mineral physics of the core

Mineral Physics of the Core

Lars Stixrude

University of MichiganGerd Steinle-Neumann, Universität BayreuthRon Cohen, Carnegie Institution of WashingtonDavid Singh, Naval Research LabsHenry Karkauer, William and Mary

Challenges for mineral

physics

Song and Richards, Nature (1996)

Origin of core structureComposition of the coreMineralogy of the inner coreTemperature at Earth’s center

Mantle

Outer Core

Inner Core

Oxides &Silicates

IronAlloy

Depth 0 660 2890 5150 6371 kmPressure 0 24 136 329 363 GPaTemperature 300 1800 3000 5500 6000 K

Earth

Solid

Solid

Liquid

Crystal structure of iron at inner core conditions

Three 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)

a

c

Theory of Planetary Materials

10-4

10-3

10-2

10-1

100

Pressure (Atomic Units)

4 6

10-22 4 6

10-12 4 6

1002

Charge Density (Atomic Units)

101

102

103

104

Pressure (GPa)

10-1

2 4 6100

2 4 6101

2 4 6

Density (Mg m-3)

JupiterEarth

Z=10

Z=26Z=1

Simple Theories FailThomas-Fermi-DiracPressure insufficient

Terrestrial pressure~ Bond deformation pressure

eV/Å3 = 160 GPa~ Bulk modulusAtomistic models will fail

What to do?Experiment (Birch, 1952)First principles theory (Bukowinski, 1977)

TheoryMany different kinds!

Quantum methodsElectronic structure computedDensity functional theoryFirst principles, ab initio

Classical methodsQM is absorbed into an approximate model of interatomic interactionsInteratomic force models/fieldsPair potentials

Hybrids

Crystal Structure of Inner Core

Ross et al., JGR, 1990Belonoshko et al., Nature, 2003

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

Could the inner core be made of bcc?

Mechanical instability of bcc iron

Stixrude et al., PRB, 1994; Stixrude & Cohen, GRL, 1995

Bains path

Origin of mechanical instability5

4

3

2

1

0

Density of States (eV

-1)

-8 -6 -4 -2 0 2 4

Energy (eV)

BCC Iron

0 Mbar

3 Mbar

Stixrude et al., US-Japan volume, 1998

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

Types of InstabilityThermodynamic 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?

Substitution mechanism?2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

Enthalpy (eV/Fe)

1.00.80.60.40.20.0

Composition FeOx

hcpnB8

iB8

rh

B1:3VO=γ−Fe4N1:1B VO

fcc

1B

:2fct VO

8:1nB VO

8:1iB VO=CdI2136 GPa

2Bbcc

:1rh VO=CdCl2

ε-Fe3N ζ-Fe2N

:1fct OFe=CuTi3

2 :1bcc OFe

2 8:3iB VO

:1fcc OFe=CuAu3

2 8:1iB VO

Fe FeO

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

a

c

HCP iron: elastic anisotropy

ρVP2 = c11 + c33 − c11( )cos2 ξ + 4 c44 − c66[ ]cos2 ξ sin2 ξ

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

IronTheory: 2 %Original xrd: 250-350 %Latest xrd: 28-64 %

4

3

2

1

0

Shear Anisotropy C

44/C

66

Sc Y

Ti Zr Hf Re

Fe Ru Co

Zn Cd

3d 4d 5dExperiment LAPW

Singh et al. 1998 α=0.5

. 1998 Singh et al α=1.0 . 1998 Mao et al

. 2004Merkel et alα=0.5

. 2004Merkel et alα=1.0

& , 1995Stixrude Cohen- . 1999 Steinle Neumann et al

Elastic anisotropy HCP iron

Stixrude & Cohen, 1995

Inner-core shear-wave splittingStixrude &

Cohen (1995)Thanks to C. Wicks for ray tracing

Influence of temperature

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

Glatzmaier & Roberts, 1996

Confirmation of high-T elastic constant prediction

Origin of texture

Inner core is not so simple!

