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
