The elastic properties of iron and iron alloys at inner core conditions evidence for partial melt

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The elastic properties of iron and iron alloys
at inner core conditions -evidence for partial
melt?

• Lidunka Vocadlo
• Earth Sciences, UCL

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• Taylor Parker

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The Earths Core

• The outer core is liquid, 10 less dense than
  Fe.
• The inner core is solid, 3-4 less dense than Fe.
• IC is crystallising from the OC.
• Light alloying elements may be things like Si, S,
  O, C, H....

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Why do we care?
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Therefore iron is a hot topic!

• To understand our planet we need an accurate
  knowledge of the physical properties and
  composition of the core.
• Therefore we need to understand the properties of
  iron and iron alloys.

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How do we find out?
Experiments using Diamond Anvil Cell
Multi-anvil press Piston cylinder
Laser Heating Synchrotron Radiation Shock
But core P/T remain challenging
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Experimental limits.

• P and T at centre of Earth 360 GPa and 6000
  K
• P and T at core/mantle 135 GPa and
  3000 to 4000 K
• Piston cylinder 4 GPa
• Multi-anvil 30 GPa (higher with
  sintered diamonds)
• Diamond-anvil 200 GPa (temperatures
  uncertain, gradients high)
• Shock guns 200 GPa (temperatures
  extremely uncertain)
• High P/T experiments are hard and can have large
  uncertainties.

Also, it can be dangerous to extrapolate
experimental data to high pressures and
temperatures.
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The dangers of relying on experiments
Unit cell volume of Fe3C as a f(T), obtained by
neutron powder diffraction Tc 483 K (Wood et
al., J. Apple. Cryst., 2004).
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Ferromagnetism also destroyed by P
Spontaneous magnetisation of Fe3C as a f(V) Tc
60 GPa (Vocadlo et al., EPSL, 2002)
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Equation of state parameters for Fe3C
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Fe3C
non-magnetic
B.Wood 1993 WRONG!
seismology
Magnetic EOS
Vocadlo et al., 2002 RIGHT!!
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The alternative to experiments is.

• Computational mineral physics!

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What can simulations predict?

• Volumes, bulk moduli
• Vibrational frequencies (phonon density of
  states)
• Elastic constants (seismic velocities)
• Heat capacities
• Free energies (phase diagrams)
• Defects
• Diffusion
• Viscosities
• Melting..

The fact that we can predict it does not make it
right!
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What does CMP Involve?

• Microscopic scale modelling of bonding in
  minerals and fluids.
• BONDING can be described by
• effective potentials - analytical functions
  approximately describing how energy varies as a
  f(atomic separation or geometry),
• quantum mechanics - calculation of energy as
  f(structure)
• Both can be very CPU intensive.

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Potential Curve
Energy (E)
Short Range Repulsion dE/dx a0
d2E/dx2 K
Atomic separation (x)
Coulombic Attraction
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Calculations on Fe, FeS, FeSi

• Quality of calculations
• EOS, phonon spectrum, DOS
• Stable phase of Fe at core conditions
• via lattice dynamics
• via molecular dynamics
• Effect of light elements
• Elastic constants gtgt VP, VS
• Birchs Law
• Anisotropy
• Melt in the inner core?

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Jurgen Furthmuller
Georg Kresse
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Ab Initio Techniques

• Numerically solving Schrodingers equation
• One approximation the effect on any one electron
  from all the other electrons is wrapped up into a
  term called Exc
• Methods used DFT, GGA, PAW
• We use an NVT ensemble, 64 atom supercell
• Considerable effort spent on convergence tests

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Quality of the simulations

• Ab initio calculations give good descriptions of
  
• EOS of bcc-Fe
• Magnetic moment of bcc-Fe
• The bcc?hcp transition pressure
• The high P density of hcp-Fe
• Phonon dispersion of bcc-Fe

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Vocadlo et al. Faraday Disc. 1997
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Vocadlo et al. PEPI, 2000
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Quality of calculations (again)

• Calculated (black dots) vibrational density of
  states for bcc and hcp iron compared with
  inelastic nuclear resonance X-ray scattering
  (open circles) (Mao et al., Science, 2000)

bcc
hcp
Number of states
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Whats the problem?

