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It is shown that statistical mechanical models with short range competing interactions, the axial Ising models, have equilibrium phase diagrams which mirror many of the features of polytypism. In particular (because of a subtle balance between energy and entropy), phases of arbitrarily long wavelength can result from interactions extending only to second neighbours. Phase diagrams of axial Ising models are described, with interactions up to third neighbours, and their relation to polytypism in silicon carbide, cadmium iodide, the spinelloids and other mineral phases are discussed.-J.M.H.

The solid inner core of the Earth and the liquid outer core consist mainly of iron(1) so that knowledge of the high-pressure thermodynamic properties of iron is important for understanding the Earth's deep interior. An accurate knowledge of the melting properties of iron is particularly important, as the temperature distribution in the core is relatively uncertain(2-4) and a reliable estimate of the melting temperature of iron at the pressure of the inner-core boundary would put a much-needed constraint on core temperatures. Here we used ab initio methods to compute the free energies of both solid and liquid iron, and we argue that the resulting theoretical melting curve competes in accuracy with those obtained from high-pressure experiments. Our results give a melting temperature of iron of similar to 6,700 +/- 600 K at the pressure of the inner-core boundary, consistent with some of the experimental measurements. Our entirely ab initio methods should also be applicable to many other materials and problems.

It is thought that the Earth's outer core consists mainly of liquid iron and that the convection of this metallic liquid gives rise to the Earth's magnetic field. A full understanding of this convection is hampered, however, by uncertainty regarding the viscosity of the outer core. Viscosity estimates from various sources span no less than 12 orders of magnitude(1,2), and it seems unlikely that this uncertainty will be substantially reduced by experimental measurements in the near future. Here we present dynamical first-principles simulations of liquid iron which indicate that the viscosity of iron at core temperatures and pressures is at the low end of the range of previous estimates - roughly 10 times that of typical liquid metals at ambient pressure. This estimate supports the promotion commonly made in magnetohydrodynamic models that the outer core is an inviscid fluid(3-5) undergoing small-scale circulation and turbulent convection(6), rather than large-scale global circulation.

We calculated thermo-elastic properties of pyrope (Mg3Al2Si3O12) at mantle pressures and temperatures using Ab initio molecular dynamic simulation. A third-order Birch-Murnaghan equation at a reference temperature of 2 000 K fits the calculations with bulk modulus, K (0)=159.5 GPa, K (0)'=4.3, V (0)=785.89 (3), Gruneisen parameter, gamma (0)=1.15, q=0.80, Anderson Gruneisen parameter delta (T) =3.76 and thermal expansion, alpha (0)=2.93x10(-5) K-1. Referenced to room temperature, where V (0)=750.80 (3), gamma (0) and alpha (0) become 1.11 and 2.47x10(-5) K-1. The elastic properties of pyrope are found to be nearly isotropic at transition zone conditions.

The thermo-elastic properties of MgSiO3 akimotoite at mantle pressure and temperature conditions are reported based on ab initio molecular dynamic simulations. A third-order Birch–Murnaghan equation at a reference temperature of 2000 K is defined by K0 = 158 GPa, K ′ 0 = 3.7 , G0 = 85.7 GPa, G ′ 0 = 4.5 , V0(2000 K) = 1100.54 A3, the Grűneisen parameter is determined to be γ(V) = γ0(V/V0(2000 K))q with γ0 = 1.84 and q = 1.84, with V(2000 K) = 1048.22 A3. An implied pressure correction is −7.6 GPa in these parameters due to GGA overestimates the pressure. The thermal expansion is determined to be α / α 0 = ( V / V 0 ( 2000 K ) ) δ T in which α0 = 3.21 × 10−5 K−1 and δT = 4.6. Akimotoite may be stable above the 660 discontinuity in relatively low temperature or low aluminium environments. The high velocity and elastic anisotropy of akimotoite provide diagnostics for its presence above the 660 km discontinuity.

