# Ni Doping: A Viable Route to Make Body-Centered-Cubic Fe Stable at Earth’s Inner Core

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{1−x}Ni

_{x}alloys (x = 0, 0.0312, 0.042, 0.0625, 0.084, 0.125, 0.14, 0.175) at 200–364 GPa and investigated their relative stability. Our thorough study reveals that the stability of Ni-doped bcc Fe is crucially dependent on the nature of the distribution of Ni in the Fe matrix. We confirm this observation by considering several possible configurations for a given concentration of Ni doping. Our theoretical evidence suggests that Ni-doped bcc Fe could be a stable phase at the Earth’s inner core condition as compared to its hcp and fcc counterparts.

## 1. Introduction

**[3,4,7,33,34,35]**, nonetheless, with pure Ni crystallizing in the fcc phase, the fcc phase also forms a part of our study. We have therefore investigated the thermodynamic stability of the bcc phase with respect to both the hcp and fcc phases of Fe/Fe-Ni. Our results strongly support the idea of bcc Fe-Ni alloy as a promising candidate for the Earth’s inner core.

## 2. Methodology

^{6}4s

^{2}) and 10 electrons for Ni (3d

^{8}4s

^{2}). To ensure the convergence of electronic energies during the self-consistent run and the calculation of Hellmann-Feynman forces on atoms during structural optimization using the conjugate gradient algorithm, we used a convergence criterion of 1 × 10

^{−8}eV and 1 × 10

^{−3}eV/Å respectively. Our calculations suggest that under Earth’s core pressure and temperature conditions, both Fe and Ni [40,41] in Fe-Ni alloy become non-magnetic [42], though that does not rule out the possibility of some disordered magnetic moments, especially at Ni sites. In all our calculations, symmetry was switched off during structural optimization, and Fermi-smearing was implemented in order to account for the contribution of electronic excitations to the free energy.

_{o}= 10.91 Å

^{3}/atom, K = 220.19 GPa, and K′ = 4.49. The P-V equation of state has been presented in the supplementary materials (Figure S1). In order to estimate the volume of hcp Fe at the Earth’s inner core pressure, we employed third-order Birch-Murnaghan’s equation of state (3 B-M EOS) [43] with V

_{o}= 10.25 Å

^{3}/atom, bulk modulus K = 290.6 GPa and K′ = 4.29 as obtained from the DFT derived energy–volume relationship of hcp Fe-6.25%Ni. The corresponding 3 B-M EOS parameters for fcc Fe-6.25%Ni alloy were: V

_{o}= 10.34 Å

^{3}/atom; K = 272.6 GPa; K′ = 4.4. A comparison of the EOS parameters to those we obtained with previously reported EOS parameters for pure and Ni-doped bcc, hcp, and fcc phases of Fe is presented in the SI (refer to S3). The elastic tensor has been determined from the stress–strain relationship [44]. The bulk and shear moduli were calculated subsequently from the elastic tensor using the Voight-Hill-Reuss approach [45,46]. Further, using the calculated bulk and shear moduli and the density, the P and S wave velocities, i.e., V

_{P}and V

_{s}, were determined using the following formulas:

_{S}) and the isothermal bulk modulus (B

_{T}) are related through

## 3. Results

## 4. Discussion

_{P}) and shear-wave (V

_{S}) sound velocities of the two comparatively more stable phases of Fe-Ni alloy, namely, hcp and bcc as a function of density (ρ), as shown in Figure 4. It is to be noted that the computed elastic properties are static; i.e., they do not include the high-temperature effect and would therefore only approximate the actual values at Earth’s core temperatures. Our calculations find that the density of the hcp phase of Fe and Fe-Ni alloy at inner core conditions is higher than that given by PREM [55] by 5.7% and 6.4%, respectively. However, bcc Fe-Ni alloy is seen to reproduce the density of the inner core within a reasonable limit. The density of bcc-Fe-Ni alloy is found to differ by 3.3% from the actual density of the inner core as given by the PREM model. The calculated phase wave velocity was also seen to be well satisfied for the case of bcc-Fe-Ni alloy, where we find that the calculated data are lower than the PREM data by 1%, whereas in the case of hcp-Fe-Ni alloy, it is off by 16%. We compared our theoretically observed sound velocity vs. density trend with the coherent experimental results using consistent pressure scale, equation of state of hcp-Fe, etc. [33,34,56,57,58,59,60,61] and shock wave determination along the Hugoniot curve at high temperatures [62]. Mao et al. (1998) obtained the compressional wave velocity on hcp-Fe at the relatively low pressure of 16.5 GPa using pulse-echo ultrasonic techniques in multi-anvil large-volume presses. Using the impulsive stimulated light scattering (ISLS) method, Crowhurst et al. (2004) measured aggregate sound velocities of hcp-Fe up to 115 GPa, whereas, by using picosecond acoustics method in diamond anvil cell, Decremps et al. [61] reported sound velocity measurements of iron up to 152 GPa. Ohtani et al. [33] determined the compressional wave velocity of powdered hcp-Fe using high-resolution IXS and in situ X-ray powder diffraction at 300 K and pressure up to 174 GPa. Antonegeli et al. [57,58] carried out sound wave velocity measurements at 300 K and pressure up to 112 GPa using inelastic X-ray Scattering (IXS) and X-ray diffraction (XRD). The linear fit to the experimental values of sound velocities is extrapolated to inner core densities and compared to PREM, which shows slightly higher values than PREM for both V

