Abstract
The dispersion of longitudinal acoustic phonons was measured by inelastic x-ray scattering in the hexagonal closed-packed (hcp) structure of iron from 19 to 110 gigapascals. Phonon dispersion curves were recorded on polycrystalline iron compressed in a diamond anvil cell, revealing an increase of the longitudinal wave velocity (V P) from 7000 to 8800 meters per second. We show that hcp iron follows a Birch law for V P, which is used to extrapolate velocities to inner core conditions. Extrapolated longitudinal acoustic wave velocities compared with seismic data suggest an inner core that is 4 to 5% lighter than hcp iron.
The knowledge of the elastic constants of the phases of iron, which makes up 70 to 90 weight % of planetary cores, is essential for comparison with global velocity models of Earth. The hcp (or ɛ) high-pressure phase of iron is stable to at least 300 GPa at ambient temperature (1). Elastic properties of hcp iron have been determined to 210 GPa by x-ray diffraction (XRD) lattice strains measurements (2, 3), but these results show discrepancies with calculations using first-principles methods (4-7), as well as with a recent experimental investigation to 42 GPa by nuclear resonant inelastic x-ray scattering (NRIXS) of synchrotron radiation (8). The most recent investigation with NRIXS (9), however, yielded results consistent with lattice strain measurements (3). Elastic properties of hcp iron determined by Raman spectroscopy to 156 GPa yielded a C 44elastic modulus that is lower than previous determinations (10). Inconsistencies among these studies might be partly attributed to the fact that none of these techniques directly measures the acoustic wave velocities of iron. This limitation can be overcome by inelastic x-ray scattering (IXS) with meV energy resolution, where the acoustic velocity can be directly derived from the dispersion of the acoustic phonon energy (11, 12).
Our IXS experiment was carried out at the inelastic scattering beamline ID28 at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. The undulator x-ray beam was monochromatized by a cryogenically cooled silicon (111) crystal and by a very high energy resolution monochromator, operating in backscattering geometry and using the silicon (888) reflection order. This beam, with an energy of 15.618 keV and an energy resolution of 3.9 meV, was focused with a gold-coated mirror down to a beam size of 270 μm by 80 μm (horizontal by vertical, full width at half maximum) at the sample location. These incident beam dimensions were further reduced by slits to avoid scattering from the high-pressure cell gasket. The scattered photons were collected by five spherical silicon crystal analyzers operating in backscattering and Rowland circle geometry at the same reflection order as the high-resolution monochromator. The momentum transfer Q [Q = 2k 0sin(θS/2), where k 0 and θS are the wave vector of the incident photon and the scattering angle, respectively] was selected by rotating the 7-m-long spectrometer arm in the horizontal plane. Spectra were collected simultaneously at five different momentum transfers [Q= 4, 6.16, 8.31, 10.46, and 12.62 nm−1]. A powdered iron sample (99.999% purity) was loaded into the 120-μm hole of a rhenium gasket and compressed between diamond anvils. Pressures were determined with the ruby fluorescence technique (13) and cross-checked by XRD (14). The maximum pressure of 112 GPa determined according to the isothermal equation of state of iron (1) agrees well with the ruby fluorescence method, which yielded 110 GPa. The experiment was performed on a polycrystalline sample of iron because it is impossible to preserve a single crystal while crossing the phase boundary from the low-pressure body-centered cubic structure (bcc, or α phase) to the hcp structure between 12 and 15 GPa (15). Data have been collected at three pressures (ambient pressure, 0.2 GPa, and 7 GPa) on the bcc structure of iron and at six pressures (19, 28, 45, 55, 64, and 110 GPa) on the hcp structure.
A typical IXS spectrum and its corresponding fits are shown as a function of transfer energy in Fig. 1. The peak centered at zero-energy transfer corresponds to the elastic contribution to the signal, whereas two other peaks are visible at higher energy transfer. The knowledge of the phonon dispersion curves of iron (16) and diamond (17) at ambient pressure allows an unambiguous assignment of these features. The inelastic signal at high-energy transfer (i.e., high acoustic wave velocity) corresponds to the transverse acoustic (TA) phonon branch of the diamond anvils, whereas the remaining peak is attributed to the longitudinal acoustic (LA) phonon of iron. These inelastic contributions shift toward higher energies with increasing Qvalues (Fig. 2), so that the inelastic contribution from diamond moves out of the energy transfer window at Q > 4 nm−1. At 8.31 and 10.46 nm−1, an additional feature is visible between the quasi-elastic line and the LA phonon of iron. Wave velocities derived from the energy position of these excitations suggest that they correspond to the TA phonon of iron. The TA phonons, however, were detected at two momentum transfers and at two pressures only, precluding any further attempt to derive the pressure dependence of the shear velocities of iron. At variance, the LA phonon branch is observed over the entire momentum and pressure range explored.
IXS spectra of hcp iron at 28 GPa and the Q value of 4 nm−1. The experimental data (open circles) are plotted along with corresponding fits. The energy positions and the widths of the excitations were fitted by using a Lorentzian model function, convoluted with the experimentally determined energy resolution function, by standard χ2minimization. Dashed lines represent the inelastic contributions of the LA phonon branch for iron and from a TA branch of the diamond anvil. The thin solid line represents the elastic contribution and the thick continuous line shows the summation of individual contributions. Error bars indicate the estimated SD of the photon-counting process.