The phase that iron adopts at the conditions of the Earth's inner core is still unknown. The two primary candidates are the hexagonal close packed (hcp) structure and the body centred cubic (bcc) structure polymorphs. Until recently, the former was favoured, but it now seems possible that bcc iron could be present. A remaining uncertainty regarding the latter phase is whether or not bcc iron is stable with respect to tetragonal strain under core conditions. In this paper, therefore, we present the results of high precision ab initio free energy calculations at core pressures and temperatures performed on bcc iron as a function of tetragonal strain. Within the uncertainties of the calculations, direct comparison of free energy values suggests that bcc may be unstable with respect to tetragonal strain at 5500 K; this is confirmed when the associated stresses are taken into account. However, at 6000 K, the results indicate that the bcc phase becomes more stable, although it is unclear as to whether complete stability has been achieved. Nevertheless, it remains distinctly possible that the addition of light elements could stabilise this structure convincingly. Therefore, bcc-Fe cannot be ruled out as a candidate inner core phase. (C) 2008 Elsevier B.V. All rights reserved.

The Earth's core is a ball of swirling hot metal at the centre of our planet, with a radius roughly one half of the Earth's radius. It is formed by two parts: a solid inner core, with a radius of 1221 km, surrounded by a shell of liquid which extends up to 3480 km from the centre. It is widely believe that the Earth's core is mainly formed by iron, or iron with up to 5-10% of nickel. It is also known that the core must contain a significant fraction of light impurities, in the region of 2-3% in the solid and 6-7% in the liquid. The nature of these light impurities is unknown. The temperature of the core is also inaccessible to direct probing. Here we present a theoretical study on the temperature and the composition of the Earth's core. The investigation is based on the application of the implementation of quantum mechanics known as density functional theory. We shall show that these techniques are very accurate at predicting the properties of iron, and therefore can be usefully used to study the properties of the core. We show that by combining these techniques with direct observations it is possible to predict the temperature of the core, in particular the temperature at the boundary between the solid and the liquid core (the ICB), and put constraints on its composition. The result of this study is that the temperature of the ICB is probably in the region of 5400-5700 K and that the outer core contains a significant fraction (8-13%) of oxygen. As the Earth cools down the solid core grows and expels oxygen in the liquid. Since oxygen is lighter than iron it rises in the liquid, and its gravitational energy is available to drive the convective motions in the liquid core that are responsible for the generation of the Earth's magnetic field.

The themoelastic properties of Mg2SiO4 ringwoodite at mantle pressure and temperature conditions are reported based on ab initio molecular dynamic simulations. A third-order Birch-Murnaghan equation at a reference temperature of 2000 K is defined by K-0 = 138 GPa, K-0' = 5.2, and V-0(2000 K) = 560 angstrom(3). The Gruneisen parameter is determined to be gamma(V) = gamma(0)(V/V-0(298 K))(q) with gamma(0) = 1.22 and q = 1.44(5), with V-0(298 K) = 524.56 angstrom(3). The thermal expansion is determined to be (alpha/alpha(0)) = (V/V-0(298 K))(delta T) in which alpha(0) = 2.74 x 10(-5) K-1 and delta(T) = 5.2(1). The bulk modulus is temperature independent at constant volume, while the shear moduli vary with temperature at constant volume. Elastic anisotropy decreases with both pressure and temperature becoming isotropic by the bottom of the upper mantle. (c) 2006 Elsevier B.V. All rights reserved.

An investigation of the relative stabilities and equations of state of possible Fe3S polymorphs was conducted using first-principles pseudopotential calculations. These calculations were based on density functional theory and performed using ultrasoft Vanderbilt pseudopotentials within the generalized gradient approximation. In accord with experiment, we found that the tetragonal Fe3P-type polymorph is the only stable phase along the 0 K isotherm as a function of pressure. Fe3S exhibits permanent magnetism at ambient conditions (Fei et al., 2000), but magnetism is suppressed by pressure and temperature, and therefore non-magnetic data are appropriate ones to use for modelling planetary interiors. For this reason, and because the Fe3P-type polymorph of Fe3S contains 32 atoms per unit cell it was impractical to incorporate magnetic properties into the simulations of this phase, we studied the behaviour of the non-magnetic phase. We obtained values of 250 GPa for the bulk modulus, K0, and 4.61 for its first derivative withrespect to pressure, K0′, by fitting a 3rd order Birch-Murnaghan equation of state to the calculated internal energy as a function of volume for the non-magnetic Fe3P-type Fe3S. This suggests that a pressure far greater than that expected in the Martian interior would be needed to achieve a density comparable to that of the Martian core. We therefore conclude that it is unlikely that the core of Mars contains significant amounts of solid Fe3S.

