Synthesis of Single Crystals of $\epsilon$-Iron and Direct Measurements of Its Elastic Constants

Seismology finds that Earth’s solid inner core behaves anisotropically. Interpretation of this requires a knowledge of crystalline elastic anisotropy of its constituents—the major phase being most likely $\epsilon$-Fe, stable only under high pressure. Here, single crystals of this phase are synthesized, and its full elasticity tensor is measured between 15 and 33 GPa at 300 K. It is calculated under the same conditions, using the combination of density functional theory and dynamical mean field theory, which describes explicitly electronic correlation effects. The predictive power of this scheme is checked by comparison with measurements; it is then used to evaluate the crystalline anisotropy in $\epsilon$-Fe under higher density. This anisotropy remains of the same amplitude up to densities typical of Earth’s inner core.

Figure 1

$\epsilon$-Fe single-crystal synthesis. (a)–(c) X-ray diffraction patterns of an iron sample collected at (a) 8.0 GPa, 770 K, (b) 9.0 GPa, 801 K, and (c) 13.5 GPa, 801 K. The diamond anvil cell has been rotated around a vertical axis by $\pm 20°$ during data collection. (d) Photograph of one sample. (e) Intensity of the ($1\overline{1}00$) diffraction peak of $\epsilon$-Fe as a function of the diamond anvil cell rotation angle. The rocking curve width is indicative of the crystal quality. (f) Iron phase diagram (after Ref. [24]) showing the approximate $P\text{−}T$ path followed for $\epsilon$-Fe single-crystal synthesis. The conditions of the patterns (a)–(c) collection are indicated with red dots.

Figure 2

Measurement of $\epsilon$-Fe single-crystal elastic constants. (a) Inelastic x-ray signal scattered close to an x-ray diffraction spot with $\mathbf{R}=(01\overline{1}\overline{1})$ at 33 GPa. The momentum $\mathbf{q}=\mathbf{Q}-\mathbf{R}$ (see text), indicating the phonon propagation direction, is indicated for each spectrum. One elastic (light gray) and two inelastic (excitation of a transverse $T2$ and longitudinal $L$ phonon in dark gray and red, respectively) contributions fit the raw spectrum. (b) $L$ and $T2$ phonon dispersion curves calculated for a propagation direction parallel to ($01\overline{1}\overline{4}$) using the measurements plotted in (a). (c) Longitudinal and transverse sound velocities, as a function of the propagation direction, measured at 33 GPa ($9.75{\AA }^3$) and calculated at $9.784{\AA }^3$ (dots, IXS data; rounded dots, IXS data from (b); dot-dashed line, fit to them) are compared to DMFT and DFT/GGA output (continuous and dotted lines).

Figure 3

(a) Calculated single-crystal elastic constants of $\epsilon$-Fe plotted vs pressure $P(V)$ calculated within the same framework. (b) Plotted vs pressure $P_{\exp }(V)$ estimated using an experimental equation of state [37]. Experimental data are represented with colored squares. $K_{S\mathrm{EOS}}$ is from Table 1. (c) Predicted acoustic wave anisotropy for $\epsilon$-Fe single crystals around 300 GPa ($6.669{\AA }^3/\mathrm{at}$) and 0 K. The velocities of the longitudinal and two transverse waves are represented vs the angle of propagation to the hexagonal $c$ axis. Literature predictions at $7.113{\AA }^3/\mathrm{at}$ [64] and $6.536{\AA }^3/\mathrm{at}$ [65] are also plotted.