Melting of Iron at Earth’s core conditions from quantum Monte Carlo free energy calculations Department of Earth Sciences & Department of Physics and Astronomy,

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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

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

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

Thermodynamic melting

The Helmholtz free energy Solids: Low T

Phonons of Fe

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

Thermodynamic integration

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

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:

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

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:

Liquid Fe

Size tests

Hugoniot of Fe

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

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

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

Thermodynamic integration, a perturbative approach:

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

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

Thermodynamic integration, a perturbative approach:

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

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 ~ times more expensive than DFT

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, 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, (R), (2004))

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)

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

Solid Fe, equation of state at 300 K

QMC correction to the DFT Fe melting curve

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