_{P}and V

_{S}, i.e., 3% to 4% above PREM, with slightly steeper slope. This is probably due to the fact that different spectrometer techniques and absolute energy calibration were used in these experimental studies. More details can be obtained in Antonangeli and Ohtani [63] for hcp-Fe under static compression by different methods on the pressure and density dependence of the sound velocities. In addition, the most accurate nuclear resonant inelastic X-ray scattering (NRIXS) measurements [34,60] were used to compare V

_{S}at ambient temperature and pressure up to 171 GPa. In addition, the temperatures in high-pressure experiments are much lower than inner core temperatures [5]. As can be inferred from Figure 4, the computed densities of Ni-doped bcc Fe are higher than the PREM. However, the densities of Ni-doped bcc Fe decrease several percent (~3%) compared to hcp Fe and hcp Fe-Ni alloy. Our calculated V

_{s}of bcc Fe-Ni alloy at inner core conditions deviates to a great extent from the PREM. This may be partly due to anharmonic and/or premelting activity at high-temperature [64,65,66] or frequency-dependent viscoelastic relaxations [67] as well as the presence of melt [68,69] in the inner core (as suggested by seismic observations and quantum mechanical calculations), which has the ability to lower the shear wave velocity.

## 5. Concluding Remarks and Implications

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Phonon density of states (DOS) for body-centered cubic (bcc)-Fe (black shaded curve), 6.25% Ni containing bcc Fe-Ni alloy in two different configurations; Model-1: uniform (orange) and Model-2: random (blue) calculated at 364 GPa. The inset shows the corresponding models of the Ni-doped cases for which the phonon DOS has been shown.

**Figure 2.**DFT calculated phonon DOS at 364 GPa for Fe-Ni alloy with varied Ni concentrations. The Ni atoms were carefully distributed in a random fashion in the bcc Fe matrix. Inset shows the phonon band structure of the 3.125% Ni-containing case.

**Figure 3.**(

**a**) Enthalpy of hexagonal closed packed (hcp)-Fe [H(hcp)] and face-centered cubic (fcc)-Fe [H(fcc)] with respect to bcc-Fe [H(bcc)] plotted as a function of pressure in the presence of 6.25% Ni doping. Black dashed line: H(bcc)-H(hcp). Red dashed line: H(bcc)-H(fcc). (

**b**) Gibbs energy of hcp-Fe [G(hcp)] and fcc-Fe [G(fcc)] with respect to bcc-Fe [G(bcc)], plotted as a function of temperature. Black dashed line: G(bcc)-G(hcp). Red dashed line: G(bcc)-G(fcc).

**Figure 4.**Calculated compressional and shear velocities of Ni-doped bcc and hcp Fe equivalents at inner core conditions as a function of density and compared with hcp-Fe and the experimental results from 300 K to 1000 K [33,34,56,57,58,59,60,61] and PREM [55] and shock wave Hugonoit measurements at high temperatures [62]. The computed densities of hcp-Fe-Ni alloy and hcp-Fe (filled green and brown circles and diamonds, respectively) are higher than PREM. However, the density of bcc-Fe-Ni alloy (filled blue circle and diamond) is close to PREM. Light grey circle: Mao et al. [56]; grey circle: Crowhurst et al. [59]; yellow circle: Antonangeli et al. [57,58]; cyan circle: Ohtani et al. [33]; magenta circle: Decremps et al. [61]; green hexagon: Brown and McQueen, [62]; red diamond: Murphy et al. [34]; grey diamond: Gleason et al. [60]. The solid lines are a linear regression across all previous experimental results and theoretical data from the present study.

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Chatterjee, S.; Ghosh, S.; Saha-Dasgupta, T.
Ni Doping: A Viable Route to Make Body-Centered-Cubic Fe Stable at Earth’s Inner Core. *Minerals* **2021**, *11*, 258.
https://doi.org/10.3390/min11030258

**AMA Style**

Chatterjee S, Ghosh S, Saha-Dasgupta T.
Ni Doping: A Viable Route to Make Body-Centered-Cubic Fe Stable at Earth’s Inner Core. *Minerals*. 2021; 11(3):258.
https://doi.org/10.3390/min11030258

**Chicago/Turabian Style**

Chatterjee, Swastika, Sujoy Ghosh, and Tanusri Saha-Dasgupta.
2021. "Ni Doping: A Viable Route to Make Body-Centered-Cubic Fe Stable at Earth’s Inner Core" *Minerals* 11, no. 3: 258.
https://doi.org/10.3390/min11030258