Earth and Planetary Science Letters 288 (2009) 534–538
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Ab initio calculations of the elasticity of hcp-Fe as a function of temperature
at inner-core pressure
Lidunka Vočadlo ⁎, David P. Dobson, Ian G. Wood
Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, UK
a r t i c l e
i n f o
Article history:
Received 14 August 2009
Received in revised form 1 October 2009
Accepted 11 October 2009
Available online 7 November 2009
Editor: R.D. van der Hilst
Keywords:
hcp-Fe
inner core
elasticity
seismic anisotropy
a b s t r a c t
Ab initio finite temperature molecular dynamics simulations have been used to calculate the elastic constants
of hexagonal-close-packed (hcp) Fe as a function of temperature at ~ 300 GPa. The longitudinal modulus c11
decreases with temperature, in stark contrast to previous calculations, but in agreement with experimental
observations on other transition metals at ambient pressures. c33 and c44 also decrease with temperature,
while c12 and c23 slightly increase. When these moduli are used to calculate P-wave velocities through the
crystal, the sense of the anisotropy is such that VP is fastest along the c-axis up to 5000 K; however, by 5500 K
the anisotropy reverses with VP becoming faster in the a–b plane. This suggests that, for an inner core
dominated by crystals of hcp-Fe aligned with the c-axis in the polar direction, the observed isotropic outer–
inner core could result from the hcp-Fe being at a temperature close to melting where the axial wave
velocities parallel and perpendicular to the c-axis become similar, while at greater depths in the inner–inner
core, where iron is further from melting, stronger anisotropy is achieved with the faster P-wave velocities
parallel to the polar axis. No other mechanisms, such as changes in composition or crystal alignment, are
therefore required to account for the observed change in seismic anisotropy of the Earth's inner core with
depth.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Seismic P-wave observations show the Earth's inner core to be
anisotropic and layered. It is well established that the inner core
exhibits significant anisotropy, with P-wave velocities ~ 3% faster
along the polar axis than in the equatorial plane (Creager, 1992; Song
and Helmburger, 1993; Song and Xu, 2002; Beghein and Trampert,
2003). More recent seismic observations suggest that the inner core
may also be layered; there is evidence for a seismically isotropic or
weakly anisotropic upper layer overlaying an irregular non-spherical
transition region to an anisotropic lower layer (Song and Helmburger,
1998; Song and Xu, 2002; Ishii and Dziewonski, 2003). The origins of
this anisotropy and layering are not properly understood but it has
generally been considered that the anisotropy reflects the preferred
orientation of the crystals present. The S-waves observed in the inner
core add to the complexity as they have unexpectedly low velocities
(Cao et al., 2005). Seismic interpretation is hampered by the lack of
knowledge of the physical properties of core phases at core conditions
and, at present, no mineralogical model exists which satisfactorily
matches the seismic data (for a recent summary of the problems of
understanding the inner core, see Vočadlo, 2007a,b).
⁎ Corresponding author.
E-mail address: l.vocadlo@ucl.ac.uk (L. Vočadlo).
0012-821X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2009.10.015
The hexagonal-close-packed (hcp) phase of pure iron is thought to
be the stable phase of iron at Earth's core conditions (Vočadlo et al.,
2000, 2003a; Nguyen and Holmes, 2004), although the addition of the
light elements likely to exist in the core may stabilise other phases such
as those with the body-centred-cubic or face-centred-cubic structures
(Vočadlo, 2007b; Vočadlo et al., 2008; Côté et al., 2008a,b, in press).
Knowledge of the elastic constants of iron at core conditions allows
determination of a number of properties including seismic wave
velocities and seismic wave anisotropy. The temperature dependence
of the elastic behaviour of hcp-Fe is thus of considerable interest,
especially since earlier calculations (Steinle-Neumann et al., 2001)
showed unexpected behaviour of the elastic stiffnesses c11 and c33 as a
function of temperature. In their paper, they used lattice dynamics
within the particle-in-a-cell method to determine the full elastic
constant tensor. They found that c11 became harder (increased) with
temperature, while c33 softened (decreased) considerably; these
changes were accompanied by an unexpectedly large increase in c12.
