The Earth's inner core is primarily composed of iron, but the stable crystalline structure of iron under core conditions still remains uncertain. The body-centered cubic (bcc) phase has been suggested as a possible candidate to explain the observed seismic complexity, but its stability at core conditions is highly disputed. In this study, we utilized thermodynamic integration techniques based on extensive first-principles molecular dynamics simulations to analyze the combining effects of high temperature and impurities on the stability of bcc structure with respect to tetragonal strain. According to our simulations, a small amount of Si/S permitted by seismological data at high temperature increases the stability of the bcc structure at high pressure, but not enough to achieve complete stability. This means the bcc-structured iron is highly unlikely to present in the Earth's inner core.
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 Although considerable effort has been made on the phase diagram of iron and its alloys under inner core conditions, the crystalline structure in the inner core is still controversial. Recently, the body-centered cubic (bcc) structured iron has been argued to be a strong candidate to interpret the seismic complexity revealed by observations with improved precision [Belonoshko et al., 2003; Mattesini et al., 2010]. However, bcc iron has long been regarded as an unstable phase at inner core conditions since it has been demonstrated to be mechanically unstable with respect to tetragonal strain at high pressures [Söderlind et al., 1996; Stixrude and Cohen, 1995a].
 The term tetragonal strain means to deform the lattice cell of bcc structure by changing the c/a ratio (i.e., change the ratio between the lengths of lattice vectors c and a) with the volume held constant. The bcc and fcc structures can be continuously deformed into one another by such a strain since they can be regarded as two special cases of the body-centered tetragonal (bct) lattice (bcc corresponds to c/a = 1.0 and fcc to c/a = √2) [Stixrude and Cohen, 1995a]. Elastic stability with respect to tetragonal strain means that the free energy of bcc phase needs to be a local minimum with respect to the tetragonal strain; otherwise, the bcc phase will transform to fcc spontaneously and therefore cannot exist. The results of previous calculations [Söderlind et al., 1996; Stixrude et al., 1994; Stixrude and Cohen, 1995a] show that the total energy of the bcc lattice decreases when the tetragonal strain is applied at high pressures (>150 GPa), which implies that the bcc structure is elastically unstable and will distort to fcc structure under core pressures. Utilizing time-consuming first-principles free-energy calculations, Vocadlo et al.  demonstrate that high temperature serves to appreciably increase the elastic stability of bcc-structured pure iron at 6000 K, although it is unclear as to whether complete stability has been achieved.
 Si, O, S, and C are the most likely light elements impurity in the core [Poirier, 1994a; Poirier, 1994b; McDonough and Sun, 1995]. These light elements and/or nickel alloyed with iron may be another factor to restabilize the bcc structure and therefore would further validate the occurrence of bcc structure in the Earth's inner core [Vocadlo et al., 2008; Lin et al., 2002; Dubrovinsky et al., 2007]. However, the combing effects of increasing temperature and alloying constituents for the elastic stability of bcc structure is unclear. To check their effects comprehensively, in this study, we carried out extensive first-principles simulations with various bct-structured iron alloys at high temperatures and utilized thermodynamic integration technique to analyze the stability of bcc structure with respect to tetragonal strain under core pressures. Since O strongly partitions into the liquid outer core [Alfe et al., 2002] and Fe-C alloys thermodynamically prefer the form of iron carbide [Huang et al., 2005], we focused on the Fe-Si/S alloys in this study.
2 Computational Method
 Phase stability at finite temperatures is fundamentally determined from free energy, which is computationally nontrivial since it is related to the volume of the entire phase space that is accessible to the system and cannot be expressed as the simple ensemble average. With a reference system of known free energy, on the other hand, it is relatively straightforward to calculate the free energy through thermodynamic integration techniques [Frenkel and Smit, 1996]. With this approach, Vocadlo et al.  calculated the free energies of bct-structured pure iron by choosing the Einstein model of harmonic crystal as the reference system and adiabatically switching the phase space with the aid of a classical potential. However, when we intended to directly apply similar routine to study the iron alloys, we found at least two inconveniences: (1) the classical potential as an important auxiliary in the integration path is much more subtle to implement for the iron alloy systems, and (2) the precision of the calculated free energies is only modest even with laborious simulations, as shown in the results of Vocadlo et al. , and may not clarify the plausible stability of bcc-Fe.
