First-principles equations of state and elastic properties of seven metals

We present a systematic comparison of first-principles zero-temperature equations of state and elastic constants of seven metals (aluminum, titanium, copper, tantalum, tungsten, platinum, and gold) with the most recent diamond-anvil-cell (DAC) experimental data, for pressures up to 150 GPa. Our calculations were performed within density functional theory, testing both the local density approximation (LDA) and the generalized gradient approximation (GGA) to the exchange-correlation term, and using several types of pseudopotentials. The obtained pressure-volume relationships show good agreement with DAC data: the difference between ab initio pressure and experiment is at most 5 GPa at 100–150 GPa except for Au and Pt. The equilibrium volumes $V_0$ and bulk moduli $K_0$ are determined within 1.5% and 6% of DAC data respectively. Experimental results are better reproduced with GGA for Al, Ti, Cu, Ta and W, but with LDA for Pt and Au, in agreement with previous theoretical studies. The predicted elastic constants are within 10% of experiment. For tantalum we have also calculated phonon spectra under pressure. They are in excellent agreement with experimental data: especially they accurately reproduced the inflexion on the longitudinal branch in the $\Gamma$-H direction, which is typical of the VB column (V, Nb, Ta).

Figure 1

Theoretical 0 K isotherm for bcc Ta compared with room temperature diamond-anvil-cell measurements (diamonds) (Ref. 1). Pseudopotentials used here are HGH with 5 (PP) or 13 (PPsc) valence electrons. In tantalum the “natural” five valence states are $5d^{3}6s^2$. The introduction of the semicore states $5s^{2}5p^6$ (see Table ) improves the results.

Figure 2

Theoretical pressure versus relative volume for Ta (GGA). The white diamonds represent the DAC data from Ref. 1, the black circles are GGA results of Ref. 29 using the full-potential linearized augmented plane wave (FP-LAPW) method. The solid lines are present PP calculations.

Figure 3

Pressure difference between ab initio results and diamond-anvil-cell measurements versus experimental pressure for Ta. DAC data are from Dewaele et al. (Ref. 1). Filled black symbols are present pseudopotential calculations in GGA. Stars represent all-electron calculations in FP-LAPW, GGA performed by Wang et al. (Ref. 29).

Figure 4

$P=0$ phonon dispersion curves for Ta and phonon density of states (extreme right), compared to the inelastic neutron diffraction data of Woods (Ref. 33).

Figure 5

$P=1000\mathrm{GPa}$ phonon dispersion curves for Ta and phonon density of states (extreme right).

Figure 6

Calculated evolution of the Grüneisen parameter with density for Ta and comparison with other approaches. The corresponding pressures can be found on the upper horizontal axis.

Figure 7

Theoretical pressure versus relative volume for Al, Cu, and W (GGA). The white diamonds represent the DAC data from Ref. 1, the black circles are FP-LAPW (GGA) results of Ref. 29. The solid lines are present PP calculations.

Figure 8

Theoretical pressure versus relative volume for Au, Pt (LDA), and Ti (GGA). The diamonds represent the DAC data from Ref. 1. Other symbols correspond to experiments carried out under different pressure media as in Fig. 7 of Ref. 38. The stars are LDA calculations of Ref. 43, using linear combinations of Gaussian-type orbitals fitting function (LCGTO-FF) method. The solid (Au, Pt, $\omega$-Ti) and dashed lines (hcp Ti) are present work.

Figure 9

Pressure difference between ab initio results and diamond-anvil-cell measurements versus experimental pressure. DAC data are from Dewaele et al. (Ref. 1). Filled black symbols are present pseudopotential calculations in LDA (Pt, Au) or GGA (Al, Cu, W).