A number of melting curves of various materials have recently been measured experimentally and calculated theoretically, but the agreement between different groups is not always good. We discuss here some of the problems which may arise in both experiments and theory. We also report the melting curves of Fe and Al calculated recently using quantum mechanics techniques, based on density functional theory with generalized gradient approximations. For Al our results are in very good agreement with both low pressure diamond-anvil-cell experiments (Boehler and Ross 1997 Earth Planet. Sci. Lett. 153 223, Hanstrom and Lazor 2000 J. Alloys Compounds 305 209) and high pressure shock wave experiments (Shaner et al 1984 High Pressure in Science and Technology ed Homan et al (Amsterdam: North-Holland) p 137). For Fe our results agree with the shock wave experiments of Brown and McQueen (1986 J. Geophys. Res. 91 7485) and Nguyen and Holmes (2000 AIP Shock Compression of Condensed Matter 505 8 1) and the recent diamond-anvil-cell experiments of Shen et al (1998 Geophys. Res. Lett. 25 373). Our results are at variance with the recent calculations of Laio et al (2000 Science 287 1027) and, to a lesser extent, with the calculations of Belonoshko et al (2000 Phys. Rev. Lett. 84 3638). The reasons for these disagreements are discussed.

We summarize the main ideas used to determine the thermodynamic properties of pure systems and binary alloys from first principles calculations. These are based on the ab initio calculations of free energies. As an application we present the study of iron and iron alloys under Earth,s core conditions. In particular, we report the whole melting curve of iron under these conditions, and we put constraints on the composition of the core. We found that iron melts at 6350士600 K at the pressure corresponding to the boundary between the solid inner core and the liquid outer core (ICB). We show that the core could not have been formed from a binary mixture of Fe with S, Si or O and we propose a ternary or quaternary mixture with 8—10% of S/Si in both liquid and solid and an additional ~8% of oxygen in the liquid. Based on this proposed composition we calculate the shift of melting temperature with respect to the melting temperature of pure Fe of ~—700 K, so that our best estimate for the temperature of the Earth's core at ICB is 5650±600 K.

The Earth has a liquid outer and solid inner core. It is predominantly composed of Fe, alloyed with small amounts of light elements, such as S, O and Si. The detailed chemical and thermal structure of the core is poorly constrained, and it is difficult to perform experiments to establish the properties of core-forming phases at the pressures (ca. 300 GPa) and temperatures (ca. 5000-6000 K) to be found in the core. Here we present some major advances that have been made in using quantum mechanical methods to simulate the high-P/T properties of Fe alloys, which have been made possible by recent developments in high-performance computing. Specifically, we outline how we have calculated the Gibbs free energies of the crystalline and liquid forms of Fe alloys, and so conclude that the inner core of the Earth is composed of hexagonal close packed Fe containing ca. 8.5% S (or Si) and 0.2% O in equilibrium at 5600 K at the boundary between the inner and outer cores with a liquid Fe containing ca. 10% S (or Si) and 8% O.

It is shown how ab initio techniques based on density functional theory can be used to calculate the chemical potentials of the leading candidate impurity elements (S, 0 and Si) in the Earth's solid inner core and liquid outer core. The condition that these chemical potentials be equal in the solid and liquid phases provides values for the ratios of the impurity mol fractions in the inner and outer core. By combining the estimated ratios with ab initio values for the impurity molar volumes in the two phases, and demanding that the resulting density discontinuity across the inner-core boundary agree with free-oscillation data, we obtain estimates for the concentrations of S, O and Si in the core. The results show that O partitions much more strongly than S and Si from solid to liquid, and indicate that the presence of O in the core is essential to account for seismic measurements. We suggest that if compositional convection drives the Earth's magnetic field, then the presence of O may be essential for this compositional convection. (C) 2002 Elsevier Science B.V. All rights reserved.

We present a new technique which allows the fully {\em ab initio} calculation of the chemical potential of a substitutional impurity in a high-temperature crystal, including harmonic and anharmonic lattice vibrations. The technique uses the combination of thermodynamic integration and reference models developed recently for the {\em ab initio} calculation of the free energy of liquids and anharmonic solids. We apply the technique to the case of the substitutional oxygen impurity in h.c.p. iron under Earth's core conditions, which earlier static {\em ab initio} calculations indicated to be thermodynamically very unstable. Our results show that entropic effects arising from the large vibrational amplitude of the oxygen impurity give a major reduction of the oxygen chemical potential, so that oxygen dissolved in h.c.p. iron may be stabilised at concentrations up a few mol % under core conditions.