• Observed anisotropy and layering in the solid
  inner core
• Origin may determine core evolution and rheology
• Composition determined by stable phase(s)
• Need to know stable phase of iron at core PT
• Need to know effect of light elements
• Need to know elastic properties of candidate
  phases
• See which matches observations

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Anisotropy and layering

• P-waves 3 faster along polar axis
• Layering could mean gt5 anisotropy in central
  region (Ouzounis Creager, 2001)
• Different phase of iron ?
• Different chemistry ?
• Different deformation ?

• Picture????

Song Helmberger 1998 Science 282, 924-927.
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Whats the problem?

• Observed anisotropy and layering in the solid
  inner core
• Origin may determine core evolution and rheology
• Composition determined by stable phase(s)
• Need to know stable phase of iron at core PT
• Need to know effect of light elements
• Need to know elastic properties of candidate
  phases
• See which matches observations

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Different phase? Even pure iron very difficult

• DAC experiments
• Laio et al.
• Alfè et al..
• Shock data
• ?Yoo et al.
• ? Brown McQueen
• ?Ahrens et al.
• ? Nguyen Holmes

Core phase? hcp, bcc, dhcp other?

• Adapted from Nguyen  Holmes 2004 Nature 427,
  339-342

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hcp
fcc
bcc
Also bct, dhcp and orthorhombically distorted hcp
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What is the core phase of iron?

• Need Gibbs free energy to obtain stable phase
• G(P,T) Ftotal(V,T) Ptotal(V,T)V
• P is first derivative of F
• F is a function of vibrational frequencies, ?i
• Use lattice dynamics to obtain ?i

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F of the harmonic solid
Use small displacement method - atoms frozen in
distorted positions gtgt residual
forces. Dispersion curves obtained by
interpolation of ?i calculated from the dynamical
matrix. S, C, E, cij, etc. f(?i) K, G, Vp, Vs
f(cij) e.g.,
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Frequency THz
Direction in crystal
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Stable phase is hcp?

• bcc and bct transform to fcc orthorhombic to
  hcp hcp fcc and dhcp remain mechanically stable
  at core pressures.
• However, fcc and dhcp are less favourable
  energetically therefore hcp is the stable phase
  in the core or is it???

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Ab Initio Molecular Dynamics

• Calculate Gibbs free energy of solid at high P
  and T with ab initio molecular dynamics
• Use technique of thermodynamic integration from
  reference system of known free energy
• Calculate time averaged stresses over gt3 ps

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bcc spontaneously distorts to the omega phase at
low T

• BUT hcp phase more thermodynamically stable at
  low T and high P

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Free energy of bcc and hcp Fe (Vocadlo et al.,
2003)

• V(Å3) T(K) Fbcc(eV) Fhcp(eV)
  ?F(meV)
• 9.0 3500 -10.063
  -10.109 46
• 8.5 3500 -9.738
  -9.796 58
• 7.8 5000 -10.512
  -10.562 50
• 7.2 6000 -10.633
  -10.668 35
• 6.9 6500 -10.545
  -10.582 37
• 6.7 6700 -10.288
  -10.321 33
• 7.2 3000 -7.757
  -7.932 175

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So..DEFINATELY (probably) hcpor is it?

• From the phonon frequencies, we know that bcc is
  unstable at 0 K and high P.
• From the free energies and analysis of atomic
  positions, we know that bcc is mechanically (if
  not thermodynamically) stable at high pressures
  and temperatures.
• BUT difference in free energies is very small
  (only 30-50 meV) and theory and experiment show
  that Si, for example, is happier in bcc-type
  structure light elements could swing it!
• (all in Vocadlo et al., Nature, 2003)

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Whats the problem?

• Observed anisotropy and layering in the solid
  inner core
• Origin may determine core evolution and rheology
• Composition determined by stable phase(s)
• Need to know stable phase of iron at core PT
• Need to know effect of light elements
• Need to know elastic properties of candidate
  phases
• See which matches observations

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So could light elements swing it?
Lin et al., Science, 295, 313-315, 2002
Phases observed in LHDAC experiments with the
starting materials of Fe(7.9 weight  Si) (8).
The slope of the phase transformation from hcp to
bcc  hcp decreases with increasing pressure.
Mixed phases are commonly observed in the heating
process, indicating broad regions of two phase
equilibria between bcc  hcp and bcc  fcc
phases. The coexistence of the bcc  hcp  fcc
phases may be due to the thermal gradient,
temperature fluctuation, or slight misalignment
of the laser beam with respect to the x-ray beam
in the LHDAC.
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Results confirmed by calculations of Côté
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• Côté finds at 330 GPa, with 5 mol. Si, bcc
  stabilised over hcp by 92 meV
• These are cold calculations there will be T
  effect.

