional theory based molecular dynamics. These data are compared to the sound velocity as measured in recent experiments. The theoretical and experimental Hugoniot and sound velocities are in very good agreement up to pressures of 210 GPa and temperatures of 3700 K on the Hugoniot. However, above that point the experiment and theory diverge. This implies that Mo undergoes a phase transition at about the same point. Considering that the melting point of Mo is likely much higher at that pressure, the related change in the sound velocity in experiment can be ascribed to a solid-solid transition.

Figure 1

The calculated dependence of pressure on density along the Hugoniot compared to experimental data. The shift, independent of density, is typical of DFT calculations on metals.

Figure 2

Elastic constants of bcc Mo along the Hugoniot as a function of pressure. The bcc phase of Mo becomes unstable above pressure 300 GPa which results in irregular behavior of elastic constants.

Figure 3

The calculated sound velocity (large diamonds) compared to experimental data (small diamonds with error bars). Up to the pressure of just above 200 GPa the experiment and theory provide very similar sound velocities (corresponding temperature is around 3700 K). At higher pressures the experiment and theory diverge, and it appears that the divergence increases with increasing pressure. The divergence cannot be explained by experimental error bars or statistical errors of our calculations. The size of the symbols corresponds approximately to the error bars. Somewhat irregular behavior of the experimental and theoretical curves also illuminates the possible error. The error bars of two computed points (circles with error bars) are obtained by the Monte Carlo method of error propagation with the errors in stresses obtained by the blocking technique (see the Appendix).

Figure 4

The histograms of longitudinal sound velocities computed at two densities as indicated in the legend. The width of the distribution represents the error in sound velocity. The error is larger at higher density that corresponds to the pressure of 257 GPa. The histograms are computed assuming errors in stresses computed by the blocking technique.