Knowledge of the composition of the Earth's core(1-3) is important for understanding its melting point and therefore the temperature at the inner-core boundary and the temperature profile of the core and mantle. In addition, the partitioning of light elements between solid and liquid, as the outer core freezes at the inner-core boundary, is believed to drive compositional convection(4), which in turn generates the Earth's magnetic field. It is generally accepted that the liquid outer core and the solid inner core consist mainly of iron(1). The outer core, however, is also thought to contain a significant fraction of light elements, because its density-as deduced from seismological data and other measurements-is 6-10 per cent less than that estimated for pure liquid iron(1-3). Similar evidence indicates a smaller but still appreciable fraction of light elements in the inner core(5,6). The leading candidates for the light elements present in the core are sulphur, oxygen and silicon(3). Here we obtain a constraint on core composition derived from ab initio calculation of the chemical potentials of light elements dissolved in solid and liquid iron. We present results for the case of sulphur, which provide strong evidence against the proposal that the outer core is close to being a binary iron-sulphur mixture(7).

In order to predict the stable polymorph of iron under core conditions, calculations have been performed on all the candidate phases proposed for inner core conditions, namely, body-centred cubic (bcc), body-centred tetragonal (bct), hexagonal close-packed (hcp), double-hexagonal close-packed (dhcp) and an orthorhombically distorted hcp polymorph. Our simulations are ab initio free energy electronic structure calculations, based upon density functional theory, within the generalised gradient approximation; we use Vanderbilt ultrasoft non-normconserving pseudopotentials to describe the core interactions, and the frozen phonon technique to obtain the vibrational characteristics of the candidate structures. Our results show that under conditions of hydrostatic stress, the orthorhombic, bce and bet structures are mechanically unstable. The relative free energies of the remaining phases indicate that dhcp and fee Fe are thermodynamically less stable than hcp Fe, therefore, we predict that the stable phase of iron at core conditions is hcp-Fe. (C) 2000 Elsevier Science B.V. All rights reserved.

Ab initio finite temperature molecular dynamics simulations have been used to calculate the free energy and elasticity of face-centred cubic (fcc) iron at a state point representative of the Earth's inner core. Whilst the free energy of this phase is found to be higher than that of hexagonal-close-packed (hcp) iron, the difference is only 14 meV/atom. It is possible that this difference might be overcome by the presence of light elements, as previous calculations at zero Kelvin have shown that the addition of elements such as silicon stabilise fec-Fe with respect to hcp-Fe by at least 40 meV/atom. The calculated elastic constants at core pressures and temperatures of pure fec-Fe, and of alloys of Fe with sulphur and nickel (Fe(3)S and Fe(3)Ni) derived from the fee structure, lead to average shear wave velocities that are considerably higher than those inferred from seismology; however, these mineralogical and seismological results could be reconciled by the presence of partial melt in the inner core. The calculated P-wave anisotropy of fec-Fe is comparable with the seismological values, but only if there is a high degree of crystal alignment, although the necessity for alignment can be reduced if a layered model for the inner core is invoked. The results presented in this paper therefore suggest that fcc-Fe cannot be ruled out as a candidate for the dominant phase of the Earth's inner core. (c) 2008 Elsevier B.V. All rights reserved.

The finite temperature density functional theory and quasiharmonic lattice dynamics have been used to compute numerous thermodynamic properties of hexagonal close packed magnesium using both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange-correlation potential. Generally, it is found that there exist only minor differences between the LDA and GGA computed properties, with both giving good agreement with experiment. The hcp-bcc phase boundary has also been computed and is found to be in agreement with experimental observation. Again, only slight differences are found between the LDA and GGA. (c) 2006 American Institute of Physics.

