GEOPHYSICAL RESEARCH LETTERS, VOL. 40, 2958–2962, doi:10.1002/grl.50582, 2013
The effect of Si and S on the stability of bcc iron with respect
to tetragonal strain at the Earth’s inner core conditions
Hang Cui,1 Zhigang Zhang,1 and Yigang Zhang1
Received 2 April 2013; revised 19 May 2013; accepted 21 May 2013; published 18 June 2013.
[1] 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. Citation: Cui, H., Z. Zhang, and Y. Zhang
(2013), The effect of Si and S on the stability of bcc iron with
respect to tetragonal strain at the Earth’s inner core conditions,
Geophys. Res. Lett., 40, 2958–2962, doi:10.1002/grl.50582.
1. Introduction
[2] 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].
[3] 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
1
Key Laboratory of the Earth’s Deep Interior, Institute of Geology and
Geophysics, Chinese Academy of Sciences, Beijing, China.
Corresponding authors: H. Cui, Key Laboratory of the Earth’s Deep
Interior, Institute of Geology and Geophysics, Chinese Academy of
Sciences, Beijing 100029, China. ([email protected])
Z. Zhang, Key Laboratory of the Earth’s Deep Interior, Institute of
Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029,
China. ([email protected])
©2013. American Geophysical Union. All Rights Reserved.
0094-8276/13/10.1002/grl.50582
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.
[2008] 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.
[4] 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
[5] 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. [2008]
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
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CUI ET AL.: STABILITY OF BCC IRON AT CORE CONDITIONS
Figure 1. (a) Stress differences of bct lattice with respect to tetragonal strain under different conditions (128 atoms in the
system). Uncertainty for each point is less than the size of the symbols. (b) Integrated free energies relative to bcc structure
under different conditions. The uncertainty corresponds to the uncertainty in stress tensor calculations plus the error during
integration, which is no more than 4.5 meV at 5000 K and 3.5 meV at 6000 K, and these are represented by the error bars
on the curve (the curves serve as guides for the reader’s eyes).
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. [2008], and
may not clarify the plausible stability of bcc-Fe.
[6] In this study, we visited the issue of elastic stability of
Fe alloys in an alternative routine through fully firstprinciples 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
[1924] 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 [2009], 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
dw ¼ Fdu ¼ Σa Aa saa dua
(1)
where the summation is taken over a = x, y, z. Aa represents
the area of the faces of lattice cell perpendicular to the axial,
saa is the principal components of stress tensor (the contributions of nonprincipal components sab are zero for they are
perpendicular to the displacements of lattice faces), and dua
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
dw ¼ a3 szz sxx þ syy =2 =t dt
(2)
[7] 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.
[8] 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.
[9] 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].
[10] 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
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CUI ET AL.: STABILITY OF BCC IRON AT CORE CONDITIONS
Figure 2. Internal energies with respect to tetragonal strain under different conditions at the same volume of 7 Å3/atom. As a
comparison, inset plot shows the internal energies at static conditions and 6.7 Å3/atom.
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. [2003] and Luo et al. [2010], 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 [1989].
3. Results
[11] The stress differences (szz – (sxx + syy)/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 (szz) first increases
until around t = 1.20 and then rapidly decreases with further
distortions. sxx and syy show the opposite trend during this
transformation and are smaller in magnitude than szz, 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
Figure 3. Correlation between the internal energy and entropy with the same volume (7 Å3/atom) under the conditions
involved in this study.
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CUI ET AL.: STABILITY OF BCC IRON AT CORE CONDITIONS
the bcc structure, their effects turn out to be weak in fully
restabilizing the bcc structure.
[12] 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.
[13] 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
[14] 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.
[15] 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. [2008] 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.
[16] 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.
[17] 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.
[18] 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].
[19] Acknowledgment. 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.
References
Alfe, D., M. J. Gillan, and G. D. Price (2002), Composition and temperature
of the Earth’s core constrained by combining ab initio calculations and
seismic data, Earth Planet. Sci. Lett., 195(1–2), 91–98.
2961
CUI ET AL.: STABILITY OF BCC IRON AT CORE CONDITIONS
Alfe, D., et al. (2003), Thermodynamics from first principles: temperature and
composition of the Earth’s core, Mineralogical Mag., 67(1), 113–123.
Bain, E. C., and N. Y. Dunkirk (1924), The nature of martensite, Transaction
AIME, 70, 25–46.
Belonoshko, A. B., R. Ahuja, and B. Johansson (2003), Stability of the bodycentred-cubic phase of iron in the Earth’s inner core, Nature, 424, 1032–1034.
