Journey to the CentreJourney to the Centreof the Earthof the Earth

(The structure of iron in the inner core)

Lidunka Vočadlo

Masters et al., 2000

Inside the EarthInside the Earth At the Earth’s surface, we experience relatively mild conditions of P and T.

But at the centre (6400 km down), the pressure reaches over 3.5 million atm and the temperature may exceed 6000oC.

Material from volcanic eruptions has come from only a few 100km down, so there remains well over 6000km to go - 90% of the Earth is effectively inaccessible.

Understanding the EarthUnderstanding the Earth To understand the Earth’s deep interior, we can perform

experiments and computer simulations on candidate minerals.

Calculations underpin experimental data and enable study at very high P/T, beyond the limitations of experimental methods.

In particular, we are working on Fe and Fe alloys under the extreme conditions of the Earth’s core where iron is squeezed to about half its normal volume.

The outer core is liquid, ~10% less dense than Fe.

The inner core is solid, 3-4% less dense than Fe.

IC is crystallising out of the OC.

TICB determined by melting of Fe alloy.

The Earth’s Core

Which Elements?Which Elements? As pointed out by

Poirier (1994), the favoured light element has varied with time, and the strength of the personalities involved!

On this basis, S, O, Si, H and C are the primary candidates.

Ni, K, etc also possible in core.

Why do we care?Why do we care? Heat generated in the Earth’s core drives

the dynamics of the planet, resulting in plate tectonics, Earthquakes & volcanoes.

Why do we care?Why do we care? Heat released from the crystallising inner core

drives convection in the liquid outer core, which in turn generates the Earth’s magnetic field.

Therefore iron is a hot topic!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.

Experiments using: Diamond Anvil Cell Multi-anvil press Piston cylinder Laser Heating Synchrotron Radiation Shock

But core pressures and temperatures remain challenging.

How do we find out?

• 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.

Experimental limits….Experimental limits….

Also, it can be dangerous to extrapolate experimental data to high pressures and temperatures….

Unit cell volume of Fe3C as a function of temperature, obtained by

time-of-flight neutron powder diffraction using the POLARIS diffractometer at the ISIS spallation neutron source. Tc = 483 K.

The dangers of relying on experiments

Spontaneous magnetisation of Fe3C as a function of unit cell volume.

Tc ~ 60 GPa

Ferromagnetism is also destroyed by pressure:

* experimental V not necessarily derived from the fit; ** data fitted to a Vinet equation of state – not BM3

Equation of state parameters for Fe3C:

V0 Å3/atom K0 GPa K’

Magnetic 9.578(37) 173.02(8) 5.79(41)

Non-magnetic 8.968(7) 316.62(2) 4.30(2)

Scott et al., 2001 9.704(9)* 175.4(35) 5.1(3)

Jephcoat, 2000** 162 6.4

The alternative to experiments is….The alternative to experiments is….

Computational mineral physics!

What can simulations predict?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 etc.

The fact that we can predict it does not make it right!

What does CMP Involve?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 function of separation or geometry),

– quantum mechanical calculation of energy as a function of structure.

Both can be very CPU intensive.

Potential CurvePotential Curve

Energy (E)

Short Range Repulsion

dE/dx a0

d2E/dx2 K Atomic separation (x) Coulombic Attraction

What is to come…What is to come… Iron phase diagram Ab initio methods Solid Fe

– Lattice dynamics– Phonon stability– Elastic properties

Liquid and anharmonic solid– Molecular dynamics– Melting– Example – aluminium– hcp-Fe

bcc-Fe instability– Mechanical stablity along Tm

– Instability at lower T

fcc hcp

bcc

What is the structure of Fe at What is the structure of Fe at core P and T?core P and T?

High P/T phase of Fe is controversial. Boehler 1993 observed a possible new phase in

DAC above ~40 GPa and ~1000 K. Andrault et al claim an orthorhombic phase. Saxena et al suggest dhcp. Shen et al fail to find it.

Brown & McQueen shock expts. claim a solid-solid phase change ~200 GPa and ~4000 K.

Nguyen and Holmes don’t find it!

Temperature /K

Pressure /GPa

liquid

hcp

bcc?bct?

fcc

bcc

2000

4000

6000

100 200 300

dhcp?hcp?