Atomic-scale materials modelling based on first-principles quantum mechanics is playing an important role in the science of the Earth and the other planets. We outline the basic theory of this kind of modelling and explain how it can be applied in a variety of different ways to probe the thermodynamics, structure and transport properties of both solids and liquids under extreme conditions. After a summary of the density functional formulation of quantum mechanics and its practical implementation through pseudopotentials, we outline the simplestway of applying first-principles modelling, namely static zero-temperature calculations. We show how calculations of this kind can be compared with static compression experiments to demonstrate the accuracy of first-principles modelling at pressures reached in planetary interiors. Noting that virtually all problems concerning planetary interiors require an understanding of materials at high temperatures as well as high pressures, we then describe how first-principles lattice dynamics gives a powerful way of investigating solids at temperatures not too close to the melting line. We show how such calculations have contributed to important progress, including the recent discovery of the post-perovskite phase of MgSiO3 in the D '' layer at the base of the Earth's mantle. A range of applications of first-principles molecular dynamics are then reviewed, including the properties of metallic hydrogen in Jupiter and Saturn, of water, ammonia and methane in Uranus and Neptune, and of oxides and silicates and solid and liquid iron and its alloys in the Earth's deep interior. Recognizing the importance of phase equilibria throughout the planetary sciences, we review recently developed techniques for the first- principles calculation of solid and liquid free energies, melting curves and chemical potentials of alloys. We show how such calculations have contributed to an improved understanding of the temperature distribution and the chemical composition throughout the Earth's interior. The review concludes with a summary of the present state of the field and with some ideas for future developments.

[1] Using first-principles molecular dynamics (MD) simulations, we study the solubility of argon in molten iron at high pressures and temperatures. In particular we explore whether the low pressure immiscibility of liquid Fe and Ar persists to high pressure (130 GPa) and temperature (4500K), or whether they mix. Starting from a variety of Fe/Ar mixtures we find that they always separate rapidly into two liquids. We conclude that there is no evidence for a significant increase in the solubility of Ar in Fe at these conditions. We cannot, therefore, attribute the lower melting temperatures of Fe obtained from DAC experiments compared to those obtained from ab initio calculations and shock experiments, to eutectic melting between Fe and the Ar pressure medium.

We report the dynamics of the structure of CaSiO(3) perovskite from ab initio molecular dynamics (AIMD) calculations at high pressure (P up to 130 GPa) and high temperature (T up to 5000 K). Our calculations indicate three separate stability fields: orthorhombic, tetragonal and cubic, with the tetragonal phase dominating the pressure and temperature region between room temperature and 4000 K. These regions are defined by the stress symmetry of the AIMD calculation. The boundary between the orthorhombic and the tetragonal structures is found to have a positive Clapyron slope and is close to room temperature at low pressure. The boundary is marked by the transition from stable, constant octahedral tilts, to dynamically varying tilts that change sign with time. The calculated atom positions indicate that the orientation of the octahedra can be noted as a(-)a(-)c(+) in the orthorhombic phase (T = 150 K). The magnitude of the octahedra rotation varies little over the entire P-T range at high T (1000 K and above) while, at elevated temperature, the rotation angles of the octahedra oscillate positively and negatively with time. The tetragonal structure is probably due to a shortened Si-O bond distance along one axis. Calculated X-ray diffraction patterns indicate small super-lattice reflections that could result from the octahedral rotations throughout the P, T region investigated. The small spontaneous strain of the tetragonal phase relative to the aristotype, cubic phase, throughout conditions appropriate to the lower mantle, creates the possibility for seismic energy absorption (low Q) in the deep Earth. (c) 2006 Elsevier B.V. All rights reserved.

Ab initio calculations of the melting properties of copper in the pressure range 0-100 GPa are reported. The ab initio total energies and ionic forces of systems representing solid and liquid copper are calculated using the projector augmented wave implementation of density functional theory with the generalized gradient approximation for exchange-correlation energy. An initial approximation to the melting curve is obtained using an empirical reference system based on the embedded-atom model, points on the curve being determined by simulations in which solid and liquid coexist. The approximate melting curve so obtained is corrected using calculated free energy differences between the reference and ab initio system. It is shown that for system-size errors to be rendered negligible in this scheme, careful tuning of the reference system to reproduce ab initio energies is essential. The final melting curve is in satisfactory agreement with extrapolated experimental data available up to 20 GPa, and supports the validity of previous calculations of the melting curve up to 100 GPa. (C) 2004 American Institute of Physics.

