Characterising the effect of pressure on the propagation of elastic waves in condensed matter is singularly important in many fundamental and applied research fields. An important case is related to the elasticity of hcp iron, the main constituent of the Earth's core. While the elastic anisotropy of Earth's inner core is now well established, its origin is still poorly understood. Theoretical predictions of the elastic moduli of iron yield very different results that are in disagreement with the scarce experimental results. Possible causes are: (i) the difficulty to perform reliable calculations at high pressures and temperatures, and (ii) the impossibility to carry out experiments on single crystal hcp iron.

Here we present the experimental determination of the five independent elastic moduli of hcp cobalt under hydrostatic compression to 39 GPa. Experimental and theoretical evidences suggest that hcp cobalt is a suitable analogue for hcp iron, with the advantage of being available as a single crystal. Thus, the knowledge of the elasticity of cobalt can be utilised to address the elastic anisotropy of hcp iron.

The experiments were carried out on the Inelastic X-ray Scattering (IXS) beamline II (ID28) utilising a focussed beam and an overall energy resolution of 3 meV. High quality single crystals (45 to 85 µm diameter, 20 µm thickness) were loaded in diamond anvil cells (DAC), using helium as pressure transmitting medium. The sound velocity of five independent acoustic phonon branches was determined as a function of pressure, permitting the derivation of the five independent elastic moduli. The results obtained compare well with ambient pressure ultrasonic measurements and, despite some discrepancies, an overall agreement with ab initio calculations [2] can be observed.

Determination of the full elastic tensor and its pressure evolution allows the mapping of the sound velocities in all along arbitrary directions in the crystal . The variation of the longitudinal acoustic sound velocity, VL{}, where is the angle from the c-axis in the meridian (a-c) plane is, in the case of iron, related to the observed seismic-wave anisotropy in Earth's inner core. Figure 16 shows VL{}, derived from IXS mesurements on cobalt, calculated at 0 K for cobalt and for iron [2], and derived from radial X-ray diffraction (RXRD) measurements on iron [1]. The results are compared at the same compression ratio 0/ = 0.86 (P~36 GPa for Co and P~39 GPa for Fe). We note a substantial agreement between the IXS results and the calculations, even if in the calculations the magnitude of the anisotropy is underestimated. This comparison validates the theoretical results and suggests a sigmoidal shape for VL{} in hcp iron as well, as indeed the calculations show.


Fig. 16: VL{}, from c- to a-axis. For a clear comparison, the velocities are normalised to one for = 90°. Solid line: IXS measurements for hcp Co; dotted line: calculations for hcp Co [2]; dashed line: calculations for hcp Fe [2]; dash-dotted line: RXRD measurements for hcp Fe [1].


The above experiments on single-crystalline Co were complemented by IXS measurements on a textured polycrystalline Fe-sample at 22 and 112 GPa. The aggregate longitudinal acoustic (LA) phonon dispersions, determined at the two pressures for two different orientations of the diamond anvil cell (at 50 and 90 respect to the compression axis of the cell), are reported in Figure 17, together with the differences in the derived sound velocities. While at 22 GPa the two dispersions almost overlap, leading to velocities equal within the error bars, at 112 GPa the difference becomes significant. Considering the known texture, with the c-axis of the crystallites preferentially oriented along the main compression axis of the cell, we can conclude that at 112 GPa the sound propagates faster by 4 to 5% at 50 from the c-axis than at 90 (i.e. in the basal plane). The measured anisotropy on a textured polycrystalline sample at 112 GPa is of the same order of magnitude as the anisotropy of seismic waves in the Earth's inner core (3-4%).


Fig. 17: LA phonon dispersions of hcp Fe at room temperature and at pressures of 22 GPa (lower curve) and 112 GPa (upper curve) for the two orientations of the DAC; full (open) circles: sound propagation at 90° (50°) to the DAC loading axis. Where not visible the errors are within the Symbols. The inset reports the relative difference in the sound speeds for the two orientations at the two pressures investigated.


Our results support the hypothesis of a sigmoidal shape of the longitudinal acoustic sound velocity in hcp iron at high pressure and ambient temperature, with a significant higher speed along the c-axis. A moderate alignment of the c-axis of the iron crystallites along the Earth's rotation axis, in an otherwise randomly oriented medium, can alone qualitatively explain compressional travel time anomalies observed in the Earth's inner core.

[1] H.K. Mao et al., Nature 396, 741 (1998); correction, Nature 399, 280 (1999).
[2] G. Steinle-Neumann et al.,Phys. Rev. B 60, 791 (1999).

Principal Publications and Authors
D. Antonangeli (a), M. Krisch (a), G. Fiquet (b), D.L. Farber (c), C.M. Aracne (c), J. Badro (b,c), F. Occelli (c), H. Requardt (a), Phys. Rev. Lett., 93, 215505 (2004); D. Antonangeli (a), F. Occelli (a), H. Requardt (a), J. Badro (b), G. Fiquet (b), M. Krisch (a), Earth Planet. Sci. Lett., 225, 243 (2004).
(a) ESRF
(b) Laboratorie de Minéralogie-Cristallographie de Paris, Institut de Physique du Globe de Paris (France)
(c) Lawrence Livermore National Laboratory (USA)