GEOPHYSICAL RESEARCH LETTERS, VOL. 26, NO. 9, PAGES 1231-1234,MAY 1, 1999
The structure of iron under the conditions of the Earth's
inner core
LidunkaVo•adlo,JohnBrodtiolt,
DarioAWe,GeoffreyD. Price
ResearchSchoolof Geologicaland GeophysicalSciences,BirkbeckCollege and University College London, UK
Michael J. Gillan
Departmentof Physicsand Astronomy,UniversityCollegeLondon,Gower Street,LondonWC1E 6BT, UK
Abstract. The inferred density of the solid inner core
indicatesthat it is predominantlymade of iron. In order to
lattice dynamicsto obtain free energiesof all the candidate
structuresproposedtbr core conditions.
interpretthe observedseismicanisotropyandunderstand
the
high pressureand temperaturebehaviourof the core, it is
CalculationMethodology
essentialto establishthe crystal structureof iron under core
conditions.On the basisof extrapolatedexperimentaldata,
a numberof candidatestructuresfor the high P/T iron phase
have been proposed, namely, body-centredcubic (bcc),
body-centredtetragonal(bct), hexagonalclose-packed(hcp),
double-hexagonal
close-packed (dhcp)
and an
orthorhombicallydistortedhcp polymorph (Matsui, 1993;
Stixrude and Cohen, 1995; Boehler, 1993; Saxena et al.,
The calculationswe shall presentare basedon densityfunctionaltheory (DFT) which is a widely used general
schemefor using electronic structurecalculationsto obtain
the total ground-stateenergy of a system as a function of
atomic positions. More specifically, we use the
pseudopotential
approachin which only valenceelectronsare
treated explicitly, with the interactions between these
electronsand the ionic coresbeingdescribedby an ab initio
pseudopotential.
The pseudopotential
approachis an alterna-
1996; Andfault et al., 1997). Here we presentthe resultsof
the first fully ab initio free energy calculationsfor all of
tive to all-electron methods; it can be as accurate as these
thesepolymorphsof iron at corepressuresandtemperatures.
methods,
while beingcomputationallymuchlessdemanding
Our resultsshow that hcp-Fe is the most stablepolymorph
of iron under the conditions of the Earth's inner core.
(Vo•adlo et al., 1997).
Our aim is to use the D FT pseudopotentialmethod to
study the relative thermodynamicstabilityof the proposed
crystal structures at core conditions. We treat electronic
exchange-correlation
energy using the generalisedgradient
approximation
(GGA),
coupled with non-norm-conserving
Since the densityof the inner core is slightly lower than
ultrasoft
Vanderbilt
pseudopotentials
(Vanderbilt, 1990) as
thatinferredfor pure iron, lighteralloyingelementsmustbe
present.However,the high P/T phasediagramof pure iron implementedin the VASP code (Vienna ab initio simulation
itselfpresentsmajor problemsand it is essentialto resolve package, Kresse and Furthmtiller, 1996). Details of the
can be found in Kresse
thesebeforetrying to understandiron alloys. Direct experi- constructionof the pseudopotentials
mentationon Fe at core conditionsis extremelydifficult, and and Hafner (1994) and specific information about Fe
there are major conflictsbetweenthe resultsof different pseudopotentialsfrom Moroni et al. (1997). The results
groups.In particular,the existenceof a solid-solidphase presentedhere were obtainedusing a pseudopotentialwith
boundarybetweenhcp-Feand anotherundefinedpolymorph the 3p electronstreatedexplicitly as valenceelectrons,i.e.,
above ~200GPa and ~4000K has been suggestedin order to with a [Ne]3s2 core, which we have previouslyfound
reconciledata from staticand shockexperiments(Anderson (Vo•adlo et al., 1997; de Wijs et al., 1998) to be essential
for the accuratedescriptionof the propertiesof solid and
and Duba, 1997); however, the results from the latest very
liquid
iron at the high pressuresof the Earth's core. The
high P/T shock experiments(Nguyen and Holmes, 1998)
DFT calculationsis
indicatethat iron doesnot undergoany very high P/T phase excellentquality of our pseudopotential
demonstrated
by
the
accurate
results
we
have found for the
change, as previously inferred by Brown and McQueen
density
and
incompressibility
of
bcc-Fe,
the
bcc-•hcp phase
(1986). Suchuncertainties
can be resolvedusingab initio
transition
pressure
(~
10
GPa
compared
to
an
experimental
calculations,
whichprovidean accuratemeansof calculating
value of 10-15 GPa), the magneticmoment of bcc-Fe (2.25
the thermoelasticpropertiesof materials at high P and T.