Hardening of c11 will increase the P-wave velocity for all directions
perpendicular to the c-axis of the hexagonal crystal, while softening of
c33 will decrease the P-wave velocity parallel to the c-axis (see, e.g.,
Pollard, 1977). The results of Steinle-Neumann et al. (2001) also gave a
bulk modulus, K, that increased with temperature, implying that the
volumetric thermal expansion coefficient, α, must also increase with
increasing pressure (e.g. Bina and Helffrich, 1992). Intuitively one would
not expect either K or c11 to increase with temperature and, indeed, this
behaviour has not been observed in other hcp transition metals at high
L. Vočadlo et al. / Earth and Planetary Science Letters 288 (2009) 534–538
535
Table 1
Isothermal elastic constants (in GPa), adiabatic incompressibility, shear modulus and sound velocities for hcp-Fe as a function of temperature.
T
(K)
c/a
ρ
(kgm‐3)
P
(GPa)
c11
c33
c12
c23
c44
c66
KS
(GPa)
G
(GPa)
VP
(kms‐1)
VS
(kms‐1)
0
2000
3000
4000
5000
5500
1.585
1.59
1.60
1.61
1.62
1.62
13698
13401
13324
13236
13154
13155
295
290
293
298
308
316
2205
1998
1915
1830
1689
1646
2418
2183
2109
1905
1725
1559
1053
1111
1121
1150
1186
1253
950
986
986
1004
990
995
479
375
313
248
216
153
576
444
397
340
252
197
1419
1412
1407
1399
1365
1363
564
444
394
328
266
208
12.59
12.23
12.05
11.78
11.44
11.17
6.42
5.75
5.44
4.98
4.50
3.97
c66 is not independent and is equal to 1/2 (c11 − c12). Uncertainties in cij b 2.5%; uncertainties in VP and VS are ~ 1%. The values for c/a were taken from Gannarelli et al. (2005).
temperatures (Ogi et al., 2004; Antonangeli and Farber, 2005). Given the
importance to the Earth science community of the elastic behaviour of
iron at high temperatures and pressures, in this paper ab initio molecular
dynamics calculations of the elastic behaviour of hcp-Fe are reported as
a function of temperature at ~300 GPa.
2. Methods
The ab initio molecular dynamics method used is based on density
functional theory (DFT) (Hohenberg and Kohn, 1964) within the
generalised gradient approximation (GGA) (Wang and Perdew,
1991). The calculations were performed using the VASP code (Kresse
and Furthmüller, 1996) with the projected-augmented wave (PAW)
(Blöchl, 1994) method to calculate the total energy of the system;
they were run on the UK high-performance computers (HecTor). The
main advantage of this code is that the ab initio energy of the system
can be combined with molecular dynamics methods to simulate the
properties of iron and its alloys at simultaneously high pressure and
high temperature. In ab initio molecular dynamics, the ions in the
system are treated as classical particles and, for each set of atomic
positions, the electronic energy and forces on the ions are calculated
within the DFT approximation which includes the thermal excitations
of the electrons. When performing calculations to determine
the elastic moduli of hcp structures it is convenient to specify the
structure in terms of a C-centred unit cell with orthogonal axes, as the
crystallographic x, y, and z axes then lie parallel to the Cartesian axes
used to define the elastic stiffness tensor (Nye, 1995). The calculations
were performed with 4 irreducible k-points in the Brillouin zone on a
64 atom supercell (4 × 2 × 2) of this C-centred cell of hcp-Fe, at inner-
Fig. 1. Elastic constants of hcp-Fe at ~300 GPa as a function of temperature (the
pressure at each temperature is listed in Table 1). c66 is not independent and is equal to
1/2 (c11 − c12). Those at 5500 K are taken from earlier work (Vočadlo, 2007a,b).
core densities equivalent to ~ 300 GPa. The temperatures were set to
values between 2000 K and 5000 K. Athermal simulations were also
performed to obtain the elastic constants at zero Kelvin.
When calculating elastic constants it is essential that the simulated
box is in hydrostatic equilibrium before any strain matrix is applied.
