## Abstract

Raman spectra of hexagonal close-packed iron (*ɛ*-Fe) have been measured from 15 to 152 gigapascals by using diamond-anvil cells with ultrapure synthetic diamond anvils. The results give a Grüneisen parameter γ_{0} = 1.68 (±0.20) and *q* = 0.7 (±0.5). Phenomenological modeling shows that the Raman-active mode can be approximately correlated with an acoustic phonon and thus provides direct information about the high-pressure elastic properties of iron, which have been controversial. In particular, the *C*
_{44} elastic modulus is found to be lower than previous determinations. This leads to changes of about 35% at core pressures for shear wave anisotropies.

Understanding recent geophysical observations of elastic anisotropy, possible superrotation, and magnetism of Earth's inner core (1) requires detailed information about the thermodynamic and elastic properties of core-forming materials under appropriate conditions. High-pressure properties of iron are crucial in this respect because the core is composed primarily of this element. Iron transforms from the body-centered cubic (bcc) phase (α-Fe) at ambient conditions to a face-centered cubic (fcc) phase (γ-Fe) at moderate pressures and temperatures and to a higher-pressure hexagonal close-packed (hcp) phase (*ɛ*-Fe) (>13 GPa) (2). The hcp phase has a wide stability field to more than 300 GPa and high temperatures (3–5). Techniques to measure lattice strains at megabar pressures (6) have determined the elastic properties of *ɛ*-Fe to 210 GPa (7). These results show discrepancies with calculations in which first-principles methods were used (8–10), in particular for shear moduli and anisotropy. Measurements and estimates of the Grüneisen parameter, an important thermodynamic property of iron that relates the thermal pressure and the thermal energy, show large discrepancies (5, 11–16). Recently, the phonon density of states of *ɛ*-Fe at high pressure has been investigated up to 42 GPa (16) and 153 GPa (17) by inelastic nuclear resonance x-ray scattering. Notable differences with the results of first-principles calculations are also found (17). Thus, there is a need to clarify the elastic and thermodynamic properties of iron at core pressures.

For many years, it was thought that vibrational Raman spectroscopy of simple metals was not possible at high pressures because of the weak scattering due to the high reflectivity and strong background from the apparatus (e.g., diamond-anvil cell). The bcc (α-Fe) and fcc (γ-Fe) phases have no first-order Raman spectra because all the atoms sit on inversion centers. The *ɛ*-Fe hcp phase has one Raman-active mode of *E*
_{2}
_{g} symmetry (18). Recent studies have shown that the*E*
_{2}
_{g} phonon mode in some hcp metals can be measured at moderate pressure in diamond cells. Measurements have been performed on Zn (19), Si (19), Zr (20), and Mg (21), but the*E*
_{2}
_{g} mode of*ɛ*-Fe was expected to be particularly weak (22) and has not been reported. Information on the frequency of the*E*
_{2}
_{g} mode of*ɛ*-Fe and its evolution with pressure provides experimental information on thermodynamic properties of the material because such vibrational frequencies are input data for construction of consistent thermodynamic models (23). Moreover, the mode correlates with a transverse acoustic phonon; thus, its frequency shift provides information on the pressure dependence of shear moduli (i.e.,*C*
_{44}), which is crucial for constraining the elastic anisotropy of iron at core pressures. Measurement of the phonon also provides a critical test of first-principles methods, which have been a difficult problem for iron.

High-purity polycrystalline Fe samples were loaded in a piston-cylinder–type diamond cell; two sets of experiments were performed at 15 to 50 GPa and 25 to 152 GPa (24). To reduce the background luminescence and scattering from the diamond anvils (which could be much stronger than the sample), we used ultrahigh-purity synthetic diamond anvils, which in previous work have been crucial for studying weak Raman excitations in the hundreds of gigapascals pressure range (25). We also used a 35° incidence angle for the exciting radiation, which prevents specular reflection from being directed to the spectrometer and reduces the signal from the diamonds. The diameter of the laser spot was less than 10 μm, which is smaller than the sample size, so effects of the pressure gradients are minimized. Signal levels were maximized with a high-throughput single-grating spectrometer with holographic notch filters.

Two Raman bands were observed at moderate pressures (between 15 and 40 GPa), with the stronger band identified as the*E*
_{2}
_{g} fundamental predicted by symmetry for the hcp lattice (26) (Fig. 1). The mode exhibits a sublinear frequency increase with pressure (Fig. 2). Raman spectra measured together with the ruby fluorescence in different locations of the sample show consistent results. Moreover, spectra measured on compression and decompression are close, which indicates that the uniaxial stress does not have an important effect on these results. The measured frequencies are lower than first-principles results (27), although the pressure dependence is similar (Fig. 2). An empirical model (28) gives much lower frequencies. The positive pressure shift (initial slope *d*ν_{0}/*dP* = 1.0 cm^{−}
^{1}/GPa) is consistent with the wide stability range of *ɛ*-Fe (3–5), in contrast to the behavior of several other hcp metals (see below).

