www.sciencemag.org/content/352/6285/580/suppl/DC1

Supplementary Materials for

Pressure-dependent isotopic composition of iron alloys

A. Shahar,* E. A. Schauble, R. Caracas, A. E. Gleason, M. M. Reagan, Y. Xiao, J. Shu, W. Mao

*Corresponding author. Email: ashahar@carnegiescience.edu

Published 29 April 2016, Science 352, 580 (2016) DOI: 10.1126/science.aad9945

This PDF file includes: Materials and Methods

Supplementary Text

Figs. S1 to S7

References

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Materials and Methods Experimental Methods We conducted high-pressure, room temperature NRIXS experiments at sector 16-ID-D (HPCAT) of the Advanced Photon Source at Argonne National Laboratory. We obtained energy spectra from -120 meV to +150 meV in steps of 0.5 meV with an energy resolution of 2 meV. The counting time varied between 6-7 seconds/point with each NRIXS scan lasting about one hour and between 19-50 scans per pressure point. 100% isotopically enriched 57Fe powder was purchased from American Elements, 57FeO was made in a gas-mixing furnace and checked for purity by x-ray diffraction and 57Fe3C was made in a piston cylinder apparatus and checked for purity by x-ray diffraction. For the hydride, pure 57Fe was loaded with fluid H2 into the gasket and reacted to form FeHx. This was checked by both x-ray diffraction and visual inspection in the diamond cell (32). The pure powders were loaded into a sample chamber drilled into a beryllium gasket in a panoramic diamond anvil cell. Pressure was calibrated using the ruby scale at HPCAT (16). Fe was measured at 17 and 40 GPa, FeHx was measured at 6, 12 and 22 GPa, FeO was measured at 5, 20, 25 and 39 GPa and Fe3C was measured at 2, 5, 19 and 36 GPa. Each pressure point was determined at least 19 times and as many as 40 times. We have also plotted data from Mao et al. (33) that we reanalyzed using the methods described below for FeHx at 4, 7, 10, 12, 22, 30, 42, 47 and 52 GPa. Density Functional Theory Methods Equilibrium mass dependent iron-isotope fractionation in metallic iron-light element alloys is estimated using the standard thermodynamic approach as extended to phonons in crystalline materials (34). These calculations require knowledge of isotope effects on vibrational frequencies in each lattice. To complement NRIXS measurements of the 57Fe component we compute the phonon density of states using first-principles calculations. We use the Density Functional Theory (DFT) and the Density Functional Perturbation Theory (DFPT) in the ABINIT and the Quantum Espresso (35) implementations. The two codes yield quasi-identical results (36) for a large variety of physical properties, including vibrational ones. We use ABINIT for pure iron in the hexagonal-closed packed (hcp) structure, in the 0 – 400 GPa pressure range for Mg0.75Fe0.25SiO3 non-magnetic bridgmanite in the 0 – 160 GPa range, and for Fe3C cementite structure in the 80 – 400 GPa range. We use Quantum Espresso for Fe3C and for FeH double-hcp (dchp) in the 0-36 GPa pressure range. Both codes employ pseudopotentials and plane-wave basis sets; we use the gradient-corrected PBE functional (37) for the exchange-correlation potential. In ABINIT we employ norm-conserving Troullier-Martins-type pseudopotentials, with kinetic energy cutoffs of 40 Hartrees (1Ha=27.2116 eV). In Quantum Espresso we use Vanderbilt-type ultrasoft pseudopotentials with kinetic cutoff energies of 20 Ha, with a higher 240 Ha cutoff energy for the charge density and potential. The scalar-relativistic Fe.pbe-nd-rrkjus.UPF pseudopotential, used in an early study of phonons in BCC iron (37), is adopted for this study. The H.pbe-rrkjus.UPF and C.pbe-rrkjus.UPF, were generated by the same authors. All three pseudopotentials are available in the online Quantum Espresso pseudopotential library (http://www.quantum-espresso.org/pseudopotentials/). The silicon pseudopotential is taken from the GBRV repository (38; https://www.physics.rutgers.edu/gbrv/). These parameters ensure

