1. Journey to the Centre of the Earth (The structure of iron in the inner core) Lidunka Vočadlo

7. Masters et al., 2000

9. Inside the Earth At the Earth’s surface, we experience relatively mild conditions of P and T. l But at the centre (6400 km down), the pressure reaches over 3.5 million atm and the temperature may exceed 6000 o C. l Material from volcanic eruptions has come from only a few 100km down, so there remains well over 6000km to go - 90% of the Earth is effectively inaccessible.

10. Understanding the Earth l To understand the Earth’s deep interior, we can perform experiments and computer simulations on candidate minerals. l Calculations underpin experimental data and enable study at very high P/T, beyond the limitations of experimental methods. l In particular, we are working on Fe and Fe alloys under the extreme conditions of the Earth’s core where iron is squeezed to about half its normal volume.

11. The outer core is liquid, ~10% less dense than Fe. The inner core is solid, 3-4% less dense than Fe. IC is crystallising out of the OC. T ICB determined by melting of Fe alloy. The Earth’s Core

12. Which Elements? As pointed out by Poirier (1994), the favoured light element has varied with time, and the strength of the personalities involved! On this basis, S, O, Si, H and C are the primary candidates. Ni, K, etc also possible in core.

13. Why do we care? Heat generated in the Earth’s core drives the dynamics of the planet, resulting in plate tectonics, Earthquakes & volcanoes.

16. Why do we care? Heat released from the crystallising inner core drives convection in the liquid outer core, which in turn generates the Earth’s magnetic field.

17. Therefore iron is a hot topic! To understand our planet we need an accurate knowledge of the physical properties and composition of the core. Therefore we need to understand the properties of iron and iron alloys.

18. Experiments using: Diamond Anvil Cell Multi-anvil press Piston cylinder Laser Heating Synchrotron Radiation Shock But core pressures and temperatures remain challenging. How do we find out?

19. P and T at centre of Earth ~ 360 GPa and ~ 6000 K P and T at core/mantle ~ 135 GPa and ~ 3000 to 4000 K Piston cylinder ~ 4 GPa Multi-anvil ~ 30 GPa (higher with sintered diamonds) Diamond-anvil ~ 200 GPa (temperatures uncertain, gradients high) Shock guns ~ 200 GPa (temperatures extremely uncertain) High P/T experiments are hard and can have large uncertainties. Experimental limits…. Also, it can be dangerous to extrapolate experimental data to high pressures and temperatures….

20. Unit cell volume of Fe 3 C as a function of temperature, obtained by time-of-flight neutron powder diffraction using the POLARIS diffractometer at the ISIS spallation neutron source. T c = 483 K. The dangers of relying on experiments

21. Spontaneous magnetisation of Fe 3 C as a function of unit cell volume. T c ~ 60 GPa Ferromagnetism is also destroyed by pressure:

22. * experimental V not necessarily derived from the fit; ** data fitted to a Vinet equation of state – not BM3 Equation of state parameters for Fe 3 C: V 0 Å 3 /atomK 0 GPaK’ Magnetic 9.578(37)173.02(8)5.79(41) Non-magnetic 8.968(7)316.62(2)4.30(2) Scott et al., 2001 9.704(9)*175.4(35)5.1(3) Jephcoat, 2000** 1626.4

23. The alternative to experiments is…. Computational mineral physics!

24. What can simulations predict? Volumes, bulk moduli Vibrational frequencies (phonon density of states) Elastic constants (seismic velocities) Heat capacities Free energies (phase diagrams) Defects Diffusion Viscosities Melting etc. The fact that we can predict it does not make it right!

25. What does CMP Involve? Microscopic scale modelling of bonding in minerals and fluids. BONDING can be described by: –effective potentials (analytical functions approximately describing how energy varies as a function of separation or geometry), –quantum mechanical calculation of energy as a function of structure. Both can be very CPU intensive.