Temperature of the inner core

• 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

5600 K

shear modulus

bulk modulus

Melting curve of iron

Alfe et al., PRB, 2002

Nguyen & Holmes, Nature, 2004

Brown & McQueen, JGR, 1986

The Geotherm

6000

5000

4000

3000

2000

1000

0

Temperature (K)

6000400020000Depth (km)

Core chemistry25 elements lighter than iron

Hypothesis testing: two extreme models of major element core composition

1. identical to that of the meteorites from which earth formed

2. 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

Future

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.

Melting on the Hugoniot

Pressure

Tem

pera

ture

Sou

nd V

eloc

ity

Solid

Liquid Hugoniot

Dynamic compression data

Hugoniot Temperature

Iron melting

Theory. Various levels of qualityElectronic. 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.

ExperimentStatic 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

Origins

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 pressureLarge decrease in ionic radiusChange in chemical affinity from lithophile to siderophile?

Bukowinski (1976) GRL 3, 491

Potassium35 GPa

Nature of Theory in Geo-Context

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

Nuclei

Point charges

Quantumobjects

Electrons&

Application of Theory

Exactly soluble only for H atomInsolubility particularly severe for real, i.e. natural, i.e. geological materialsBasic difference in approach between earth science and physics/chemistry

"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

€

−∇2 + V[ ]ψ i = ε iψ i

Kinetic Potential

Wavefunction Energy

The Schrödinger Equation

Size of System

One challenge of natural systems is encapsulated by the concept of size. Aspects of natural systems that lead to large size

Structural complexityImpuritiesDefectsSolid solutionTemperature

NOT number of atoms in a sample O(1023) Theory deals with systems that are infinite and periodicSize means size of periodically repeating unit, i.e. unit cell.

Approaches to Large SystemsDensity functional theory

Exact in principleMust 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

€

E[n(r)] = T[n(r)] +1

2

Zα Zβ

Rβ − Rαα ,β

∑ +Zα n(r)

Rα − r∫

α

∑ dr

+n(r)n( ′ r )

′ r − rdrd ′ r ∫ + Exc[n(r)]

Spac

kman

et a

l., (

1987

)

Illustration: Solid Solutions

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

Illustration: Solid Solutions

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.

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

Liquid and hcp Fe:O,Si,S

Illustration: Influence of Temperature

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 nucleiIntegrate Newton’s 2nd law

Lattice DynamicsExpand energy to second order in displacementsFind normal modes of vibration

Energy

Displacement

How to detect melt in static compression?

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

Inner Core Anisotropy

ρVP2 = c11 + c33 − c11( )cos2 ξ + 4 c44 −

1

4c33 + c11 − 2c13( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥cos2 ξ sin2 ξ

ρVP2 = c11 + c33 − c11( )cos2 ξ + 4 c44 − c66[ ]cos2 ξ sin2 ξ

Origin of Magnetism

Ferromagnet

Paramagnet

PauliParamagnet

electrons=±1/2

atomic or localS=2

Bulkf(V)

Magnetic CollapseOrigin

Levels

Bands

Low Pressure High Pressure

Magnetic Collapse

Cohen, Mazin, Isaak, Science, (1997) Steinle-Neumann, Stixrude, Cohen, Phys Rev B (1999)

Challenges for mineral physics

Relate structure to processThermal evolutionTemperature in the inner coreChemical evolutionComposition of the coreMagnetic field generationMineralogy 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.

What is Earth made of?

Structure of hcp iron: c/a• 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

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

Ideal

Inner core density

Temperature of core?

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

Other approaches?

Elasticity of inner core

Composition & Temperature

1200

1000

800

600

400

200

Elastic Modulus (GPa)

10098969492908886

Volume (Bohr3)

Re

C11

C33

C12

C13

C44

Duffy et al. PRB 1999Manghnani et al., 1974

Elastic constants by x-ray diffraction