Of particular importance in this case is the use of the correct equilibrium
c/a ratio for hcp-Fe, which itself changes as a function of temperature;
the values employed here were taken from Gannarelli et al. (2005). The
calculations at high temperature were performed at constant cell
parameter, a (=2.158 Å), varying c/a as described above, so there is a
slight variation in density (decreasing) and pressure (increasing) as T is
raised due to the varying c/a ratio (see Table 1). For the athermal
calculations a value of a = 2.145 Å was used so as to obtain the required
pressure. This approach enabled approximately constant pressures
(300± 10 GPa) to be achieved as a function of temperature (truly
isobaric calculations can only be attained by trial and error); however,
we do not expect these small variations in pressure to significantly affect
the results since, for example, it has previously been shown that
between 300 GPa and 360 GPa the pressure derivatives of c11 and c33
are almost identical, such that the calculated anisotropy should be
invariant (Söderlind et al., 1996). The elastic constants were determined
from the calculated stresses associated with strains applied to
hydrostatically equilibrated supercells. A deformation matrix was
applied to enable distortions, δ, of ±4% to be applied, as shown
below. Simulation times from 5ps to 10ps were used, depending on the
Fig. 2. a) Incompressibility (upper line) and shear modulus (lower line) of hcp-Fe;
b) average seismic wave velocities VP (upper line) and VS (lower line) of hcp-Fe at
~ 300 GPa as a function of temperature (the pressure at each temperature is listed in
Table 1).
536
L. Vočadlo et al. / Earth and Planetary Science Letters 288 (2009) 534–538
time taken for the simulated system to equilibrate. Convergence tests in
k-point sampling and cell-size had been carried out previously (Vočadlo,
2007b) to ensure that the uncertainties in the time-averaged stresses
from the molecular dynamics simulation were negligible.
Representing the hexagonal structure using a C-centred orthogonal unit cell in order to follow the standard convention for the axes of
the elastic constant tensor (Nye, 1995), two deformation matrices are
required in order to obtain the five elastic constants of the hexagonal
Fig. 3. Single-crystal P-wave velocities as a function of propagation direction for hcp-Fe at ~ 300 GPa and temperatures of 2000, 3000, 4000, 5000 and 5500 K. The results at 5500 K are
taken from earlier work (Vočadlo, 2007a,b). Note that the velocities in the [hk0] directions, perpendicular to the c-axis, are gradually becoming more equal to that along the c-axis
until above 5000 K the direction of maximum VP changes from [001] to [hk0]. Figure generated by Unicef Careware (Mainprice, 1990).
L. Vočadlo et al. / Earth and Planetary Science Letters 288 (2009) 534–538
phase. These matrices, used to define the axes of the deformed unit
cell, are as follows:
0
1+δ
@ 0
0
p0ffiffiffi
3
δðc = aÞÞ
10
0 ffiffiffi
1
p
δ 3 A @0
ðc = aÞ
0
1
0
p0ffiffiffi
A
3
0
0 ðc = aÞð1 + δÞ
Using the time-averaged stresses, σij, from the calculations, the
elastic constants were obtained from the standard relations σij =cijklεkl.
Having obtained the elastic constants it is straightforward to determine
the isothermal incompressibility, KT, for a polycrystalline aggregate by
using the Voigt average (see e.g., Simmons and Wang, 1971), which, for
a hexagonal crystal is given by:
KT = ð2 = 9Þðc11 + c12 + 2c23 + ð1 = 2Þc33 Þ
The adiabatic incompressibility was then obtained from:
KS = KT ð1 + αγTÞ
using values of the volumetric thermal expansion coefficient, α = 1 ×
10− 5 K− 1 and the Grüneisen parameter, γ = 1.5 (Vočadlo et al., 2003b).
The Voigt average for the shear modulus of a hexagonal crystal is
given by:
G=
2ðc11 + c33 Þ c12 + 2c23
3ð2c44 + ð1 = 2Þðc11 −c12 ÞÞ
+
−
15
15
15
The average seismic P-wave velocity, VP, and shear-wave velocity, VS,
are then readily obtained from standard relations. Single-crystal elastic
wave velocities were calculated and plotted using Unicef Careware
(Mainprice, 1990) Uncertainties in the elastic constants were estimated
from the calculation of c23, which is obtainable from both deformation
matrices; this led to an uncertainty of ±2.5%, in agreement with the
estimated uncertainty in previous similar calculations (Vočadlo, 2007b;
Vočadlo et al., 2008). Comparison with previous work indicates that the
errors in the axial cij will be smaller than this value, while that in c44 may
be slightly larger, b4% (Vočadlo, 2007b; Vočadlo et al., 2008). However,
uncertainties of this order in the values of the elastic constants propagate
through to uncertainties in the elastic bulk and shear moduli and in VP
and VS of only ~1%, in agreement with previous work (Vočadlo 2007b;
Vočadlo et al., 2008).
