Mineral Physics of the Core. Lars Stixrude University of Michigan. Gerd Steinle-Neumann, Universit ä t 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
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University of Michigan
Gerd Steinle-Neumann, Universität Bayreuth
Ron Cohen, Carnegie Institution of Washington
David Singh, Naval Research Labs
Henry Karkauer, William and Mary
Origin of core structure
Composition of the core
Mineralogy of the inner core
Temperature at Earth’s center
Song and Richards, Nature (1996)
Depth 0 660 2890 5150 6371 km
Pressure 0 24 136 329 363 GPa
Temperature 300 1800 3000 5500 6000 K
aCrystal structure of iron at inner core conditions
Three known phases
But: no experimental determinations of structure at inner core conditions (yet)
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
Stixrude et al., PRB, 1994; Stixrude & Cohen, GRL, 1995
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
Vocadlo et al, Nature (2003)
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).
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?
LAPW: Stixrude & Cohen, Science, 1995; Steinle-Neumann et al., PRB, 1999
XRD: Mao et al., Nature, 1998
Small anisotropy, assume C12≈C13
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
Less than 50 % for all hcp transition metals stable at ambient conditions
Theory: 2 %
Original xrd: 250-350 %
Latest xrd: 28-64 %
Stixrude & Cohen, 1995
Thanks to C. Wicks for ray tracing
Steinle-Neumann et al., Nature, 2001
Confirmation of high-T elastic constant prediction
Origin of texture
Inner core is not so simple!
Glatzmaier & Roberts, 1996
Alfe et al., PRB, 2002
Nguyen & Holmes, Nature, 2004
Brown & McQueen, JGR, 1986
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
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.
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.
Static compression. How to detect melt?
Dynamic compression. How to determine temperature?
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 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
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
"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
One challenge of natural systems is encapsulated by the concept of size.
Aspects of natural systems that lead to large size
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
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.
Evaluate forces acting on nuclei
Integrate Newton’s 2nd law
Expand energy to second order in displacements
Find normal modes of vibration
X-ray diffraction. Re-crystallization. Absence of evidence
Cohen, Mazin, Isaak, Science, (1997)
Steinle-Neumann, Stixrude, Cohen, Phys Rev B (1999)
Relate structure to process
Temperature in the inner core
Composition of the core
Magnetic field generation
Mineralogy of the core
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.
Steinle-Neumann, Stixrude, Cohen, Gulseren, Nature (2001)
Uncertainties in freezing point depression now outweigh uncertainties in melting curve of iron
Elasticity of inner core
Duffy et al. PRB 1999
Manghnani et al., 1974