Thermodynamiccalculationson hcp-Fe andfcc-Fe at high idatom comparedto the experimentalvalueof 2.12 tdatom),
P/T have already been reported (Stixrude et al. 1997; and the elastic constantsof bcc-Fe (a third order BirchWassermanet al., 1996) in which ab initio calculationswere Murnaghanfit givesa value tbr the incompressibility,K, of
184 GPa comparedto the experimentalvalue of 173 GPa).
used to parameterisea tight-bindingmodel; the thermal
Theseresultsare alsoin excellentagreementwith thosefrom
propertiesof this model were then obtainedusing the
particle-in-a-cellmethod.The calculationspresentedhere, the best all-electron first-principles methods currently
available (Stixrude et al., 1994; S6derlind et al., 1996).
usingtheCray T3E at Edinburgh,are the first in whichfully
The most stable crystal structureat a given P and
ab initiomethodsare usedin conjunctionwith quasiharmonic
T is the one with the lowestGibbsfree energy,G, given by:
Introduction
Copyright1999 by theAmericanGeophysical
Union.
G
=
F+PV
(1)
Papernumber1999GL900214.
0094-8276/99/1999GL900214505.00
where F is the Helmholtz free energyand V is the volume.
1231
1232
VOI•ADLO ET AL.' THE STRUCTURE OF IRON AT INNER CORE CONDITIONS
frequencies,
andhencefree energy,is small( < 0.3 %); more
importantly,this is a systematiceffect with supercellsize,
andthereforeits influenceon the differencein free energy
betweenpolymorphswill be negligible.
Our strategyis to start by calculatingF as a functionof V
andT, andto obtainP from the volumedependence
of F by
usingthe relation:
(2)
we may then constructG from F, P and V.
The Helmholtz free energy of a vibrating crystal can be
expressedas:
Results
Spinpolarizedsimulations
wereinitiallyperformedon all
the candidatephasesat core pressures(whichrangefrom
~325-360 GPa). These revealed, in agreement with
S6derlindet a/.(1996), that under these conditionsbcc and
bct-Fehadonly a smallmagneticmoment(~•A•B/atom)and
Ftotd(V,T
) = F•,erf(V,T
) +Fvib(V,T
)
(3) all otherphaseshad zero magneticmoment. We foundthat
at thesepressures
the bcc, bct and the recentlysuggested
polymorphof iron (Andfaultet al., 1997) are
whereF•,er
f andFvibare the freeenergyof therigidperfect orthorhombic
lattice and the free energy due to atomic vibrations.At zero mechanically
unstable.
The bccandbctphases
continuously
temperature,
Fp•rfis just the total energyof the perfect transformto the fcc phase (confirmingthe findingsof
lattice, but at the high T consideredhere, it must also StixrudeandCohen,1995), while the orthorhombic
phase
containthe contributionto free energy from the electronic spontaneously
transformsto the hcp phasewhen allowed to
excitations.
In DFT, thiselectronic
contribution
to F•,er
f is relaxto a stateof isotropicstress.In contrast,hcp,dhcpand
calculatedby minimisingthe Mermin functionalin the Kolm- fcc Fe remainmechanically
stable'
at corepressures,
andwe
Sham formalism (see, e.g., de WJjs et al, 1998).
werethereforeableto calculatetheirphononfrequencies
and
To calculatethe vibrationalterm, includingthezero point free energies.
energy, we usethe harmonicapproximation,in whichF,• is
Althoughno experimental
dataexistfor phononsat high
given by a standard formula involving a sum over the P, thequalityof our calculations
canbe gaugedby comparBrillournzone of the free energy of eachphononmode. The ing thecalculated
phonondispersion
for bcc-Fe(doneusing
phononfrequencieswere obtainedusing the frozen-phonon fully spinpolarisedcalculations),
at ambientpressure,where
technique: starting from the perfect lattice, an atom is experimentaldata do exist. Figure 1 showsthe phonon
displacedfrom equilibrium, and the resultingforceson all dispersioncurve for magneticbcc-Feat ambientconditions
atoms are calculated.For a small displacement,the forces comparedwith inelasticneutronscattering
experiments(see
are proportionalto the displacement,and the coefficientsof Gat et al., 1993); the calculatedfrequenciesare in excellent
proportionality constitute the force-constantmatrix. The agreement with the experimental values which further
calculationsof the phonon frequencies from the force- confirmsthe quality of the pseudopotential
used in this
constantmatrix is standard(e.g., Born and Huang, 1954). study.
