1. Melting of Iron at Earth’s core conditions from quantum Monte Carlo free energy calculations Department of Earth Sciences & Department of Physics and Astronomy, Materials Simulation Laboratory & London Centre for Nanotechnology University College London Dario ALFÈ

4. Birch (1952) - “The Core is iron alloyed with a small fraction of lighter elements” Nature of light element inferred from: Cosmochemistry Meteoritics Equations of state Core composition

6. Temperature of the Earth’s core Exploit solid-liquid boundary Exploit core is mainly Fe Melting temperature of Fe

7. Thermodynamic melting

8. The Helmholtz free energy Solids: Low T

9. Phonons of Fe

10. The Helmholtz free energy Solids: Liquids: Low T High T

11. Thermodynamic integration

12. Example: anharmonic free energy of solid Fe at ~350 GPa

13. Improving the efficiency of TI F is independent on the choice of U ref, but for efficiency choose U ref such that: is minimum. For solid iron at Earth’s core conditions a good U ref is:

14. Improving the efficiency of TI (2) At high temperature we find c 1 = 0.2, c 2 = 0.8

15. F independent on choice of U ref, but for efficiency choose U ref such that is minimum. For liquid iron a good U ref is: Free energy for liquid Fe:

16. Liquid Fe

17. Size tests

18. Hugoniot of Fe

19. Alfè, Price,Gillan, Nature, 401, 462 (1999); Phys. Rev. B, 64, 045123 (2001); Phys. Rev. B, 65, 165118 (2002); J. Chem. Phys., 116, 6170 (2002 ) The melting curve of Fe

20. NVE ensemble: for fixed V, if E is between solid and liquid values, simulation will give coexisting solid and liquid Melting: coexistence of phases

22. Alfè, Price,Gillan, Nature, 401, 462 (1999); Phys. Rev. B, 64, 045123 (2001); Phys. Rev. B, 65, 165118 (2002); J. Chem. Phys., 116, 6170 (2002 ) The melting curve of Fe Free energy approach and Coexistence give same result (as they should !)

23. Thermodynamic integration, a perturbative approach:

24. Only need to run simulations with one potential (the reference potential for example).

25. Melting of Fe from QMC: Free energy corrections from DFT to QMC:

26. Thermodynamic integration, a perturbative approach:

27. Extracting the ground state: substitute  = it Beyond DFT, Diffusion Monte Carlo:

28. Imaginary time Schroedinger equation with V = 0: Diffusion equation in a 3-N dimensional space: Brownian particles (walkers)  distribution function Potential energy V --> source or sink of walkers Problems: Fixed nodes approximation: Pseudopotentials (locality approximation) DMC is ~ 10 3 -10 4 times more expensive than DFT

29. QMC on Fe, technical details CASINO code: R. J. Needs, M. D. Towler, N. D. Drummond, P. Lopez-Rios, CASINO user manual, version 2.0, University of Cambridge, 2006. DFT pseudopotential, 3s 2 3p 6 4s 1 3d 7 (16 electrons in valence) Single particle orbitals from PWSCF (plane waves), 150 Ry PW cutoff. Then expanded in B-splines. (D. Alfè and M. J. Gillan, Phys. Rev. B, 70, 161101(R), (2004))

30. Blips Storing the coefficients: avc(x,y,z,ib) [old] avc(ib,x,y,z) [new] New is faster on large systems, but slower on small systems (cutoff ~ 250 electrons)

31. Solid (h.c.p.) Fe, finite size Ester Sola

32. Solid Fe, equation of state at 300 K

33. QMC correction to the DFT Fe melting curve

35. Conclusions Melting temperature of Fe at 330 GPa = 6800 +- 400 K Melting point depression due to impurities ~ 800 K Probable temperature of the Earth’s core is ~ 6000 K