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precision of the total energy on the order of 1 mHa/formula unit or better. The reciprocal space is sampled using regular grids of high-symmetry Monkhorst-Pack k points. Their density varies from 20x20x16 for hcp Fe, to 8x8x8 for bridgmanite, 6x4x6 for Fe3C-cohenite, or 16x16x5 for dhcp-FeH. Each crystal structure is relaxed, allowing atomic positions and lattice constants to vary until the unit cell strain is within 0.05 GPa of the target pressure, and the forces acting on each atom in each Cartesian direction are on the order of 10–5 Ha/bohr. For the models with norm-conserving pseudopotentials, the dynamical matrices are directly calculated on regular symmetric grids of q points, of the Monkhorst-Pack type, spanning the first Brillouin zone. Then we use Fourier-interpolation techniques (19) to obtain the dispersion of the phonon bands in the entire Brillouin zone and interpolate them. This type of model has yielded results of useful accuracy in previous studies of iron isotope fractionation in nonmetallic crystals (35). For the models with ultrasoft pseudopotentials, the phonon density of states is directly sampled by density functional perturbation theory at 1-6 discrete phonon wave vectors forming a regular grid in the Brillouin zone (6 for BCC-Fe, 4 for the 4-atom Fe3Si unit cell, 2 for the 8-atom dhcp-FeH unit cell, and 1 for the 16-atom Fe3C-cohenite unit cell). All phonon wave vector grids are offset from the center of the Brillouin zone. Based on a comparison between the model frequencies and inelastic neutron scattering measurements in BCC-iron at five vectors in the Brillouin zone at 1 atm (38), there is no evidence of systematic error in calculated frequencies (ωDFT = 1.004±0.015 x νObs.; R2 = 0.98). There will be frequency errors associated with the neglect of anharmonic effects in the calculations, but much larger systematic deviations of up to ~5% are observed in in electronic insulators and molecules using the same method, indicating that the close match between model and observed frequencies for iron may be due to a fortuitous cancellation of errors. We have not applied any frequency scale factors to the calculated frequencies in DFT-based calculations of fractionation factors. The data in figure 3 are reported at the density corresponding to a static pressure of 60 GPa, i.e. they were not corrected for the thermal dilation. However, we have calculated the effect to be no more than a couple of percent for the materials studied. Indeed a 2% density correction between 0K and mantle temperatures (39) yields a correction in the beta factors on the order of less than 2% for any one specific phase. The thermal density correction is within the same order of magnitude for the mantle silicates and the iron alloys (40). This implies that the thermal correction on the partitioning functions should be negligible, i.e. less than 1%, compared to the pressure effect. Supplementary Text Calculating Force Constants and Beta Factors

In order to calculate the beta-factors from the NRIXS spectra, the force constant needs to be determined. There are currently several different methods for determining the force constants, which lead to different results for the beta factors. The PHOENIX (41) program analyzes the NRIXS data and calculates three different force constant values. The first determination of the force constant uses the moment approach wherein the normalized excitation probability is computed using the raw data and the experimental resolution function. The outcome is based on the raw data and therefore multi-phonon

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contributions are included in the analysis. The second determination of the force constant is also based on this moment approach however the data is first ‘refined’ that is, the multi-phonon contribution is corrected for and S(E) is extrapolated (where S(E) represents the moments of the NRIXS spectrum). Dauphas et al. (15, 42) argue that this moment approach is the best method for determining the force constant as it has smaller uncertainties, is not as dependent on the background, and it not as sensitive to asymmetric scans. The third determination is based on the partial density of states (DOS, Figure S1) that is computed from the raw data. The force constant is calculated based on only one-phonon contributions and therefore does not need to be corrected for the possible multiple phonon contributions. Murphy et al. (43) argue that using the DOS is the best approach because when they plot the force constant using the moment approach and the phonon DOS approach they find more scatter in the moment approach. They suggest that the extra scatter results from multi-phonon contributions beyond the scanned energy range, that are not taken into account in the moment-based algorithm. This contribution is not important in the DOS determination of the force constant because the phonon DOS is determined by single phonon contributions only. In order to obtain the most precise force constants as a function of pressure and to determine the effect of the various variables, we have truncated our data so as to span a variety of energy intervals, assumed a range of values for the background correction, and used both the moment approach and DOS approach. As discussed above, there are several ways in which to calculate the force constants from the spectra, but in all cases the elastic peak is first subtracted in order to obtain the normalized spectrum. In order to subtract the elastic peak, a proper background must be chosen. However, Figure S2 shows how the background is not symmetrical around the elastic peak, making the determination of a background non-trivial. The asymmetrical background increases with pressure thus creating a situation where high-pressure data are more susceptible to errors introduced by background fitting than low pressure data. This difference in background is one of the reasons that Dauphas et al. (40) suggest using their SciPhon program. Figure S3 shows how the choice of background correction affects the force constant produced by PHOENIX, when using the DOS approach. When background is known, however, this is not as much of an issue, and in our measurements the background is always determined before and after scans. The main test we used to determine which force constants were the most accurate was changing the range of energy used to calculate the force constants. We truncated the data at different maximum energies and then calculated the force constant. For example, if the scan was originally measured to a maximum energy of 150 meV, truncated sets are processed with cut-offs set at 100 meV, 110 meV, etc. The influence of the maximum energy truncation is shown in Figures S4, S5 and S6. For data process with the PHOENIX algorithms (red and yellow symbols) there is a typical progression: at low cutoff energies the force constants are low, initially increasing as more high-energy data are included, then flattening out for cutoffs in the 110-120 meV range, and then increasing again at the highest cutoff energies. If one takes an "average" force constant over the physically plausible range of cutoff energies, this roughly matches the 110 - 120 meV cutoff result. It is unclear from this analysis what is causing this difference in estimated force constants, and which energy range constitutes the ‘correct’ value. While Dauphas et al. (42) suggest that the largest energy ranges are best, it is clear that the force constants calculated from spectra acquired at >120 meV in this study are