We model the inner core by an alloy of iron and 8 per cent sulphur or silicon and the outer core by the same mix with an additional 8 per cent oxygen. This composition matches the densities of seismic model, Preliminary Reference Earth Model (PR-EM). When the liquid core freezes S and Si remain with the Fe to form the solid and excess 0 is ejected into the liquid. Properties of Fe, diffusion constants for S, Si, 0 and chemical potentials are calculated by first-principles methods under the assumption that S, 0, and Si react with the Fe and themselves, however, not with each other. This gives the parameters required to calculate the power supply to the geodynamo as the Earth's core cools. Compositional convection, driven by light O released at the inner-core boundary on freezing, accounts for half the entropy balance and 15 per cent of the heat balance. This means the same magnetic field can be generated with approximately half the heat throughput needed if the geodynamo were driven by heat alone. Chemical effects are significant: heat absorbed by disassociation of Fe and 0 almost nullify the effect of latent heat of freezing in driving the dynamo. Cooling rates below 69 K Gyr(-1) are too low to maintain thermal convection everywhere; when the cooling rate lies between 35 and 69 K Gyr(-1) convection at the top of the core is maintained compositionally against a stabilizing temperature gradient; below 35 K Gyr(-1) the dynamo fails completely. All cooling rates freeze the inner core in less than 1.2 Gyr, in agreement with other recent calculations. The presence of radioactive heating will extend the life of the inner core, however, it requires a high heat flux across the core-mantle boundary. Heating is dominated by radioactivity when the inner core age is 3.5 Gyr. We, also, give calculations for larger concentrations of O in the outer core suggested by a recent estimation of the density jump at the inner-core boundary, which is larger than that of PREM. Compositional convection is enhanced for the higher density jumps and overall heat flux is reduced for the same dynamo dissipation, however, not by enough to alter the qualitative conclusions based on PREM. Our preferred model has the core convecting near the limit of thermal stability, an inner-core age of 3.5 Gyr and a core heat flux of 9 TW or 20 per cent of the Earth's surface heat flux, 80 per cent of which originates from radioactive heating.

The high-pressure melting curve of iron is of major importance to the Earth's sciences, as it provides a close estimate of the temperature of the Earth's core. Despite being studied experimentally for more than a decade and, more recently, using theoretical quantum mechanics techniques, there are still large discrepancies between different groups. In this article, we will describe our theoretical approach to the problem and discuss the reason of the discrepancies with other theoretical calculations. (C) 2004 Elsevier Ltd. All rights reserved.

The Earth's core is largely composed of iron (Fe). The phase relations and physical properties of both solid and liquid Fe are therefore of great geophysical importance. As a result, over the past 50 years the properties of Fe have been extensively studied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (similar to6000 K) found in the core provide a major experimental challenge, and it is not surprising that there are still considerable discrepancies in the results obtained by using different experimental techniques. In the past 15 years quantum mechanical techniques have been applied to predict the proper ties of Fe. Here we review the progress that has been made in the use of first principles methods in the study of Fe, and focus upon (i) the structure of Fe under core conditions, (ii) the high P melting behaviour of Fe, (iii) the thermodynamic properties of hexagonal close-packed (hcp) Fe, and (iv) the rheological and thermodynamic properties of high P liquid Fe. (C) 2003 Elsevier B.V. All rights reserved.

SUMMARY The power required to drive the geodynamo places significant constraints on the heat passing across the core‐mantle boundary and the Earth’s thermal history. Calculations to date have been limited by inaccuracies in the properties of liquid iron mixtures at core pressures and temperatures. Here we re-examine the problem of core energetics in the light of new firstprinciples calculations for the properties of liquid iron. There is disagreement on the fate of gravitational energy released by contraction on cooling. We show that only a small fraction of this energy, that associated with heating resulting from changes in pressure, is available to drive convection and the dynamo. This leaves two very simple equations in the cooling rate and radioactive heating, one yielding the heat flux out of the core and the other the entropy gain of electrical and thermal dissipation, the two main dissipative processes. This paper is restricted to thermal convection in a pure iron core; compositional convection in a liquid iron mixture is considered in a companion paper. We show that heat sources alone are unlikely to be adequate to power the geodynamo because they require a rapid secular cooling rate, which implies a very young inner core, or a combination of cooling and substantial radioactive heating, which requires a very large heat flux across the core‐mantle boundary. A simple calculation with no inner core shows even higher heat fluxes are required in the absence of latent heat before the inner core formed.