Early Shock experiments

Latest Shock experiments

Latest DAC experiments

Latest Ab Initio Calculations

(Nguyen and Holmes, 1998)

(Brown and McQueen, 1986)

(Vo adlo et al., 2000)è

(Saxena et al., 1996)

(Andrault et al., 1997)

Ab Initio TechniquesAb Initio Techniques Numerically solving Schrodinger’s equation.

One major approximation; the effect on any one electron from all the other electrons (a very serious many body problem) is wrapped up into a term called the exchange-correlation.

Methods used:– Density Functional Theory– Generalised Gradient Approximation for Exc

– Ultrasoft, non-norm-conserving pseudo-potentials and/or PAW for the interactions between valence electrons and the tightly bound core electrons

Real and pseudo wavefunction

5d-orbitals in Au.

From the web page of Andrew M. Rappe

All electron Pseudowavefunction

Ab Initio TechniquesAb Initio Techniques

Code used was VASP, running on CSAR T3E and UCL-Bentham.

DFT with PAW and/or PP.

We use an {NVT} ensemble, 64+ atom supercell.

ConsiderableConsiderable effort spent on convergence tests in cell size, k-point sampling, etc. to minimise error in free energies to just a few meV per atom.

Quality of the simulationsQuality of the simulations

Ab initio calculations give good descriptions of:

EOS of bcc-Fe

Magnetic moment of bcc-Fe

The bcchcp transition pressure

The high P density of hcp-Fe

Phonon dispersion of bcc-Fe

Solid IronSolid Iron

Need Gibbs free energy to obtain stable phase in the core

G(P,T) = Ftotal(V,T) + Ptotal(V,T)V

P is first derivative of F

F is a function of vibrational frequencies

Use lattice dynamics to obtain ωi

F of the harmonic solidF of the harmonic solid

Ftotal(V,T) = Fperfect(V,T)+Fvibrational(V,T)

Fperfect from static electronic minimisation calculation including thermal electronic excitations via:

Fperfect (V,T) = U0(V)+ Uel(V,T) -TSel(V,T)

Fvibrational requires ωi and vibrational DOS to put into statistical mechanics equations.

F of the harmonic solidF of the harmonic solid

Use small displacement method - atoms frozen in distorted positions >> residual forces.

Dispersion curves obtained by interpolation of ωi calculated from the dynamical matrix.

S, C, E, cij, etc. = f(ωi)

K, G, Vp, Vs = f(cij)

e.g.,

√√↵

−+=

−

Tk

h

i B

iB

B

i

eTk

hTkF

ωω

1ln2

Results!Results! What is the phase of iron in the core?

Mechanical instability of bcc-Fe at high P

Thermodynamic stability of hcp-Fe at high P

Phonon DOS of bcc and hcp compared with expts

Elastic properties of hcp-Fe

The stable phase The stable phase in the core is in the core is hcphcp

Spin polarised calculations on all phases at core P reveal reduced μ for bcc/bct and zero μ for other phases no magnetism.

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.

(harmonically at this point!)

Calculated vibrational density of states compared with inelastic nuclear resonance X-ray scattering (open circles;

Mao et al., 2000)

Elastic ConstantsElastic Constants Elastic constants determined from dispersion curves.

Γ K or Γ M:

c11 = ρ vL2

½ ( c11 - c12 ) = ρ vT12

c44 = ρ vT22

Γ A:

c33 = ρ vL2

c44 = ρ vT12 = ρ vT2

2

Γ 45o between K and A:

½ (c11 + c22 + 2c44 ) ± [ ¼ (c11 - c33 )2 + ( c13 + c44 )2 ]½ = 2ρ vT12 = 2ρ vT2

2

Calculated (dotted line) thermodynamic properties compared with inelastic nuclear resonance X-ray scattering experiments (open circles; Mao et al., 2000).

Calculated (dotted line) elastic properties compared with inelastic nuclear resonance X-ray scattering experiments (open circles; Mao et al., 2000) and all-electron calculations (dashed line; Steinle-Nemann,1999).

