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
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)
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
c a 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)
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
Mechanical instability of bcc iron Bains path Stixrude et al., PRB, 1994; Stixrude & Cohen, GRL, 1995
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?
c 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
Inner-core shear-wave splitting Stixrude & 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 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
Melting curve of iron Alfe et al., PRB, 2002 Nguyen & Holmes, Nature, 2004 Brown & McQueen, JGR, 1986
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.
Melting on the Hugoniot Liquid Hugoniot Temperature Solid Sound Velocity Pressure
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
35 GPa 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
Nature of Theory in Geo-Context 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
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 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. 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
Spackman et al., (1987) 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
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
Alfé, Gillan, Price (2002) EPSL 195, 91 Liquid and hcp Fe:O,Si,S 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.
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 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
Bulk f(V) Origin of Magnetism electron s=±1/2 atomic or local S=2 Ferromagnet Paramagnet Pauli Paramagnet
Magnetic CollapseOrigin Levels High Pressure Low Pressure Bands
Magnetic Collapse Cohen, Mazin, Isaak, Science, (1997) Steinle-Neumann, Stixrude, Cohen, Phys Rev B (1999)