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Whats the problem?

• Observed anisotropy and layering in the solid
  inner core
• Origin may determine core evolution and rheology
• Composition determined by stable phase(s)
• Need to know stable phase of iron at core PT
• Need to know effect of light elements
• Need to know elastic properties of candidate
  phases
• See which matches observations

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Elastic properties of Fe and Fe-X

• Zero Kelvin ab initio calculations to get Cij of
  bcc-Fe, hcp-Fe, FeS and FeSi
• Ab initio MD calculations to get Cij at high P
  and T
• Deform supercell via strain matrix
• Resulting stresses as a f(strain)
• Stress/strain relations lead to Cij
• Comparison with experiment
• Use these to verify or otherwise Birchs Law
• Infer something about the inner core..?

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Simplest case - bcc box

• Only C11, C12 and C44
• Deformation matrix
• For hcp need two strain matrices in order to get
  C11, C12 C13, C33 and C44

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C11 ( s11/e11) for hcp-Fe (cold)

• Calculated Cij (two different methods) compare
  well with others

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Ab Initio Molecular Dynamics

• Calculate properties of solid at high P and T
  with ab initio molecular dynamics
• Deform solid to get stresses on box
• Calculate time averaged stresses over gt3 ps

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Time evolution of hcp-Fe stresses for e33
s33
s11 s22
s12 s23 s32 0
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From stresses to sound velocities

• Having got Cij for single crystals, use Voigt
  average to get K and µ of randomly oriented
  polycrystalline aggregate (other averages, e.g.
  Reuss, give effectively identical values)
• Then use standard relations

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Birchs Law

• Birchs law says that there is a linear
  relationship between sound velocities and
  density
• where and Vp, Vs, K (Cij)
• Important because, if true, laboratory
  experiments can be extrapolated to inner core
  densities

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Badros experiments (IXS)
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Badros experiments extrapolated
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Calculations vs. experiment
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Structure of Fe not important? Or T?
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Birchs Law
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Birchs Law
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What does all this tell us?

• Excellent experimental agreement
• Calculated VF(?) almost independent of T
• Difficult to discriminate between an inner core
  consisting of hcp or bcc iron
• Anisotropy..

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bcc-Fe
FeSi
FeS
hcp-Fe
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Core solutions? Or more problems.

• P-wave anisotropy 4-6 for hcp, bcc, FeSi, FeS
• Can account for isotropic upper layer via
  randomly oriented crystals of any or all of the
  above
• Time orients Fe(X?) giving textured anisotropic
  central region
• But shear modulus in all cases gt e.g., PREM,
  ak95, iasp91 by gt10

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• Calculated shear modulus in all
• cases gt seismology by gt10
• Lowest G 203 GPa
• Highest obs 175 GPa
• Vs (PREM) 3.5-3.67 kms-1
• Vs (calcs) gt 4 kms-1
• Vs (BrownMcQueen) 4.04 kms-1

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So we have a problem.

• Calculations wrong? No no no no no.. Surely not!
• Seismology wrong? (or poorly constrained)
• Vs(FeFe-X) non-linear?
• X some other light element e.g. H, C..??
• Current seismological and mineralogical models
  cannot be reconciled

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Is it because of anelasticity..

• The reduction in VS due to shear wave attenuation
  is given by
• For the inner core
• Quality factor, Q 100 (Resovsky et al., 2005)
• freq dep of Q, a 0.2-0.4 (Jackson et al.,
  2000)
• This results in a decrease in the shear velocity
  of only 0.5-1.5, nowhere near the gt8 difference
  between the seismological observations and the
  calculated materials properties.

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A partially molten inner core?

• Estimated amount of melt with Hashin-Shtrikman
  bound for the effective shear modulus of
  two-phase media of volume fraction f (using
  µs200 GPa)
• 8 melt in the inner core
• (Vocadlo 2006 EPSL Accepted)

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Summary

• The bcc phase of iron becomes stable at high T
• The phase of iron in the inner core if it was
  pure iron would be hcp
• The presence of light elements favours bcc
• Anisotropy can be accounted for by bcc with the
  body-diagonal
• aligned N-S or hcp with the c-axis in the
  equatorial plane
• The high shear modulus for all phases is
  incompatible with seismology
• Everything works if there is gt8 melt in the
  inner core

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The End

• Thanks to
• Ian Wood
• David Price
• Frederik Simons
• David Dobson
• John Brodholt