The nature of the stable phase of iron in the Earth's solid inner core is still highly controversial. Laboratory experiments 1 suggest the possibility of an uncharacterized phase transformation in iron at core conditions and seismological observations(2-4) have indicated the possible presence of complex, inner-core layering. Theoretical studies(5,6) currently suggest that the hexagonal close packed (h. c. p.) phase of iron is stable at core pressures and that the body centred cubic (b. c. c.) phase of iron becomes elastically unstable at high pressure. In other h. c. p. metals, however, a high-pressure b. c. c. form has been found to become stabilized at high temperature. We report here a quantum mechanical study of b.c.c.-iron able to model its behaviour at core temperatures as well as pressures, using ab initio molecular dynamics free-energy calculations. We find that b.c.c.-iron indeed becomes entropically stabilized at core temperatures, but in its pure state h.c.p.-iron still remains thermodynamically more favourable. The inner core, however, is not pure iron, and our calculations indicate that the b. c. c. phase will be stabilized with respect to the h. c. p. phase by sulphur or silicon impurities in the core. Consequently, a b.c.c.-structured alloy may be a strong candidate for explaining the observed seismic complexity of the inner core(2-4).

Ab initio techniques based on density functional theory in the projector-augmented-wave implementation are used to calculate the free energy and a range of other thermodynamic properties of liquid iron at high pressures and temperatures relevant to the Earth’s core. The ab initio free energy is obtained by using thermodynamic integration to calculate the change of free energy on going from a simple reference system to the ab initio system, with thermal averages computed by ab initio molecular dynamics simulation. The reference system consists of the inverse-power pair-potential model used in previous work. The liquid-state free energy is combined with the free energy of hexagonal close packed Fe calculated earlier using identical ab initio techniques to obtain the melting curve and volume and entropy of melting. Comparisons of the calculated melting properties with experimental measurement and with other recent ab initio predictions are presented. Experiment-theory comparisons are also presented for the pressures at which the solid and liquid Hugoniot curves cross the melting line, and the sound speed and Gruneisen parameter along the Hugoniot. Additional comparisons are made with a commonly used equation of state for high-pressure–high-temperature Fe based on experimental data.

We report phonon densities of states (DOS) of iron measured by nuclear resonant inelastic x-ray scattering to 153 gigapascals and calculated from ab initio theory. Qualitatively, they are in agreement, but the theory predicts density at higher energies. From the DOS, we derive elastic and thermodynamic parameters of iron, including shear modulus, compressional and shear velocities, heat capacity, entropy, kinetic energy, zero-point energy, and Debye temperature. In comparison to the compressional and shear velocities from the preliminary reference Earth model (PREM) seismic model, our results suggest that Earth's inner core has a mean atomic number equal to or higher than pure iron, which is consistent with an iron-nickel alloy.

A general set of methods is presented for calculating chemical potentials in solid and liquid mixtures using {\em ab initio} techniques based on density functional theory (DFT). The methods are designed to give an {\em ab initio} approach to treating chemical equilibrium between coexisting solid and liquid solutions, and particularly the partitioning ratio of solutes between such solutions. For the liquid phase, the methods are based on the general technique of thermodynamic integration, applied to calculate the change of free energy associated with the continuous interconversion of solvent and solute atoms, the required thermal averages being computed by DFT molecular dynamics simulation. For the solid phase, free energies and hence chemical potentials are obtained using DFT calculation of vibrational frequencies of systems containing substitutional solute atoms, with anharmonic contributions calculated, where needed, by thermodynamic integration. The practical use of the methods is illustrated by applying them to study chemical equilibrium between the outer liquid and inner solid parts of the Earth's core, modelled as solutions of S, Si and O in Fe. The calculations place strong constraints on the chemical composition of the core, and allow an estimate of the temperature at the inner-core/outer-core boundary., Comment: 19 pages, two figures

The free energy and other thermodynamic properties of hexagonal-close-packed iron are calculated by direct ab initio methods over a wide range of pressures and temperatures relevant to the Earth’s core. The ab initio calculations are based on density-functional theory in the generalized-gradient approximation, and are performed using the projector augmented wave approach. Thermal excitation of electrons is fully included. The Helmholtz free energy consists of three parts, associated with the rigid perfect lattice, harmonic lattice vibrations, and anharmonic contributions, and the technical problems of calculating these parts to high precision are investigated. The harmonic part is obtained by computing the phonon frequencies over the entire Brillouin zone, and by summation of the free-energy contributions associated with the phonon modes. The anharmonic part is computed by the technique of thermodynamic integration using carefully designed reference systems. Detailed results are presented for the pressure, specific heat, bulk modulus, expansion coefficient and Gruneisen parameter, and comparisons are made with values obtained from diamond-anvil-cell and shock experiments.