The ab initit calculation of the force-constant matrix was
The temperature
dependence
of theHelmholtzfreeenergy
performedover the three densities13.64, 13.44 and 13.00 andthethermal
pressure
(i.e. Ptotd(V,T)- P•,•rfm(V,T=0))
gcm'3 usingsupercelIs
of 16 or 32 atoms(our currentlimit are shownin Figures2 and 3. The thermalpressureat core
with available resources).The electronick-point sampling conditionshas been estimatedto be 58 GPa (Anderson,
(between6 and 48 k-points in the irreducibleBrillouin zone 1995) and >50 GPa (Stixrude et al., 1997); this is in
dependingon symmetry)andplane-wavecutoffswere chosen excellentagreementwith our calculatedthermalpressurefor
to give a convergencein the energy difference between thehcpanddhcpstructures
(58 GPa and49 GPa respectively
polymorphs of better than 0.004 eV/atom. As will be at 6000K); however, the calculatedthermal pressure is
discussedlater, such highly converged calculations are considerably
highertbrJ•c-Fe, and we find that thisthermoimportant since the free energy dift•rences between the dynamicallydestabilisesthis phaseat core conditionswith
polymorphsbecomesvery small (---0.05 to 0.01 eV/atom) at respectto dhcp and hcp-Fe.
core conditions.
10.0
It is possiblethat there will be a thermalcontributionto
the electrondensityand resultinginteratomicforcesdue to
electronic excitations which will affect the phonon frequencies.However, althoughit is relatively straightforward
tocalculate
Fperf
asa function
of electronic
temperature,
it is
impracticable to calculate phonon frequencies of large
supercelIsas a derailedfunctionof electronictemperature.
Therefore, the supercellcalculationswere all performedwith
an electronictemperatureof 6000K, whichwe considerto be
representativeof the possiblerange of temperaturesto be
8.0
o
o
•:6.0
•
øø
4.0
found in the inner core.
In our calculations,the phononfrequencieswere obtained
from diagonalisationof the dynamicalmatrix which is the
Fourier transform of the Ibrce constant matrix. Since this has
been calculated to first nearest neighhours, the phonon
frequenciesare exact only at the gamma point and zone
boundaries,the interpolationbeing pertbrmedthroughouta
19x19x19grid. The eft•ct of supercellvolume (e.g. 16 or 32
atoms for an hcp supercell) on the calculatedvibrational
2.0
0.0
F
L
F
x
Figure 1. Calculatedphonondispersioncurve for magnetic
bcc-Fe under ambient conditionscompared with inelastic
neutronscatteringexperiments(seeGat et al., 1993) (open
circles).
VOCADLO ET AL.' THE STRUCTURE OF IRON AT INNER CORE CONDITIONS
Ahcp
edhcp
7.500
•
-8.o
•
-10.0
1233
'
2.500
o
-12.0
-14.0
0.0
'
•
2000.0
'
C•
40 0.0
'
•
6000.0
'
-5.000
8000.0
0.000
4000.000
Temperature /K
8000.000
12000.000
Temperature/K
Figure 2. Calculated temperature dependenceof the
Helmholtz free energy as a function of temperature at
V=7.134•3(lowercurve)
andV=6.8•• (upper
curve).
Figure4. Calculated
Gibbsfreeenergyforhcp-FeanddhcpFe asa furlctionof temperature
at P = 325GPa(lowercurves)
and P=360 GPa (upper curves); hcp-Fe is more stable
throughoutthe whole P/T spaceshown.
