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## Mineral Physics of the Core

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### Mineral Physics of the Core

### Influence of temperature

### Origins

### Nature of Theory in Geo-Context

### Application of Theory

### Size of System

### Approaches to Large Systems

### Illustration: Solid Solutions

### Illustration: Solid Solutions

### Illustration: Influence of Temperature

### Magnetic Collapse

Lars Stixrude

University of Michigan

Gerd Steinle-Neumann, Universität Bayreuth

Ron Cohen, Carnegie Institution of Washington

David Singh, Naval Research Labs

Henry Karkauer, William and Mary

Challenges for mineral physics

Origin of core structure

Composition of the core

Mineralogy of the inner core

Temperature at Earth’s center

Song and Richards, Nature (1996)

Mantle

Oxides &

Silicates

Outer Core

Solid

Iron

Alloy

Liquid

Solid

Inner Core

Depth 0 660 2890 5150 6371 km

Pressure 0 24 136 329 363 GPa

Temperature 300 1800 3000 5500 6000 K

a

Crystal structure of iron at inner core conditionsThree known phases

- Body-centered cubic (bcc)
- Observed to 10 GPa
- Face-centered cubic (fcc)
- Observed to ~60-100 GPa
- Hexagonal close-packed (hcp)
- Only phase observed above 100 GPa

But: no experimental determinations of structure at inner core conditions (yet)

Theory of Planetary Materials

- Simple Theories Fail
- Thomas-Fermi-Dirac
- Pressure insufficient
- Terrestrial pressure
- ~ Bond deformation pressure
- eV/Å3 = 160 GPa
- ~ Bulk modulus
- Atomistic models will fail
- What to do?
- Experiment (Birch, 1952)
- First principles theory (Bukowinski, 1977)

TheoryMany different kinds!

- Quantum methods
- Electronic structure computed
- Density functional theory
- First principles, ab initio
- Classical methods
- QM is absorbed into an approximate model of interatomic interactions
- Interatomic force models/fields
- Pair potentials
- Hybrids

Crystal Structure of Inner Core

Some soft-sphere interatomic potentials predict bcc stable at high temperatures

Could the inner core be made of bcc?

Ross et al., JGR, 1990

Belonoshko et al., Nature, 2003

Origin of mechanical instability

BCC phase is unique in having a large peak in the electronic density of states at the fermi level

Two stabilization mechanisms:

Low P: Magnetism

High P: Distortion

Stixrude et al., US-Japan volume, 1998

Types of Instability

- Thermodynamic instability
- At least one other phase with lower Gibbs free energy.
- Phase may still exist in a metastable state (kinetics).
- Phase occupies local minimum on energy surface.
- Examples: Quenchable phases, Metamorphic rocks
- Mechanical instability
- Phase spontaneously decays.
- Occupies local maximum or saddle point on energy surface.
- Phase is not observable.
- Examples: Many displacive phase transformations

BCC IRON

Influence of temperature?

Vocadlo et al, Nature (2003)

Thermal restabilization of bcc? No…

In the canonical ensemble (NVT fixed) a condition of hydrostatic stress is a necessary but not sufficient condition for mechanical stability

The stress tensor of bcc iron at static conditions (where all agree on mechanical instability) is hydrostatic!

The fact that the stress tensor of bcc iron in a canonical md simulation is hydrostatic is therefore not a demonstration of mechanical stability

Previous arguments that the instability is much too large to be overcome by temperature are not contradicted.

Test: compute stress tensor and/or free energy in a strained configuration (as was done in the static calculations).

Chemical stabilization of the bcc structure?

Lin et al. (2002) find that addition of Si expands bcc stability field

Maximum pressure < 1Mbar

Vocadlo et al. (2003) find that substitution of Si, S is more favorable in bcc phase

Which substitution mechanism?

a

Iron at inner core conditions- Hexagonal close-packed (hcp) structure
- Two repeat distances
- a - close-packed planes
- c - spacing between planes
- Ideal Ratio
- c/a=√8/3≈1.633
- Elastic wave speed
- Compare with inner core
- Anisotropy
- Temperature

HCP iron: elastic anisotropy

LAPW: Stixrude & Cohen, Science, 1995; Steinle-Neumann et al., PRB, 1999

XRD: Mao et al., Nature, 1998

Small anisotropy, assume C12≈C13

Elasticity by x-ray diffraction

State of stress in the diamond anvil cell is non-hydrostatic

D-spacing may depend on orientation

Amount of variation depends on several factors including the elastic constants

Elastic anisotropy of hcp transition metals

Less than 50 % for all hcp transition metals stable at ambient conditions

Iron

Theory: 2 %

Original xrd: 250-350 %

Latest xrd: 28-64 %

Elastic anisotropy HCP iron

Stixrude & Cohen, 1995

Steinle-Neumann et al., Nature, 2001

Anisotropy of inner core

- Compute single crystal elasticity
- Assume polycrystalline texture
- Compute travel times of seismic waves
- Compare with seismological observation
- Implies dynamical process capable of texturing

Remaining issues

Confirmation of high-T elastic constant prediction

Origin of texture

Inner core is not so simple!