3. Results and conclusion
The calculated elastic properties and average seismic velocities
from the present study are listed in Table 1 and shown in Figs. 1
and 2; values from previous work at 5500 K (Vočadlo, 2007b) have
also been included. It can be seen that the longitudinal modulus c11
decreases with temperature, in contrast to previous calculations
(Steinle-Neumann et al., 2001), but in agreement with experimental
observations on other transition metals at ambient pressures (Ogi
et al., 2004). c33 and c44 also decrease with temperature, while both
c12 and c23 slightly increase. The incompressibility, K, decreases only
very slightly with temperature, being almost invariant, while the
shear modulus decreases smoothly, following the requirement that
it must fall to zero at melting. Both VP and VS decrease smoothly with
temperature. Our elastic constants are broadly similar to those found
in a very recent computational study of hcp-Fe to 4000 K by Sha and
Cohen (2009), who also found that both c11 and c33 decrease with
temperature, with c33 N c11 at 0 K.
Fig. 3 shows the calculated single-crystal P-wave anisotropy as a
function of temperature, in the range of 2000–5000 K, from this study
together with that from previous work on hcp-Fe at 5500 K (Vočadlo,
2007b). The sense of the anisotropy is such that VP is fastest along the
c-axis up to 5000 K; however, by 5500 K, the anisotropy has reversed,
with VP becoming faster in the a–b plane. This suggests that, for an
537
inner core dominated by aligned crystals of hcp-Fe, with the c-axis
aligned in the polar direction, the observed seismically isotropic
outer–inner core could result from the hcp-Fe being at a temperature
close to melting, where the axial wave velocities in [001] and [hk0]
directions become similar, while at greater depths in the inner–inner
core, where iron is further from melting, stronger anisotropy will be
achieved, with the fast P-wave direction parallel to the polar axis.
Recent estimates based on the melting curve of pure iron put the
temperature at the inner-core boundary at ~ 6400 K (Alfè, 2009)
which is much higher than the temperature in the present study at
which the sense of the P-wave anisotropy reverses; however, these
results can be reconciled if the addition of alloying elements to hcp-Fe
were to lower the melting temperature without greatly affecting the
elastic properties. No other mechanisms, such as changes in composition
or crystal alignment, are therefore required to account for the observed
change in seismic anisotropy of the Earth's inner core with depth.
References
Alfè, D, 2009. Temperature of the inner core boundary of the Earth: melting of iron at high
pressures from first-principles coexistence simulations. Phys. Rev. B 79, 060101.
Antonangeli, D, Farber, DL, 2005. Elasticity of hcp cobalt at high pressure and
temperature. ESRF Experimental Report HS-2906.
Beghein, C., Trampert, J., 2003. Robust normal mode constraints on inner core
anisotropy from model space search. Science 299, 552–555.
Bina, C.R., Helffrich, G.R., 1992. Calculation of elastic properties from thermodynamic
equation of state principles. Ann. Rev. Earth Planet. Sci. 20, 527–552.
Blöchl, PE, 1994. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979.
Cao, A.M., Romanowicz, B., Takeuchi, N., 2005. An observation of PKJKP: inferences on
inner core shear properties. Science 308, 1453–1455.
Côté, A.S., Vočadlo, L., Brodholt, J., 2008a. The effect of silicon impurities on the phase
diagram of iron and possible implications for the Earth's core structure. J. Phys.
Chem. Solids 69, 2177–2181.
Côté, A.S., Vočadlo, L., Brodholt, J., 2008b. Light elements in the core: effects of
impurities on the phase diagram of iron. Geophys. Res. Lett. 35, L05306.
Côté AS, Vočadlo L, Dobson DP, Alfè D, Brodholt J. in press. 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. Int. doi:10.1016/
j.pepi.2009.07.004.
Creager, K.C., 1992. Anisotropy of the inner core from differential travel times of the
phases PKP and PKIKP. Nature 356, 309–314.