 In this study, we visited the issue of elastic stability of Fe alloys in an alternative routine through fully first-principles simulations by noticing that the free-energy difference rather than the absolute free energy determines the stability. As first pointed out by Bain and Dunkirk  and mentioned in the previous section, the bcc↔fcc transformation can be continuously linked through a simple tetragonal path (often named as Bain path) without crossing any first-order phase transition. Therefore, in the spirit of the thermodynamic integration technique, the free-energy differences ΔF can be calculated by integrating the reversible work driving the system from the reference to the target. To facilitate the integration, according to the methodology developed by Ozolins , it is more convenient to introduce a tetragonal distortion parameter t that is equivalent to the c/a ratio mentioned above, and the bct lattice with a volume of a3 can be described by the vectors a = (a/t1/2, 0, 0), b = (0, a/t1/2, 0), and c = (0, 0, at). Obviously, t = 1 for the bcc structure, and t = 21/3 for the fcc structure. Thus, the work done corresponding to infinitesimal tetragonal strain is
where the summation is taken over α = x, y, z. Aα represents the area of the faces of lattice cell perpendicular to the axial, σαα is the principal components of stress tensor (the contributions of nonprincipal components σαβ are zero for they are perpendicular to the displacements of lattice faces), and duα is the elastic displacement of lattice faces. Since Ax = Ay = a2t1/2, Az = a2/t, ux = uy = a/t1/2, and uz = at, equation (1) reduces to
 By taking the integral on both sides of equation (2), we can obtain the free energies of bct-structured phases along the Bain path if we choose the bcc phase as the reference system.
 We carried out extensive first-principles simulations based on the density functional theory [Kohn and Sham, 1965] for the integrations of equation (2). The simulations were implemented with the efficient VASP code [Kresse and Furthmüller, 1996], incorporating projected augmented wave method (PAW) [Blochl, 1994] and PBE exchange-correlation functional [Perdew et al., 1996]. Both static calculations and molecular dynamics simulations have been carried out in this study.
 In the static calculations, we used a 2 × 2 × 2 bcc supercell containing 16 Fe atoms to facilitate substitutions of different amounts of Si. In the iron alloys systems, we separated the Si/S atoms as far as possible since their contact in the iron solution is energetically unfavorable [Alfe et al., 2003]. To keep balance between accuracy and computation cost, we chose 7 × 7 × 7 k-points sampling grid mesh and a cut-off energy of 550 eV, which is found to be accurate enough with an energy convergence of no more than 1.2 meV/atom. The spin polarization was included since the bcc phase has a significant residual magnetic moment at static conditions and core pressures [Söderlind et al., 1996].
 In the molecular dynamics simulations, we used supercells of 128 atoms (4 × 4 × 4 supercell of the cubic 2-atom box; we have tried several supercells containing 32, 64, 128, and 192 atoms, and the obtained stress differences are well converged with respect to the size of the simulation box) and sampled the Brillouin zone at the Γ-point only with a cut-off energy of 550 eV. The thermal equilibrium between ions and electrons is assumed via the Mermin functional [Mermin, 1965]. Molecular dynamics trajectories were propagated in the canonical ensemble (NVT) with the Nose thermostat [Nosé, 1984] with a time step of 1 fs and a volume of 7 Å3/atom (corresponding to ~330 GPa at 6000 K, roughly the temperature and pressure condition at the boundary between the inner and the outer core (ICB)). We have not counted the spin polarization at high temperatures to accelerate simulations since the residual magnetic moment would be essentially destroyed by the thermal excitation of electrons [Vocadlo et al., 2003]. The simulations lasted 2000 steps for equilibriation and 10000 (10 ps) steps for statistical sampling. Consistent with Vocadlo et al.  and Luo et al. , we have verified the dynamical stability for all the high-temperature conditions involved in this study through careful analysis using all the molecular dynamics trajectories. The uncertainties were estimated by the blocking average method provided by Flyvbjerg and Petersen .
 The stress differences (σzz – (σxx + σyy)/2) of various bct lattice structures along the Bain path are shown in Figure 1a. The bcc (with t = 1) and fcc (with t = 21/3) phases are found to be in hydrostatic state with essentially vanished stress differences. Over the transformation from bcc to fcc, the magnitude of the stress in z-direction (σzz) first increases until around t = 1.20 and then rapidly decreases with further distortions. σxx and σyy show the opposite trend during this transformation and are smaller in magnitude than σzz, which means that the system output works and therefore is released to a lower energetic state along the path. This is more clearly shown in Figure 1b by integrating equation (2) with respect to tetragonal strain. The fcc-iron is demonstrated to be stable with strong resistance to the increased temperature and impurities. From 5000 K to 6000 K, the free-energy difference ΔFbcc–fcc decreased from 0.0625 eV/atom to 0.038 eV/atom but is obviously still far from reversing. Although the alloyed Si and S more or less further decrease the ΔFbcc–fcc and plausibly create a shallow local minimum in the vicinity of the bcc structure, their effects turn out to be weak in fully restabilizing the bcc structure.
 In Figure 2, we inspect the variations of internal energy along the Bain path at various temperatures and impurities. As a comparison, we also plot the counterpart at static conditions with a smaller volume (corresponding to the typical pressure of around 330 GPa of ICB). Ionic vibrations and finite temperatures are found to significantly and monotonously decrease the internal energy difference ΔUbcc–fcc from 0.48 eV/atom to 0.166 eV/atom. The effects of Si and S, on the other hand, are related with the temperature. At static conditions, 6.25 at.% S further decreases the ΔFbcc–fcc by 0.046 eV/atom, and the same amount of Si decreases it even further by 0.088 eV/atom. Over the tetragonal distortion, Si and S show noticeable differences in altering the internal energy of the system. Nevertheless, when temperature is increased to as high as 6000 K, the differences are remarkably diminished, and the ΔUbcc–fcc for both alloys is similar or slightly smaller than that of pure iron.