Glatzmaier & Roberts, 1996

Temperature of the inner core

5600 K

- Compare elastic moduli of
- hcp iron (theory)
- inner core (seismology)
- Estimate consistent with those based on
- Iron melting curve
- Mantle temperatures, adiabatic outer core, …
- Implies relatively large component of basal heating driving mantle convection

bulk modulus

shear modulus

Core chemistry

25 elements lighter than iron

Hypothesis testing: two extreme models of major element core composition

identical to that of the meteorites from which earth formed

Set by equlibration with the mantle after core formation

Can we eliminate either of these on the basis of property matching alone?

Lee et al., GRL, 2004

Conclusions

Inner core is likely to be made of hcp iron. Caveat: light element stabilization of a different phase cannot be ruled out at present.

Iron is elastically anisotropic at inner core conditions. Magnitude is at least as large as that seen seismologically. Sense appears to depend on temperature.

Estimates of inner core temperature based on elasticity and melting are converging to a value near 5600 K.

Iron melting

Theory. Various levels of quality

Electronic. Quantum, First principles, ab initio, self-consistent (Alfe)

Atomistic. Classical potetential, Pair potential, interatomic forces, embedded atom potential (Belonoshko)

Hybrid. “Optimal potential” Laio et al.

Experiment

Static compression. How to detect melt?

Dynamic compression. How to determine temperature?

Iron Melting Summary

High quality theory and most recent experiment in perfect agreement.

Melting curve consistent with that found by Brown and McQueen (1986)

No solid-solid phase transformation along Hugoniot

Potassium

Potassium shows a fundamental change in its electronic structure at high pressure, from that of an alkali metal to that of a transition metal.

4s electrons are more strongly influenced by compression than the initially unoccupied 3d states, which are increasingly populated at high pressure

Large decrease in ionic radius

Change in chemical affinity from lithophile to siderophile?

Bukowinski (1976) GRL 3, 491

Nuclei

&

Electrons

Pressure in Earth is large enough to fundamentally alter the electronic structure…

but low enough that complete ionization or alteration of nuclear structure do not occur.

Both the traditional ionic model and jellium models are limiting

Quantum

objects

Point

charges

"The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the application of these laws leads to equations much too complicated to be soluble." - Dirac (1929) Proc. Roy. Soc (London) 123, 714

The Schrödinger Equation

Exactly soluble only for H atom

Insolubility particularly severe for real, i.e. natural, i.e. geological materials

Basic difference in approach between earth science and physics/chemistry

Wavefunction

Energy

Kinetic

Potential

NOT number of atoms in a sample O(1023)

- Theory deals with systems that are infinite and periodic
- Size means size of periodically repeating unit, i.e. unit cell.

One challenge of natural systems is encapsulated by the concept of size.

Aspects of natural systems that lead to large size

Structural complexity

Impurities

Defects

Solid solution

Temperature

Density functional theory

Exact in principle

Must approximate many-body interactions (LDA, GGA)

Charge density is a scalar function of position (and observable).

Pseudopotential theory: Replace “frozen” core and nucleus with “softer” potential

Structural relaxation and dynamics: Hellman-Feynman theorem allows computation of forces and stresses

Coexistence of long-range disorder with possible short-range order requires special techniques.

Interpolate among a finite number of first principles calculations with a model of the effective interactions among solution atoms.

Evaluate thermodynamic quantities via Monte Carlo simulations over a convergently large domain

Alfé, Gillan, Price (2002) EPSL 195, 91

Liquid and hcp Fe:O,Si,S

What is the light element in the core?

Compute chemical potentials of light elements in liquid and solid iron.

Predict equilibrium partitioning between liquid and solid phases and the density contrast.

Compare with seismological density jump at inner core boundary.

Precise description demands analysis of each snapshot of dynamical system.

Vibrations increase the size of the system by breaking the symmetry of snapshots.

Molecular Dynamics

Evaluate forces acting on nuclei

Integrate Newton’s 2nd law

Lattice Dynamics

Expand energy to second order in displacements

Find normal modes of vibration

How to detect melt in static compression?

X-ray diffraction. Re-crystallization. Absence of evidence

Cohen, Mazin, Isaak, Science, (1997)

Steinle-Neumann, Stixrude, Cohen, Phys Rev B (1999)

Challenges for mineral physics

Relate structure to process

Thermal evolution

Temperature in the inner core

Chemical evolution

Composition of the core

Magnetic field generation

Mineralogy of the core

What to do?

Experiment (Birch, 1952)

Because simple theories fail, in situ experimental measurement at high pressure is essential.

Intelligent, semi-empirical methods of interpolation and extrapolation of limited data are also critical, e.g. finite strain theory.

First principles theory (Bukowinski, 1976)

Must go beyond “back-of-the-envelope” model of electronic structure for the earth.

Replace simple model of the charge density with self-consistent quantum mechanical treatment of charge density and potential.

This cannot be done exactly.

Density functional theory appears to be sufficiently accurate to address key geophysical questions.

Structure of hcp iron: c/a

Inner core

density

- Increases with increasing temperature
- Values much greater than ideal
- Anticipate slower elastic wave propagation along c
- Computation of full elastic constant tensor confirms ~12% slower

Ideal

Steinle-Neumann, Stixrude, Cohen, Gulseren, Nature (2001)

Temperature of core?

Uncertainties in freezing point depression now outweigh uncertainties in melting curve of iron

Other approaches?

Elasticity of inner core

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