Dispersion of the iron LA phonon with increasingQ values (in nm−1). LA phonons of iron are indicated by ticks. A TA mode detected at Q = 8.31 nm−1 and at Q = 10.46 nm−1 is indicated by broken ticks. Data are normalized to the intensity of the iron LA phonon peak. The integration time for each point was of the order of 600 to 700 s, obtained by a summation of four to six scans in the range of 0 to 50 meV. The energy position of the phonons could be determined with a relative error of typically 3%. Error bars indicate the estimated SD of the photon-counting process. a.u., arbitrary units.
The LA wave velocity V
P was determined at each pressure by fitting the dispersion curve with a sine function
(1)from which V
P as well as the position of the edge of the first Brillouin zone, Q
MAX, can be derived (Fig. 3). Data recorded at four to five momentum transfers have been used in each dispersion curve to constrainV
P within an estimated error of 3%, with the exception of the data point at 110 GPa, for which only two momentum transfer data points could be used. The observed values ofQ
MAX are in agreement with those calculated after (1). At 110 GPa, the acoustic wave velocity was determined with a fixed Q
MAX calculated after (1). LA wave velocities as a function of pressure are summarized in Table 1. Our results for the low-pressure bcc structure agree with the ultrasonic data collected to 1 GPa (18) and extrapolated to 10 GPa within 1%, thus attesting to the reliability of these measurements (19).
LA phonon dispersion curves of iron at different pressures. Lines represent the results of the fit of Eq. 1. Solid symbols and dashed lines stand for measurements carried out on the bcc phase at 0.2 and 7 GPa. Open symbols and solid lines correspond to the pattern recorded on the hcp structure of iron at 19, 28, 45, 55, 64, and 110 GPa from bottom to top, respectively. The energy position of the phonons could be determined within 3% (error bars).
Acoustic sound velocities of hcp and bcc iron at 298 K and high pressures.
Our results for the high-pressure hcp structure of iron (Fig. 4) compared with shock wave measurements (20) show that hcp iron follows a Birch law (21) for V P, which provides a convenient relation for extrapolating our measurements to higher pressures. Seismic data (22) do not fit the experimental extrapolation, suggesting that Earth's inner core is slightly lighter than hcp iron, as proposed in earlier work (1, 23). The density differences are 4 to 5%.
LA wave velocities of hcp iron [open squares (this work)] and solid diamonds [shock wave Hugoniot measurements (20)] as a function of specific mass. Preliminary Reference Earth Model seismic data are represented by open diamonds (22). As shown by the dashed line, the experimental points for pure iron move along a straight line. This linear relation between velocity and density, known as Birch's law, is described in detail in (21). Error bars indicate the error inV P as obtained from Eq. 1.
Our measurements are consistent with the ultrasonic data (3), the XRD measurements (3), and the NRIXS data (9) below 100 GPa (Fig. 5). Above 100 GPa, however, our extrapolation departs from these measurements, yielding lower acoustic velocities than those derived from XRD (3) and NRIXS experiments (9). In the same manner, one observes a substantial discrepancy with results from theoretical calculations (4–7) at pressures of 210 GPa.
LA wave velocities (V P) of iron as a function of pressure for the present work (•) and extrapolated at higher pressure after a Birch fit to our data (solid line) and plotted along with ultrasonic (○) and XRD measurements (⊞) (3), XRD measurements (×) (2), NRIXS data (□) (9), shock wave Hugoniot measurements not reduced to 300 K (⧫) (20), observations for the inner core (◊) (22), and ab initio calculations (▿) (4–7). The possible effects of preferred orientations as estimated from (3) and (24) are within the displayed error bars.
Our IXS experiment on a polycrystalline iron sample only allows us to determine the orientationally averaged dispersion curves for the LA phonon branch. The experiment is therefore sensitive to preferred orientations of crystals in the sample, when reciprocal lattice vectors are not randomly oriented in comparison withQ. XRD measurements carried out in parallel to our IXS study (14) indicate randomly oriented iron crystals at pressures below 40 GPa. At low pressure, we find an excellent agreement between our V P measurements and orientationally averaged ultrasonic data (3). At pressures higher than 50 GPa, our diffraction data show that hcp iron displays a concentration of c axes parallel to the diamond anvil cell compression axis, in agreement with recent XRD texture measurements (24). Those XRD measurements (24) predict a large anisotropy for V P, withP waves traveling 18% faster at 45° from the caxis than either in the ab plane or along the caxis. According to our XRD measurements, the x-ray inelastic scattering by acoustic phonons is made preferentially in the ab plane, because momentum transfer lies perpendicular to the incident x-ray beam and to the c axis. Consequently, one could have underestimated V P as preferred orientations develop. When anisotropy curves are considered (3,24), however, the orientationally averaged values ofV P are similar, within a few percent, to the values corresponding to a predominant but not complete preferential orientation of the c axis [figure 4 in (3)]. Taking into account experimental error bars, the values ofV P measured in this study should therefore be indistinguishable from orientationally averaged values. Up to 100 GPa, the agreement with measurements of V P by XRD data (3) is good (Fig. 5), suggesting that the isostress assumption used for interpreting such data, experimentally validated for cubic phases of iron and iron oxide only (2), might be valid for hcp iron as well. A direct comparison with NRIXS data (8, 9) is more difficult. In such experiments, the strong elastic line has to be subtracted in order to perform a meaningful parabolic fit of the low-energy part of the density of states. This yields an average Debye phonon velocity, which has to be converted into V P and shear wave velocityV S through an a priori averaging scheme. We note, however, that our V P measurements are encouragingly consistent with V P values derived from vibrational densities of state measured by NRIXS (9), although extrapolation of the trends outside of the actual measurement ranges would yield very different values of V Pat inner core conditions.
↵* To whom correspondence should be addressed. E-mail: fiquet{at}lmcp.jussieu.fr