Thus far we have obtainedG(V,T), but in order to make
any predictionsaboutthe relativestabilityof the dhcpand
(i) The results show that the free energy difference
hcp phasesat core conditions,we require G(P,T). By between the hcp and dhcp phases is small, as we would
analysingthe total pressureas a functionof temperature expect for two very similar close packed isochemical
obtainedfrom our calculationsfor thesetwo phases,we are structures.We calculate AG ~0.05eV/atom at P = 325GPa,
able to ascertain the temperature for each of the three
volumes considered at which P =325GPa
and P = 360GPa -
and ranges from AG~0.05eV/atom at P=360GPa and
T=4000K
to AG~0.01eV/atom
at P=360GPa
and 8000K.
pressuresspanningthe inner core range of pressures- and
thereforedeterminethe Gibbsfree energyat thoseT and P.
Figure 4 showsthe Gibbsfree energyas a functionof T for
Althoughthesedifferencesare small, they are significantly
that (•}G/•}T)T=o=0
and (•}G/•}T)T•o=(Stot•t)c•tca,t•a.
Over the
would be desirableto samplethe phononspectrummore
effectivelyby performingcalculations
on significantlylarger
larger than the 0.004eV/atom precision of our calculations
limitedby k-pointsamplingandplane-wavecut off discussed
thesetwo phasesat 325 GPa and 360 GPa; thesewere fitted
earlier. In addition, although we expect the effect of
toanexponential
function
satisfying
theentropic
requirement supercellsize tO be systematicbetweenthe two phases,it
whole P-T spaceinvestigatedthe hcp phasehas the lower
Gibbsfree energyand is therefore.predictedto be the stable
structure at core conditions.
supercelIs.
(ii) As mentionedearlier, the phononfrequencieswere
calculatedusinga fixed electronictemperatureof 6000K. In
order to determinethe validity of this constraint,we per-
Discussion
formed indicative
calculations
on the effect of electronic
The validity of this importamresult dependsupon three
temperatureon phononfrequencies.We foundthatthe effect
key issueswhich we discussbelow: (i) the convergence
of electronictemperatureon free energy in the expected
criteria used in the calculations, (ii) the use of a fixed
range of core temperatures(4000 to 8000 K) is generally
electronictemperature,and (iii) the affect of anharmonicity
significantlysmallerthanthe free energydifferencesbetween
at thesehigh P/T conditions.
150.0
,
,
'
,
the phases(a maximum of 16 % of thesedifferencesbetween
4000K and 7000K), and only becomes comparable in
magnitudeto the calculatedfree energydifferencebetween
thedhcpandhcppolymorphsasthe temperature
approaches
'
ßdhcp
ß
ß hcp
•. 100.0
.,
ßfcc
-"
.
.-='
(D
-'
u)
'
8000K
at P = 360 G Pa.
(iii) As indicatedabove, thesecalculationsdo not take into
accountpossibleartharmoniceffects.Over muchof the (P,T)
range investigated,we believe that we are justified in
assumingthat the motion of the atoms can be treated as
harmonicvibrations,and that all thermodynamicproperties
can reasonablybe calculatedfrom the energy of the static
perfect crystal and the harmonic lattice vibrational frequencies(Matsui et al., 1994). This assumptionis validated
by our calculatedthermal expansioncoefficientfor hcp-Fe at
~5100 K and 325 GPa of ~0.96 x 10'5 K 'l, in excellent
agreementwith the valuesinferred from shockexperiments
(Duffy and Ahrens, 1993) and thermoelasticitycalculations
•50.0 ,-'
.-"
,.",•'",•'••
0.0 '•'=
0.0
I
2000.0
,
I
4000.0
=
I
6000.0
=
8000.0
Temperature /K
Figure 3. Calculated thermal pressure as a function of
temperature;fcc-Fe is thermodynamicallydestabilisedwith
respectto the other two phasesdue to its relatively high
thermal pressure.
(Wasserman
et al., 1996)of ~0.9 x 10-5K'•. At thehighest
temperatures,however, artharmoniceffects may become
significant,but given the structuralsimilarity of hcp and
dhcpFe, we wouldexpectanharmoniceffectson bothphases
1234
VOI•.ADLO ET AL.' THE STRUCTURE OF IRON AT INNER CORE CONDITIONS
to be comparableand we suggestthat they would have a
small, or even negligible,effect on the relative stabilityof
these two polymorphs.