26. Potential Curve Energy (E) Short Range Repulsion dE/dx a 0 d 2 E/dx 2 K Atomic separation (x) Coulombic Attraction

27. What is to come… Iron phase diagram Ab initio methods Solid Fe –Lattice dynamics –Phonon stability –Elastic properties Liquid and anharmonic solid –Molecular dynamics –Melting –Example – aluminium –hcp-Fe bcc-Fe instability –Mechanical stablity along T m –Instability at lower T

28. fcc hcp bcc

29. What is the structure of Fe at core P and T? High P/T phase of Fe is controversial. Boehler 1993 observed a possible new phase in DAC above ~40 GPa and ~1000 K. Andrault et al claim an orthorhombic phase. Saxena et al suggest dhcp. Shen et al fail to find it. Brown & McQueen shock expts. claim a solid- solid phase change ~200 GPa and ~4000 K. Nguyen and Holmes don’t find it!

31. Ab Initio Techniques Numerically solving Schrodinger’s equation. One major approximation; the effect on any one electron from all the other electrons (a very serious many body problem) is wrapped up into a term called the exchange- correlation. Methods used: –Density Functional Theory –Generalised Gradient Approximation for E xc –Ultrasoft, non-norm-conserving pseudo-potentials and/or PAW for the interactions between valence electrons and the tightly bound core electrons

32. Real and pseudo wavefunction

33. 5d-orbitals in Au. From the web page of Andrew M. Rappe All electronPseudowavefunction

34. Ab Initio Techniques Code used was VASP, running on CSAR T3E and UCL-Bentham. DFT with PAW and/or PP. We use an {NVT} ensemble, 64+ atom supercell. Considerable Considerable effort spent on convergence tests in cell size, k-point sampling, etc. to minimise error in free energies to just a few meV per atom.

35. Quality of the simulations Ab initio calculations give good descriptions of:  EOS of bcc-Fe  Magnetic moment of bcc-Fe  The bcc  hcp transition pressure  The high P density of hcp-Fe  Phonon dispersion of bcc-Fe

38. Solid Iron Need Gibbs free energy to obtain stable phase in the core G(P,T) = F total (V,T) + P total (V,T)V P is first derivative of F F is a function of vibrational frequencies Use lattice dynamics to obtain ω i

39. F of the harmonic solid F total (V,T) = F perfect (V,T)+F vibrational (V,T) F perfect from static electronic minimisation calculation including thermal electronic excitations via: F perfect (V,T) = U 0 (V)+ U el (V,T) -TS el (V,T) F vibrational requires ω i and vibrational DOS to put into statistical mechanics equations.

40. F of the harmonic solid Use small displacement method - atoms frozen in distorted positions >> residual forces. Dispersion curves obtained by interpolation of ω i calculated from the dynamical matrix. S, C, E, c ij, etc. = f(ω i ) K, G, V p, V s = f(c ij ) e.g.,

41. Results! What is the phase of iron in the core?  Mechanical instability of bcc-Fe at high P  Thermodynamic stability of hcp-Fe at high P  Phonon DOS of bcc and hcp compared with expts  Elastic properties of hcp-Fe

43. The stable phase in the core is hcp Spin polarised calculations on all phases at core P reveal reduced μ for bcc/bct and zero μ for other phases  no magnetism. bcc and bct transform to fcc, orthorhombic to hcp; hcp, fcc, and dhcp remain mechanically stable at core pressures. However, fcc and dhcp are less favourable energetically; therefore hcp is the stable phase in the core. (harmonically at this point!)

44. Calculated vibrational density of states compared with inelastic nuclear resonance X-ray scattering (open circles; Mao et al., 2000)

45. Elastic Constants Elastic constants determined from dispersion curves. Γ  K or Γ  M: c 11 = ρ v L 2 ½ ( c 11 - c 12 ) = ρ v T1 2 c 44 = ρ v T2 2 Γ  A: c 33 = ρ v L 2 c 44 = ρ v T1 2 = ρ v T2 2 Γ  45 o between K and A: ½ (c 11 + c 22 + 2c 44 ) ± [ ¼ (c 11 - c 33 ) 2 + ( c 13 + c 44 ) 2 ] ½ = 2ρ v T1 2 = 2ρ v T2 2

47. Calculated (dotted line) thermodynamic properties compared with inelastic nuclear resonance X-ray scattering experiments (open circles; Mao et al., 2000).

