Ab initio calculations of the B1-B2 phase transition in MgO

We present an ab initio study of MgO at high temperature and pressure, around the phase transition between the B1 and B2 phases. By means of ab initio molecular dynamic calculations, the thermal evolution of vibrational properties and thermodynamic quantities is obtained. We carefully compare our results with previous theoretical works on the phase transition curve and we analyze the differences among them. We show that anharmonic effects have been underestimated in the quasiharmonic approximation and that their inclusion in the free energy strongly straightens up the transition curve. Then, we use our B1-B2 phase boundary and our calculated Hugoniot to analyze recent decaying shock experiments on MgO. We also provide important thermodynamic quantities as the Grüneisen parameter and sound velocities and we discussed their temperature dependence.

Figure 1

Theoretical B1-B2 phase boundary of MgO. Dotted lines: quasiharmonic approximation of Root et al. [12] (black line), Belonoshko et al. [9] (blue line), and Cebulla et al. [11] (red line). The black and red lines include anharmonic corrections beyond the quasiharmonic approximation. The green squares are the results of Boates et al. [10] using velocity autocorrelation function and the green line a Kechin fit to these data. The inset shows the renormalization of the thermal transition pressures by the athermal one.

Figure 2

PDOS of the B1 and B2 phases of MgO at $6.75\mathrm{g}/{\mathrm{cm}}^3$ and for temperatures of 1000, 5000, and 8000 K.

Figure 3

Comparison of the vibrational free energies of the B1 (in red) and B2 (in blue) phases of MgO obtained with DFPT (lines) and TDEP (filled circles) at $7.75\mathrm{g}/{\mathrm{cm}}^3$. Inset: difference between the free energies of the B1 and the B2 phases in QHA (solid line) and with the TDEP method (filled circles and dotted line)

Figure 4

Vibrational free-energy differences between the B1 and B2 phases at 2000 and 8000 K for the QHA (black circles) and with the TDEP method as a function of the supercell size.

Figure 5

Same as Fig. 1, with our QHA results (purple open circles and dotted line) and TDEP results (purple filled circles and solid line).

Figure 6

Internal anharmonic energy $U_{\text{anh}}$ for the B1 (in black) and B2 (in red) phases as a function of temperature obtained with Eq. (7) at $6.75\mathrm{g}/{\mathrm{cm}}^3$. Inset: $F_{\text{anh}}$ as a function of temperature.

Figure 7

Entropy difference between the B1 and B2 phases at $7.75\mathrm{g}/{\mathrm{cm}}^3$ as a function of the temperature. The solid line corresponds to the QHA, the filled black circles to the TI of Eq. (8), and the red squares to TDEP. The blue squares are similar to the red ones with the correction $F_{\text{anh}}$ of Eq. (6), while the open black circles correspond to a TI starting at 5000 K. The green triangles are the results of the VACF at 2000 and 8000 K. The error bars have been estimated with respect to calculation with supercell sizes of 216 atoms and 250 atoms for the B1 and the B2 structures, respectively.

Figure 8

Experimental shock temperature data in comparison with our predicted B1-B2 phase boundary. Purple, magenta, and orange dots: Bolis et al. [15]; green dots: McWilliams et al. [14]. The black solid line with filled circles is the B1-B2 boundary obtained with TDEP and the blue line the calculated Hugoniot. The dashed blue line is an extension of our Hugoniot curve for the liquid phase in the stability domain of the solid phase. The red curve is the calculated melt line of Root et al. [12] for the B2 structures and the red dashed one is the calculated melt line of de Koker et al. [53] for the B1 structure. The red cross indicates the melting of an AIMD with B2 as the starting structure of the simulation.

Figure 9

Elastic constants $C_{11}, C_{12}$, and $C_{44}$, isotropic bulk ($K_T$), and shear ($G$) moduli of MgO of the B1 and B2 phases as a function of pressure and temperature. The dashed lines are the results of Karki et al. [60, 61] at 0 K (black) and at 2000 K (red) using the QHA.

Figure 10

Grüneisen parameter of MgO. The black open symbols are the effective Grüneisen parameter of Oganov et al. [69]. The lines are Grüneisen parameters used in semiempirical equation of state [8, 65, 66, 68].

Figure 11

Temperature dependence of the Grüneisen parameter of MgO for the B1 (squares) and B2 (circles) phases at $6.25\mathrm{g}/{\mathrm{cm}}^3$ (red) and $7.75\mathrm{g}/{\mathrm{cm}}^3$ (blue). The dashed lines are linear fits of these data.

Figure 12

Compressional and shear wave velocities of MgO as a function of pressure at 0 (black), 2000 (red), 5000 (blue), and 8000 K (green). The kink in the curves indicates the phase transition between B1 and B2.