Ab initio molecular dynamics calculations, based upon density functional theory within the generalised gradient approximation (GGA) using ultrasoft non-norm conserving Vanderbilt pseudopotentials, have been used to predict the transport properties of liquid Fe–S. In order to compare our simulations with experimental data, the simulations were performed for the eutectic composition of Fe–S at the experimentally accessible conditions of 5 GPa and of 1300 and 1500 K. Our results give values for Fe and S diffusion of a few times 10−5 cm2 s−1. Our calculated viscosities, obtained directly from the simulations, are 11±5 and 4±1 mPa s at 1300 and 1500 K, respectively. Our calculated diffusion and viscosity coefficients agree well with recent experiments at similar pressures and temperatures, supporting a high diffusivity and low viscosity in liquid Fe–S at temperatures up to a few hundred Kelvin above the eutectic temperature. Furthermore, an extensive study of the liquid structure shows no evidence for sulphur polymerisation or the existence of any large viscous flow units. These results are in direct conflict with the previously reported experimental results of Le Blanc and Secco [LeBlanc, G.E., Secco, R.A., 1996. Viscosity of an Fe–S liquid up to 1300°C and 5 GPa, Geophys. Res. Lett., 23, 213–216.].

First principles electronic structure calculations based on density functional theory have been used to study the thermodynamic, structural and transport properties of solid solutions and liquid alloys of iron and oxygen at Earth's core conditions. Aims of the work are to determine the oxygen concentration needed to account for the inferred density in the outer core, to probe the stability of the liquid against phase separation, to interpret the bonding in the liquid, and to find out whether the viscosity differs significantly from that of pure liquid iron at the same conditions. It is shown that the required concentration of oxygen is in the region 25-30 mol percent, and evidence is presented for phase stability at these conditions. The Fe-O bonding is partly ionic, but with a strong covalent component. The viscosity is lower than that of pure liquid iron at Earth's core conditions. It is shown that earlier first-principles calculations indicating very large enthalpies of formation of solid solutions may need reinterpretation, since the assumed crystal structures are not the most stable at the oxygen concentration of interest., 21 pages, 12 figures

The inferred density of the solid inner core indicates that it is predominantly made of iron. In order to indicates that it is predominantly made of iron. In order to interpret the observed seismic anisotropy and understand the high pressure and temperature behaviour of the core, it is essential to establish the crystal structure of iron under core conditions. On the basis of extrapolated experimental data, a number of candidate structures for the high PIT iron phase have been proposed, namely, body-centred cubic (bcc), body-centred tetragonal (bct), hexagonal close-packed (hcp), double-hexagonal close-packed (dhcp) and an orthorhombically distorted hcp polymorph (Matsui, 1993; Stixrude and Cohen, 1995; Boehler, 1993; Saxena et al., 1996; Andrault et al., 1997). Here we present the results of the first fully ab initio free energy calculations for all of these polymorphs of iron at core pressures and temperatures. Our results show that hcp-Fe is the most stable polymorph of iron under the conditions of the Earth's inner core.

The pressure as a function of volume and temperature has been investigated for B2-type NaCl over the pressure range of 20-360 GPa and at temperatures between 300 and 3000 K. The simulations were performed using ab initio molecular dynamics method within the density-functional theory framework. A Vinet equation of state fitted to the 300 K data yielded a bulk modulus of B(Ta)=128.66 GPa and a pressure derivative of B(Ta)'=4.374 at standard state pressure of 30 GPa. The thermal pressure contribution was determined to be of the form Delta P(th)=[alpha B(T)(V(a))+(partial derivative B(T)/partial derivative T)(V) ln(V(a)/V)]Delta T. When alpha B(T)(V(a)) is assumed to be constant, the fit to the data yielded alpha B(T)(V(a))=0.0033 GPa/K at standard volume, corresponding to the pressure of 30 GPa. In contrast, the volume dependence of the thermal pressure was very small, and fitting yielded (partial derivative B(T)/partial derivative T)(V)=0.000 87. (c) 2008 American Institute of Physics.