Liquid Fe and the anharmonic solidLiquid Fe and the anharmonic solid

We cannot use lattice dynamics for:

– the high T solid which departs from harmonicity

– the liquid system where there is no long range order

– bcc-Fe, which is only stable at high T

For these we use molecular dynamics

Ab InitioAb Initio Molecular Dynamics Molecular Dynamics

Calculate the energy of liquids and anharmonic systems with ab initio molecular dynamics

Simulate the properties of materials at high temperatures

Calculate the energy of a configuration ab initio with DFT, then move the atoms classically according to Newtons Laws.

Calculation Strategy for TCalculation Strategy for Tmm

Calculate the Gibbs free energy of both the solid and liquid as a function of P and T.

At each chosen P obtain Tm as the point at which GS(P,Tm) = GL(P,Tm).

In fact, we calculate F(V,T) and calculate G(P,T) from G=F+PV where P=-(F/V)T

F of Liquid & Anharmonic SolidF of Liquid & Anharmonic Solid Cannot calculate F(V,T) directly since this is not an

ensemble average - thermodynamic integration.

Start from known F of simple model system and switch the PE function continuously to real system.

PE between states I and II:

( )01

0

01 UUdt

ddtFFF

simT

−=−=Δ ==

λλλ

10)1( UUU λλλ +−=

The reference systemThe reference system

For liquid, start with free energy of a simple IP, for anharmonic solid start from a combination of the IP and harmonic solid.

Only repulsive term; bonding term depends strongly on V and T, but not on atomic positions.

For Fe, Γ=1.77 eVÅ, α=5.86

α

√↵

Γ

=r

rU 4)(

Calculating Tm at the ICB

Using first principles molecular dynamics and thermodynamic integration, we can calculate the Gibbs free energy of both the solid and liquid systems as f(P,T); melting occurs when GS=GL.

(Example for aluminium)

Results!Results!

Melting:Example – aluminium

hcp-Fe

bcc-Fe:Mechanical stablity along Tm

Instability at lower T

TTmm of of hcphcp-Fe-Fe

At the P conditions existing at the inner core/outer core boundary, pure iron melts at 6400 C.

However, the presence of alloying elements such as S, Si and O will probably reduce this temperature to 5500 C.

This is very hot, comparable with temperatures on the surface of the Sun.

(Dario Alfè et al.)

So iron in the core is all sewn up So iron in the core is all sewn up ….or is it?!!….or is it?!!

Many groups favour bcc-Fe as the stable phase - there may be a break in slope of Tm at high P.

It is possible that bcc could be stabilised by T.

Therefore we have performed free energy calculations on bcc-Fe as a f(V,T).

But this is difficult - how do you get harmonic F?

Answer: use TI between mechanically stable reference system and high T system.

Temperature /K

Pressure /GPa

liquid

hcp

bcc?bct?

fcc

bcc

2000

4000

6000

100 200 300

dhcp?hcp?

Early Shock experiments

Latest Shock experiments

Latest DAC experiments

Latest Ab Initio Calculations

(Nguyen and Holmes, 1998)

(Brown and McQueen, 1986)

(Vo adlo et al., 2000)è

(Saxena et al., 1996)

(Andrault et al., 1997)

Relative stability along TRelative stability along Tmm(P)(P)

V T Fbcc (eV) Fhcp (eV) ΔF (meV)

9.0 Å 3500 K -10.063 -10.109 45

8.5 Å 3500 K -9.738 -9.796 58

7.8 Å 5000 K -10.512 -10.562 50

7.2 Å 6000 K -10.633 -10.668 35

6.9 Å 6500 K -10.545 -10.582 37

6.7 Å 6700 K -10.228 -10.321 38

So..DEFINATELY So..DEFINATELY (probably)(probably) hcphcp……but…when does but…when does bccbcc become stable? become stable?

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.

What happens to the system as we lower T?– Stresses on the box– Atomic deviation from bcc structure

SummarySummary Calculations and experiments are both very useful tools for

probing the Earth’s deep interior.

Calculations can be as accurate as experiments, although care needs to be taken to ensure a meaningful comparison is made.

The methodology for determining melting curves works well for aluminium; there is still some controversy surrounding the iron melting curve.

The stable phase in the core is hcp-Fe and not the high temperature bcc-Fe.

I have not failed 10,000 times,I have not failed 10,000 times,

I have successfully found 10,000 ways I have successfully found 10,000 ways that will not work.that will not work.

Thomas A. EdisonThomas A. Edison