We examined the thermal Grüneisen parameter γ_{th} at high pressure. Despite the importance of the Grüneisen parameter of *ɛ*-Fe, experimental data have been limited and do not agree (Table 1). The mode-Grüneisen parameter γ_{i} of the Raman mode is defined as(1)where ν is the frequency of the mode and *V*is the volume. It provides an approximate means for calculating γ_{th} with γ_{i} = 〈γ_{i}〉 ∼ γ_{th}. This assumes that the γ_{i} for the mode is representative of all the vibrations of the crystal (23). We can write(2)where γ_{0} represents the extrapolated value of γ_{th} at zero pressure with the volume dependence of γ_{th} explicitly given by the parameter *q*. Using this relation and the equation of state of *ɛ*-Fe (3, 29), we calculate γ_{0} = 1.68 (±0.20) and *q* = 0.7 (±0.5). Assuming *q* = 1, we have γ_{0}= 1.81 (±0.03). The results agree with the analysis of recent high-pressure, high-temperature x-ray diffraction measurements (5), which suggests that γ_{i} of the*E*
_{2}
_{g} mode provides a good approximation for 〈γ_{i}〉. Our analysis is also compatible with that obtained from measurements of the phonon density of state of *ɛ*-Fe between 20 and 40 GPa, where*q* = 0 was assumed (16) (Table 1).

In *ɛ*-Fe, the *C*
_{44} elastic modulus and the *E*
_{2}
_{g} Raman mode are properties of the same phonon branch. Specifically, the*E*
_{2}
_{g} mode correlates with a transverse acoustic phonon and *C*
_{44} represents the slope of this branch at the center of the Brillouin zone. With a phenomenological model, they may be related as(3)where *M* is the atomic mass of iron,*c* and *a* are unit cell parameters, and ν is the frequency of the optical phonon (30). Applying this relation to our data, we deduce the pressure dependence of the*C*
_{44} of *ɛ*-Fe (Fig. 3). Notably, our data are lower than the results of lattice-strain experiments (7). The analysis used in these experiments assumes that the state of stress on all crystallographic planes is identical. However, recent work has shown that this may not be satisfied in a material undergoing anisotropic deformation, specifically in many hcp transition metals (31). The first-principles results (9, 10) are also higher than those obtained here. Application of first principles for magnetic metals such as iron has been problematic, partly because differences between theory and experiments for the pressure-volume relation at low pressures (32) may be associated with changes of the magnetic properties within *ɛ*-Fe (9). The use of Raman spectroscopy to determine the pressure dependence of*C*
_{44} is exact for a sine dispersion relation; if the phonon branch does not deform with pressure, this assumption can be supported by theory (27, 28).

The results have important geophysical implications. Although knowledge of *C*
_{44} is not sufficient to calculate the full seismic anisotropy parameters, the shear wave polarized perpendicular to the basal plane (S1) and that polarized parallel to the basal plane (S2) have the anisotropies(4)Therefore, the lower value of*C*
_{44} found here would increase Δ_{S1}and decrease Δ_{S2}. Assuming no difference in the estimation of the other elastic moduli (7) arising from the change of *C*
_{44}, we calculate Δ_{S1} ≈ 0.65 and Δ_{S2} ≈ 1.44 at 39 GPa, corresponding to changes of about 35%. This also improves the agreement with the first-principles calculations (33).

Finally, we can compare the results with those of other hcp structured transitions metals that have been studied recently by high-pressure Raman spectroscopy—for example, Zn (19), Si (19), Zr (20), and Mg (21). The behavior of the *E*
_{2}
_{g}phonon with pressure differs appreciably from one to another. Zn, Si, and Mg show an increase of frequency with pressure with initial slopes (*d*ν_{0}/*dP*) of about 3.3, 3.1, and 3 cm^{−}
^{1}/GPa, respectively. In contrast,*ɛ*-Fe also shows a positive pressure shift but with a*d*ν_{0}/*dP* of 1.0 cm^{−}
^{1}/GPa. Comparing γ_{i} values, we find γ_{i}(Zn) ≈ 2.4 between ambient pressure and 12 GPa, γ_{i}(Mg) ≈ 1.6 at ambient pressure, and γ_{i}(Si) ≈ 6 near 40 GPa compared with γ_{i}(Fe) = 1.44 (±0.03) between 20 and 150 GPa. Zirconium appears to be a unique case: mode softening is observed for the *E*
_{2}
_{g} phonon of α-Zr, with *d*ν_{0}/*dP* = −0.7 cm^{−}
^{1}/GPa (20). Moreover, the relation between elastic modulus *C*
_{44} and the frequency of the *E*
_{2}
_{g} mode (Eq. 3) can be examined for Zr, Mg, and Zn because they crystallize in the hcp structure at zero pressure and reliable experimental measurements of *C*
_{44} are available (Table 2). The results agree to within 15% for Zr and Mg, but a large discrepancy is found for Zn, which undergoes higher-pressure phase transition and anomalous changes in electronic structure (34) (not evident in Fe). Direct comparison of the measured and calculated frequency of the*E*
_{2}
_{g} phonon at different pressures is possible for Mg (35) and Zr (36). In both cases, we find that the measured frequency is lower than that calculated by first-principles methods, which is consistent with results obtained for *ɛ*-Fe (Figs. 2 and 3). This study and analysis open the possibility of Raman investigations of the vibrational dynamics of other metals, including planetary core-forming materials at megabar to multimegabar pressures.