Elasticity of Hexagonal-Closed-Packed Cobalt at High Pressure and Temperature: A Quasiharmonic Case

We performed high-resolution inelastic x-ray scattering measurements on a single crystal of hcp cobalt at simultaneous high pressure and high temperature, obtaining 4 of the 5 independent elements of the elastic tensor. Our experiments indicate that the elasticity of hcp-Co is well described within the quasiharmonic approximation and that anharmonic high-temperature effects on the elastic moduli, sound velocities, and elastic anisotropy are minimal at constant density. These results support the validity of Birch’s law and represent an important benchmark for ab initio thermal lattice dynamics and molecular-dynamics simulations.

Figure 1

Representative IXS spectra of single-crystal hcp-Co at high pressure and high temperature (experimental data and fits). The spectra have been collected at constant density ($\rho =9.40\mathrm{g}/{\mathrm{cm}}^3\leftrightarrow V/V_0\simeq 0.947$) and at the reported temperatures. The vertical lines are guides to the eyes fixed at zero energy (quasielastic line) and at the energies of the Stokes and anti-Stokes $\mathrm{LA}[1.1,0,0]$ phonons at ambient temperature.

Figure 2

Temperature evolution at constant density ($\rho =9.40\mathrm{g}/{\mathrm{cm}}^3\leftrightarrow V/V_0\simeq 0.947$) of selected elastic moduli of hcp cobalt. The lines through the experimental points are guides to the eyes fixed to the mean values of the moduli over the investigated temperature range.

Figure 3

Anharmonic factor $\alpha \times \gamma$ as a function of the compression ratio $x=V/V_0={\rho }_0/\rho$ for hcp-Co (solid lines [34, 35]) and hcp-Fe (open circle [25], upside-down triangles [22], triangles [24], dotted line [20, 21], diamond [23], squares [38], dashed line [39], dash-dotted line [26], dash-dot-dotted line [9]). When high-compression data were not available, low pressure results have been extrapolated (lines in the figure) according the relations $\gamma ={\gamma }_0{x}^q$ and $\alpha ={\alpha }_0\exp [-\frac{\delta }{\kappa }(1-x^k)]$ [39], where $q$ is a positive number $\sim 1$, $\delta$ is the Anderson-Grüneisen parameter, and $\kappa$ is a positive number $\sim 1.4$. When numbers were not specifically provided in the individual work, we used $q=1$, $\kappa =1.4$, and $\delta =3.63$ for cobalt [35] and $q=1$, $\kappa =1.5$, and $\delta =6.1$ for iron [26, 39]. Extrapolations with $0.6\le q\le 1.4$ or $\alpha ={\alpha }_0{x}^{\delta }$, with $\delta =\mathrm{const}$, $5\le \delta \le 7$, provide comparable results, and do not change our qualitative conclusions.