Gannarelli, C.M.S., Alfè, D., Gillan, M.J., 2005. The axial ratio of hcp iron at the conditions
of the Earth's inner core. Phys. Earth Planet. Int. 152, 67–77.
Hohenberg, P., Kohn, W., 1964. Inhomogeneous electron gas. Phys. Rev. 136, B864–871.
Ishii, M., Dziewonski, A.M., 2003. Distinct seismic anisotropy at the centre of the Earth.
Phys. Earth Planet. Int. 140, 203–217.
Kresse, G., Furthmüller, J., 1996. Efficient iterative schemes for ab initio total-energy
calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186.
Mainprice, D, 1990. A FORTRAN program to calculate seismic anisotropy from the
lattice preferred orientation of minerals. Comput. Gesosci. 16, 385 ftp://www.gm.
univ-montp2.fr/mainprice//CareWare_Unicef_Programs/.
Nguyen, J.H., Holmes, N.C., 2004. Melting of iron at the physical conditions of the Earth's
core. Nature 427, 339–342.
Nye, J.F., 1995. Physical Properties of Crystals. Clarendon Press, Oxford.
Ogi, H., et al., 2004. Titanium's high-temperature elastic constants through the hcp-bcc
phase transformation. Acta Materialia 52, 2075–2080.
Pollard, H.F., 1977. Sound Waves in Solids. Methuen, New York.
Sha X and Cohen RE 2009 First-principles thermal equation of state and thermoelasticity of hcp_Fe at high pressures. Submitted to Phys. Rev. B.
Simmons, G., Wang, H., 1971. Single Crystal Elastic Constants and Calculated Aggregate
Properties; a Handbook, 2nd edn. MIT Press, Cambridge Mass.
Söderlind, P., Moriarty, J.A., Wills, J.M., 1996. First principles theory of iron up to
Earth-core pressures: structural, vibrational and elastic properties. Phys. Rev. B
53, 14063–14072.
Song, X.D., Helmburger, D.V., 1993. Anisotropy of Earth's inner core. Geophys. Res. Lett.
20, 2591–2594.
Song, X.D., Helmburger, D.V., 1998. Seismic evidence for an inner core transition zone.
Science 282, 924–927.
Song, X.D., Xu, X.X., 2002. Inner core transition zone and anomalous PKP(DF)
waveforms from polar paths. Geophys. Res. Lett. 29, 1042.
Steinle-Neumann, G., Stixrude, L., Cohen, RE, Gülseren, O, 2001. Elasticity of iron at the
temperature of the Earth's inner core. Nature. 413, 57–60.
Vočadlo, L, 2007a. The Earth's core: iron and iron alloys. In: Price, GD (Ed.), Treatise on
Geophysics Vol 2, Mineral Physics. Elsevier, Amsterdam.
Vočadlo, L., 2007b. Ab initio calculations of the elasticity of iron and iron alloys at inner
core conditions: evidence for a partially molten inner core. Earth Planet. Sci. Lett.
254, 227–232.
Vočadlo, L., Brodholt, J.P., Alfe, D., Gillan, M.J., Price, G.D., 2000. Ab initio free energy
calculations on the polymorphs of iron at core conditions. Phys. Earth Planet. Int.
117, 123–137.
538
L. Vočadlo et al. / Earth and Planetary Science Letters 288 (2009) 534–538
Vočadlo, L., Alfè, D., Gillan, M.J., Wood, I.G., Brodholt, J.P., Price, G.D., 2003a. Possible
thermal and chemical stabilisation of body-centred-cubic iron in the Earth's core.
Nature 424, 536–539.
Vočadlo, L., Alfè, D., Gillan, M.J., Price, G.D., 2003b. The properties of iron under core
conditions from first principles calculations. Phys. Earth Planet. Int. 140, 101–125.
Vočadlo, L., Wood, I.G., Alfè, D., Price, G.D., 2008. Ab initio calculations on the free
energy and high P–T elasticity of face-centred-cubic iron. Earth Planet. Sci. Lett.
268, 444–449.
Wang, Y., Perdew, J., 1991. Correlation hole of the spin-polarized electron gas, with
exact small-wave-vector and high-density scaling. Phys. Rev. B 44, 13298–13307.