48. Calculated (dotted line) elastic properties compared with inelastic nuclear resonance X-ray scattering experiments (open circles; Mao et al., 2000) and all-electron calculations (dashed line; Steinle-Nemann,1999).

49. Liquid Fe and the anharmonic solid We cannot use lattice dynamics for: –the high T solid which departs from harmonicity –the liquid system where there is no long range order –bcc-Fe, which is only stable at high T For these we use molecular dynamics

50. Ab Initio Molecular Dynamics Calculate the energy of liquids and anharmonic systems with ab initio molecular dynamics Simulate the properties of materials at high temperatures Calculate the energy of a configuration ab initio with DFT, then move the atoms classically according to Newtons Laws.

51. Calculation Strategy for T m Calculate the Gibbs free energy of both the solid and liquid as a function of P and T. At each chosen P obtain T m as the point at which G S (P,T m ) = G L (P,T m ). In fact, we calculate F(V,T) and calculate G(P,T) from G=F+PV where P=-(  F/  V) T

52. F of Liquid & Anharmonic Solid Cannot calculate F(V,T) directly since this is not an ensemble average - thermodynamic integration. Start from known F of simple model system and switch the PE function continuously to real system. PE between states I and II:

53. The reference system For liquid, start with free energy of a simple IP, for anharmonic solid start from a combination of the IP and harmonic solid. Only repulsive term; bonding term depends strongly on V and T, but not on atomic positions. For Fe, Γ=1.77 eVÅ, α=5.86

54. Calculating T m at the ICB Using first principles molecular dynamics and thermodynamic integration, we can calculate the Gibbs free energy of both the solid and liquid systems as f(P,T); melting occurs when G S =G L. (Example for aluminium)

55. Results! Melting:  Example – aluminium  hcp-Fe bcc-Fe:  Mechanical stablity along T m  Instability at lower T

58. T m of hcp-Fe At the P conditions existing at the inner core/outer core boundary, pure iron melts at 6400 C. However, the presence of alloying elements such as S, Si and O will probably reduce this temperature to 5500 C. This is very hot, comparable with temperatures on the surface of the Sun. (Dario Alfè et al.)

59. So iron in the core is all sewn up ….or is it?!! Many groups favour bcc-Fe as the stable phase - there may be a break in slope of T m at high P. It is possible that bcc could be stabilised by T. Therefore we have performed free energy calculations on bcc-Fe as a f(V,T). But this is difficult - how do you get harmonic F? Answer: use TI between mechanically stable reference system and high T system.

62. Relative stability along T m (P) VTF bcc (eV) F hcp (eV) ΔF (meV) 9.0 Å 3500 K-10.063-10.10945 8.5 Å 3500 K-9.738-9.79658 7.8 Å 5000 K-10.512-10.56250 7.2 Å 6000 K-10.633-10.66835 6.9 Å 6500 K-10.545-10.58237 6.7 Å 6700 K-10.228-10.32138

63. So..DEFINATELY (probably) hcp… but…when does bcc become stable? From the phonon frequencies, we know that bcc is unstable at 0 K and high P. From the free energies and analysis of atomic positions, we know that bcc is mechanically (if not thermodynamically) stable at high pressures and temperatures. What happens to the system as we lower T? –Stresses on the box –Atomic deviation from bcc structure

67. Summary Calculations and experiments are both very useful tools for probing the Earth’s deep interior. Calculations can be as accurate as experiments, although care needs to be taken to ensure a meaningful comparison is made. The methodology for determining melting curves works well for aluminium; there is still some controversy surrounding the iron melting curve. The stable phase in the core is hcp-Fe and not the high temperature bcc-Fe.

68. I have not failed 10,000 times, I have successfully found 10,000 ways that will not work. Thomas A. Edison