2. 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

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

4. Mantle Outer Core Inner Core Oxides & Silicates Iron Alloy Depth 0 660 2890 5150 6371 km Pressure 0 24 136 329 363 GPa Temperature 300 1800 3000 5500 6000 K Earth Solid Liquid

5. 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

6. Theory of Planetary Materials Simple Theories Fail Thomas-Fermi-Dirac Pressure insufficient Terrestrial pressure ~ Bond deformation pressure eV/Å 3 = 160 GPa ~ Bulk modulus Atomistic models will fail What to do? Experiment (Birch, 1952) First principles theory (Bukowinski, 1977)

7. Theory Many 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

8. Crystal Structure of Inner Core Ross et al., JGR, 1990 Belonoshko et al., Nature, 2003 Some soft-sphere interatomic potentials predict bcc stable at high temperatures Could the inner core be made of bcc?

9. Mechanical instability of bcc iron Stixrude et al., PRB, 1994; Stixrude & Cohen, GRL, 1995 Bains path

10. Origin of mechanical instability 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

11. 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

12. Influence of temperature? Vocadlo et al, Nature (2003)

13. 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).

14. 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?

15. Substitution mechanism?

16. 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

17. HCP iron: elastic anisotropy LAPW: Stixrude & Cohen, Science, 1995; Steinle-Neumann et al., PRB, 1999 XRD: Mao et al., Nature, 1998 Small anisotropy, assume C 12 ≈C 13

18. 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

19. 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 %

20. Elastic anisotropy HCP iron Stixrude & Cohen, 1995

21. Inner-core shear-wave splitting Stixrude & Cohen (1995) Thanks to C. Wicks for ray tracing

22. Influence of temperature Steinle-Neumann et al., Nature, 2001

23. 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 

24. Remaining issues Glatzmaier & Roberts, 1996 Confirmation of high-T elastic constant prediction Origin of texture Inner core is not so simple!

25. 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

26. Melting curve of iron Alfe et al., PRB, 2002 Nguyen & Holmes, Nature, 2004 Brown & McQueen, JGR, 1986

27. The Geotherm

28. Core chemistry 25 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

29. Future

30. 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.

32. Melting on the Hugoniot Pressure Temperature Sound Velocity Solid Liquid Hugoniot

33. Dynamic compression data

34. Hugoniot Temperature

35. 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?

36. 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

38. 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 pressure Large decrease in ionic radius Change in chemical affinity from lithophile to siderophile? Bukowinski (1976) GRL 3, 491 Potassium 35 GPa

39. 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 Quantum objects Electrons&

40. Application of Theory 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 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 KineticPotential WavefunctionEnergy The Schrödinger Equation

41. 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 NOT number of atoms in a sample O(10 23 ) Theory deals with systems that are infinite and periodic Size means size of periodically repeating unit, i.e. unit cell.

42. 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 Spackman et al., (1987)

43. 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

44. 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

45. 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 2 nd law Lattice Dynamics Expand energy to second order in displacements Find normal modes of vibration

46. How to detect melt in static compression? X-ray diffraction. Re-crystallization. Absence of evidence

47. Inner Core Anisotropy

48. Origin of Magnetism Ferromagnet Paramagnet Pauli Paramagnet electron s=±1/2 atomic or local S=2 Bulk f(V)

50. Magnetic Collapse Origin Levels Bands Low Pressure High Pressure

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

52. 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

53. 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.

54. What is Earth made of?

57. 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

58. Temperature of core? Uncertainties in freezing point depression now outweigh uncertainties in melting curve of iron Other approaches? Elasticity of inner core

59. Composition & Temperature

60. Duffy et al. PRB 1999 Manghnani et al., 1974 Elastic constants by x-ray diffraction