Finally, it has been suggested(Matsui and Anderson,
1997) that althoughnot mechanicallystableat zero Kelvin,
the bccphasecouldbe entropicallystabilisedat high T in the
inner core, sincebcc retainssomemagneticmomentat core
pressures (S6derlind et al., 1996), and it is theretbre
conceivablethat magnetic entropy effects may be nonnegligible. However, Moroni et al. (1996) have estimated
that the maximumcontributionof magneticentropyat core
conditionsis of the order of 0.3R per mole at 6000K. The
maximumlikely contribution
to vibrationalentropyis
comparableto the melting entropy(Stixrudeet al., 1994),
whichtendsto Rln2 for close-packed
metalsat high pressures (see Poirier, 1991). Therefore, the total maximum
entropiccontributionis ~R. We calculatethat the metastable
initiototalenergycalculations
usingplanewavebasisset,Phys.
Rev. B, 54, 11169-11186, 1996.
Kresse, G. and Hafner, J. Norm conservingand ultrasoft
pseudopotentials
for first row elementsand transitionmetals,J.
Phys.' Condens.Matter, 6, 8245-8257, 1994.
Matsui,M. Moleculardynamicsstudyof iron at Earth'sinnercore
conditions,
AIP Conference
Proceedings,
AmericanInstituteof
Physics,887-891, 1993.
Matsui,M. andAnderson,O.L. The casefor a bodycentredcubic
phase(o•')for iron at innercoreconditions,
Phys.EarthPlanet.
Int., 103, 55-62, 1997.
Matsui, M., Price, G.P. and Patel, A. Comparisonbetweenthe
latticedynamicsand moleculardynamicsmethods:calculation
resultsfor MgSiO3perovskite,Geophys.Res. Lett., 21, 16591662, 1994.
Moroni,E.G. andJarlborg,T. Bee andhopphasecompetition
in
Fe, Europhys.Lett., 33, 223-228, 1996.
Moroni, E.G.,
Kresse, G., Furthmfiller, J. and Hafner, J.
Pseudopotentials
appliedto magnet.it
Fe, Co andNi' fromatoms
to solids, Phys.Rev. B., 56, 15629-15646,1997.
enthalpydifferencebetweenbcc-Feandhcp-Feat a density Nguyen, J.H. and Holmes, N.C. Iron soundvelocitiesin shock
waveexperiments
up to 400 GPa. A GU Abstracts,79, T21D-06,
of 13 g/cc is ~l.leV. For thislatticeenergydifferenceto be
over come, the entropydifference.between
the phaseswould
needto exceed~2R, and it is thereforeimprobablethat the
magneticandvibrationalentropycouldovercomethisat core
1998.
Poirier, J.P. Introductionto the physicsof the Earth's interior,
CUP, 1991.
Stixrude,L. and Cohen, R. E. Constraintson the crystalline
conditions.
•uucLu• m the innercore:mechanical
instabilityof BCC iron at
In conclusion,theretbre, our calculations,combinedwith
highpressure,Geophys.Res. Lett., 22, 125-128, 1995.
otherevidence,indicatethat hcp is the thermodynamically Saxena,S. K., DubrovinskyL. S. andH/iggkvistP. X-ray evidence
for the new phaseof 13-ironat high temperature
and high
most stablephaseof Fe at the conditionsbelieved to exist in
pressure,Geophys.Res. Lett., 23, 2441-2444, 1996.
the Earth's inner core. Our work has established the essential
S6delind,P:, Moriarty, J. A. and Wills, J. M. First principles
basisfor a further studywhich is now neededto establishthe
theoryof ironup to Earthcorepressures:
structural,vibrational
equationsof stateand thermodynamic
stabilityof alloysof
and elasticproperties,Phys.Rev. B, 53, 14,063-14,072, 1996.
h, Iron at highpressure:
Fe to establishwhich phase and which light alloying Stixrude,L., R. E. Cohen,andD. J. Sing,
linearised
augmented
planewavecomputations
in thegeneralised
elementsare indeed presentin the Earth's core.
gradientapproximation,
Phys.Rev. B, 50, 6442-6445, 1994.
Stixrude,L., Wasserman,E. and Cohen, R.E. Compositionand
Acknowledgements.The authorsacknowledge
receipt of a
temperatureof the Earth's inner core, J. Geophys.Res., 102,
researchgrantfrom the NERC andwe alsothankI.G.Wood andG.
24,729-24,739. 1997.
Kresse for useful discussion.
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