 With free energies and internal energies at hand for the bct structures along the tetragonal distortion Bain path, it is straightforward to get the entropies through the thermodynamic principle of S = (U – F)/T. Similar with tungsten [Ozolins, 2009], the entropies are interestingly found to be in direct correlation with the internal energies (Figure 3), which can be very well represented with second-order polynomials. Increasing temperature slightly stiffens the slope of ΔU/ΔS while the impurity slightly flattens it. In the units of Figure 3 and taking all the data points into account, ΔUbct–bcc ≈ 0.53ΔSbct–bcc – 0.45(ΔSbct–bcc)2.
4 Discussions and Concluding Remarks
 The bcc-structured iron, stable at low pressures as compared with fcc and hcp structures, has long been found to be thermodynamically and mechanically unstable at core pressures [Stixrude et al., 1994; Vocadlo et al., 1999]. Our calculations at static conditions verify that the global minimum along the Bain path is moved from bcc with c/a = 1(or t = 1) at ambient pressure to fcc with c/a = 21/2 (or t = 21/3) at 330 GPa. The Si impurity can restabilize the bcc structure at high pressures, consistent with previous studies [e.g., Cote et al., 2010], but more than 18.75 at.% silicon impurity is needed, and this will lead to too much density deficit in the inner core [Birch, 1964]. The same is true for other impurities possibly alloyed with iron, e.g., we found that even with 50 at.% Ni, the bcc-iron has not been stabilized.
 At finite temperatures, entropic effects become important and could in principle restabilize the bcc structure. While the sole effects of temperature turn out to be insufficient to restabilize the bcc structure, as shown in Vocadlo et al.  and in this study (Figure 1b), the alloyed impurities should at least further decrease the energetic gap between bcc and fcc structures since the entropy would be increased by impurities and takes more effects. This has always been argued to be another factor that would reinforce the stability of bcc-iron at core conditions [Dubrovinsky et al., 2007; Vocadlo et al., 2008; Cote et al., 2010]. Our analysis in this study reveals that it is true that the impurities will enhance the entropic effects, but impurities may not be as determinative as previous expectations. From Figure 2, we found that the effect of Si/S impurity on the internal energy with respect to tetragonal strain at high temperature is much smaller than that at static condition. This should be closely related with the increasingly metallic and isotropic nature of Fe-Si/S bonds and will compensate the entropic effects of restabilization. Therefore, we argue that the effects of impurities cannot be simply accounted with those inspected at static or lower temperatures.
 The calculations in this study reveal an almost excellent direct correlation between internal energy and entropy, which is almost independent of temperature and amount and type of impurities. Since the internal energy can be straightforwardly retrieved from a standard molecular dynamics simulation, the relation would greatly facilitate an at least rough estimation of the entropy and free energy of the system without extensive computations along the integration path. With the polynomial fitted equation mentioned at the end of the previous section, since ΔSbct–bcc is always negative in the bcc→fcc transformation, a temperature of over 7000 K is needed to fully restabilize the bcc structure. At such a high temperature, with the supposition that the iron has not been melted, as argued above, the effects of impurities would be further diminished.
 Our calculations in this study for high temperatures involve only one volume of 7 Å3/atom and one atomic concentration of alloyed Si/S (6.25%, i.e., 8 out of 128 atoms in the system). The volume was selected since the corresponding pressures (~315 GPa at 5000 K and ~325 GPa at 6000 K) along the Bain path are similar or a little smaller than the typical pressure at ICB (330 GPa), and the concentration is specified to the lowest possible counterpart in static simulations (1 out of 16 atoms). While it is worthwhile to investigate the free-energy profiles at other volumes/pressures and smaller concentrations (to typically 2–3% as constrained from seismic observations) using the approach in this study, the conclusions would not be affected since the increased pressure and decreased amount of impurities will further destabilize the bcc structure.
 According to seismic observations, the inner core is complex and exhibits a significant degree of layering, which is difficult to explain using the hcp phase alone; therefore, the possible existence of bcc-structured iron alloy in the inner core has been suggested to explain the observed seismic complexity [Belonoshko et al., 2003; Mattesini et al., 2010]. However, with extensive molecular dynamics simulations and thermodynamic integrations, we show in this study that the combined effects of temperature and impurities are still not able to restabilize the bcc structure under the Earth's inner-core conditions. The complexity and anisotropy of the inner-core seismic structures should be reconsidered with the fcc-hcp coexisting phase [Cote et al., 2010] or even the conventionally believed hcp phase [Stixrude and Cohen, 1995b].
 This research is supported by the National Natural Science Foundation of China (grants 41020134003, 90914010, and 40973048). Thanks to Prof. Wysession and John Brodholt and an anonymous reviewer for their comments and suggestions. All the simulations were performed in the facilities of Computer Simulation Lab in IGGCAS.