Mineral Physics of the Core

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Mineral Physics of the Core

• 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
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Challenges for mineral physics
Origin of core structure Composition of the
core Mineralogy of the inner core Temperature at
Earths center
Song and Richards, Nature (1996)
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Earth
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
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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)

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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)
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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
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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
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Mechanical instability of bcc iron
Bains path
Stixrude et al., PRB, 1994 Stixrude Cohen,
GRL, 1995
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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
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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
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Influence of temperature?
Vocadlo et al, Nature (2003)
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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).
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Chemical stabilization of the bcc structure?
Lin et al. (2002) find that addition of Si
expands bcc stability field Maximum pressure lt
1Mbar Vocadlo et al. (2003) find that
substitution of Si, S is more favorable in bcc
phase Which substitution mechanism?
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Substitution mechanism?
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Iron at inner core conditions

• Hexagonal close-packed (hcp) structure
• Two repeat distances
• a - close-packed planes
• c - spacing between planes
• Ideal Ratio
• c/av8/31.633
• Elastic wave speed
• Compare with inner core
• Anisotropy
• Temperature

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HCP iron elastic anisotropy
LAPW Stixrude Cohen, Science, 1995
Steinle-Neumann et al., PRB, 1999 XRD Mao et
al., Nature, 1998
Small anisotropy, assume C12C13
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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
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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
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Elastic anisotropy HCP iron
Stixrude Cohen, 1995
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Inner-core shear-wave splitting
Stixrude Cohen (1995)
Thanks to C. Wicks for ray tracing
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Influence of temperature
Steinle-Neumann et al., Nature, 2001
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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

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Remaining issues
Confirmation of high-T elastic constant
prediction Origin of texture Inner core is not
so simple!
Glatzmaier Roberts, 1996
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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
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Melting curve of iron
Alfe et al., PRB, 2002
Nguyen Holmes, Nature, 2004
Brown McQueen, JGR, 1986
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The Geotherm
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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
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Future
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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.
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Melting on the Hugoniot
Liquid
Hugoniot
Temperature
Solid
Sound Velocity
Pressure
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Dynamic compression data
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Hugoniot Temperature
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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?
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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
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Origins
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
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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
Electrons

Quantum objects
Point charges
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Application of Theory
"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

• 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

The Schrödinger Equation
Wavefunction
Energy
Kinetic
Potential
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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 complexity
• Impurities
• Defects
• Solid solution
• Temperature

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Approaches to Large Systems

• 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

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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

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Illustration Solid Solutions
Liquid and hcp FeO,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.

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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 nuclei
• Integrate Newtons 2nd law
• Lattice Dynamics
• Expand energy to second order in displacements
• Find normal modes of vibration

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How to detect melt in static compression?
X-ray diffraction. Re-crystallization. Absence
of evidence
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Inner Core Anisotropy
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Origin of Magnetism
Bulk f(V)
electron s1/2
atomic or local S2
Ferromagnet
Paramagnet
Pauli Paramagnet
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Magnetic CollapseOrigin
Levels
High Pressure
Low Pressure
Bands
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Magnetic Collapse
Cohen, Mazin, Isaak, Science, (1997)
Steinle-Neumann, Stixrude, Cohen, Phys Rev B
(1999)
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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
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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.
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What is Earth made of?
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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)
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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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Composition Temperature
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Elastic constants by x-ray diffraction
Duffy et al. PRB 1999 Manghnani et al., 1974