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extremely high when using the PHOENIX data reduction algorithms. Data processed with SciPhon, however, do seem to converge to a consistent force constant value at higher energy cutoffs. SciPhon determines the background on each side of the elastic peak and subtracts them individually instead of using one background value for the whole spectral range. The broadest spectral energy ranges allow for the most accurate determination of the background correction, apparently reducing overall uncertainty. As can be seen in Figures S4-S6 there are some data sets and cutoff energies for which the two programs produce the same force constant, and others, which yield substantially different results -- and no obvious criterion for predicting whether a particular spectrum will fall into either group. As far as we know, no one has conducted a similar comparison of these two programs, in part because SciPhon is so new. Electronic structure models of Fe1-δO at room pressure, based on density functional theory, suggest a force constant of 142 N/m [Reference] at ambient pressure, suggesting that the values obtained by PHOENIX (~180 N/m) and SciPhon (~165 N/m) are too high. Force constants estimated with SciPhon are in general lower than the PHOENIX values, and therefore are closer to the theoretical prediction in this case. In this study, we have chosen to use the SciPhon values (with the entire data range) as the ‘correct’ values for the force constants of the phases measured. We have chosen to use these values because SciPhon:

1. uses more accurate background subtraction, 2. gives values closer to theory where there are theoretical values to compare

with, 3. and is more consistent over differing energy ranges.

Comparison to Previous Data As a way to check whether our values were consistent with previous studies, we plot the data of Mao et al. (33) for FeHx (Figure S7) along with our current data. We reanalyzed the data files with SciPhon and found excellent agreement between the two studies.

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Fig. S1. PDOS of FeO at the four pressures analyzed in this study, 5, 20, 25 and 39 GPa as an example. As expected, the spectral features are shifted to higher energies with increasing pressure.

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Fig. S2 A close-up of the background surrounding the elastic peak of 57FeO at pressures of: a) 5 GPa, b) 20 GPa, c) 25 GPa and d) 39 GPa. At 5 GPa the background on the low and high-energy ends of the scan range is relatively constant, though a slight increase is observed. However as the pressure increases the background becomes significantly asymmetric, and is higher on the high energy side of the peak.

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Fig. S3 A plot of the beta factor obtained as a function of the background value used in the PHOENIX code. The red circles show force constants derived after refinement, and the yellow circles show force constants derived from the partial phonon Density of States (DOS). It can be seen that the partial DOS analysis is sensitive to the background value chosen for each experiment while the other two methods are not. The data are from the 57FeO experiment at: (a) 5 GPa and (b) 25 GPa with an energy cut-off at 120 meV.

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Fig. S4 Force constants of Fe3C determined by PHOENIX and SciPhon as a function of the energy range used in the analysis at four different pressures: a) 2 GPa, b) 5 GPa, c) 19 GPa and d) 36 GPa. The red circles show the force constants derived after refinement, the yellow circles show force constants derived from the partial DOS, both from PHOENIX; the pink circles show results reduced using SciPhon. The PHOENIX data show greater sensitivity to the choice of energy range sampled, and in all cases have a very large increase for the largest energy range. The SciPhon data show some spread, however mostly when a narrower sample of energies are included.

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Fig. S5 The two panels show the force constant of FeHx determined by PHOENIX and SciPhon as a function of the energy range used in the analysis at two different pressures: a) 12 GPa and b) 22 GPa. The red circles show force constants derived after refinement, the yellow circles show force constants calculated from the partial DOS, both using the PHOENIX code; the pink circles show force constants calculated with the SciPhon code.

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Fig. S6 The four panels show the force constant of FeO determined by PHOENIX and SciPhon as a function of the energy range used in the analysis at the four different pressures: a) 5 GPa, b) 20 GPa, c) 25 GPa and d) 39 GPa. The red circles derive the force constant after refinement, the yellow circles use the partial DOS to derive the force constant both from PHOENIX; the pink circles use SciPhon.

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Fig. S7 A comparison of our recent FeHx results to Mao et al. 2006, showing the good agreement between the two studies. The data sets were (re)analyzed with SciPhon for this plot.

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