Birch, F. (1964), Density and composition of the mantle and core,
J. Geophys. Res., 69, 4377–4388.
Blochl, P. E. (1994), Projector augmented-wave method, Phys. Rev. B, 50,
17953–17979.
Cote, A. S., L. Vocadlo, D. P. Dobson, D. Alfe, and J. P. Brodholt (2010), Ab
initio lattice dynamics calculations on the combined effect of temperature
and silicon on the stability of different iron phases in the Earth’s inner
core, Phys. Earth Planet. In., 178(1–2), 2–7.
Dubrovinsky, L., et al. (2007), Body-centred cubic iron-nickel alloy in
Earth’s core, Science, 316, 1880–1883.
Flyvbjerg, H., and H. G. Petersen (1989), Error-estimates on averages of
correlated data, J. Chem. Phys., 91, 461–466.
Frenkel, D., and B. Smit (1996), Understanding Molecular Simulation,
Academic, Boston.
Huang L., N. V. Skorodumova, A. B. Belonoshko, B. Johansson, and
R. Ahuja (2005), Carbon in iron phase under high pressure, Geophys.
Res. Lett., 32, L213, doi:10.1029/2005GL024187.
Kohn, W., and L. J. Sham (1965), Self–Consistent Equations Including
Exchange and Correlation Effects, Phys. Rev., 140, A1133–A1138.
Kresse, G., and J. Furthmüller (1996), Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set, Phys. Rev. B, 54, 11169–11186.
Lin, J. F., D. L. Heinz, A. J. Campbell, J. M. Devine, and G. Y. Shen (2002),
Iron-silicon alloy in the Earth’s core?, Science, 295, 313–315.
Luo, W., J. Borje, O. Eriksson, S. Arapan, P. Souvatzis, M. Katsnelson, and
R. Ahuja (2010), Dynamical stability of body center cubic iron at the
Earth’s core conditions, PNAS, 107, 9962–9964.
Mattesini, M., et al. (2010), Hemispherical anisotropic patterns of the Earth’s
inner core, PNAS, 107(21), 9507–9512.
McDonough, W. F., and S.-S. Sun (1995), The composition of the Earth,
Chem. Geol., 120, 223–253.
Mermin, N. D. (1965), Thermal properties of inhomogeneous electron gas,
Phys. Rev., 137(5A), A1441–A1443.
Nosé, S. (1984), A molecular-dynamics method for simulations in the
canonical ensemble, Mol. Phys., 52, 255–268.
Ozolins, V. (2009), First-Principles Calculations of Free Energies of
Unstable Phases: The Case of fcc W, Phys. Rev. Lett., 102, 065702.
Perdew, J. P., K. Burke, and M. Ernzerhof (1996), Generalized gradient
approximation made simple, Phys. Rev. Lett., 77, 3865–3868.
Poirier, J. P. (1994a), Light elements in the Earth’s outer core: a critical
review, Phys. Earth Planet. Inter., 85, 319–337.
Poirier, J. P. (1994b), Phsical-properties of the Earth core, C. R. Acad. Sci. II,
318, 341–350.
Söderlind, P., J. A. Moriarty, and J. M. Willis (1996), First-principles theory
of iron up to Earth-core pressures: structural, vibrational and elastic
properties, Phys. Rev. B, 53, 14063–14072.
Stixrude, L., and R. Cohen (1995a), Constraints on the crystalline-structure
of the inner-core mechanical-instability of bcc iron at high-pressure,
Geophys. Res. Lett., 22(2), 125–128.
Stixrude, L., and R. E. Cohen (1995b), High-pressure elasticity of iron and
anisotropy of earth’s inner core, Science, 267, 1972–1975.
Stixrude, L., et al. (1994), Iron at high-pressure linearized-augmented-planewave computations in the generalized-gradient approximation, Phys. Rev.
B, 50(9), 6442–6445.
Vocadlo, L., J. Brodholt, D. Alfe, M. J. Gillan, and M. D. Price (1999), The
Structure of Iron under the Conditions of the Earth’s Inner Core, Geophys.
Res. Lett., 26, 1231–1234.
Vocadlo, L., D. Alfe, M. J. Gillan, I. G. Wood, J. P. Brodholt, and G. D. Price
(2003), Possible thermal and chemical stabilization of body-centred-cubic
iron in the Earth’s core, Nature, 424, 536–539.
Vocadlo, L., et al. (2008), The stability of bcc-Fe at high pressures and
temperatures with respect to tetragonal strain, Phys. Earth Planet. In.,
170(1–2), 52–59.
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