Composition and temperature of the Earth's core constrained by combining ab initio calculations and seismic data

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, O 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. ß 2002 Elsevier Science B.V. All rights reserved. Keywords: chemical composition; core; seismology; simulation Ever since the discovery of the Earth's core 90 years ago, its composition and temperature have been controversial. The work of Birch [1] and others in the 1950s made it clear that the core consists mainly of iron with a minor fraction of nickel, but that unidenti¢ed light impurities must also be present, since the density of the core is signi¢cantly lower than that of iron: modern estimates [2^5] indicate that the outer core is lower in density by 5^10%, and the inner core by 2^3%. The main obstacle to further progress has been the lack of reliable thermodynamic data for the relevant iron alloys under core conditions, and it has been stressed recently [6] that ab initio calculations of thermodynamic functions can make an important contribution. We show here how ab initio calculations of the chemical potentials of the main impurity candidates can provide strong new constraints on both the composition and the temperature of the core, and we present numerical values both for the impurity mol fractions in the inner and outer core and for the temperature at the inner-core/outer-core boundary (ICB). 0012-821X / 02 / $ ^ see front matter ß 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 2 8 2 1 X ( 0 1 ) 0 0 5 6 8 4 * Corresponding author. Fax: +44-20-7679-1360. E-mail address: [email protected] (M.J. Gillan). EPSL 6066 22-1-02 Earth and Planetary Science Letters 195 (2002) 91^98 www.elsevier.com/locate/epsl Cosmochemical abundances of the elements, combined with models of the Earth's history, limit the possible impurities to a few candidates [2]. The most often discussed are S, O and Si, and we shall con¢ne ourselves to these three here, though C and H have sometimes been proposed [2]. Our strategy for constraining the impurity fractions and the temperature is based on the fact that the solid inner core has grown from the liquid outer core over many hundreds of millions of years, so that the solid and liquid should be in thermodynamic equilibrium at the ICB. This implies that the chemical potentials of Fe and of each impurity must be equal on the two sides of the ICB. If the core consisted of pure Fe, equality of the chemical potential (Gibbs free energy in this case) would tell us only that the temperature at the ICB is equal to the melting temperature Tm of Fe at the ICB pressure of 330 GPa. With impurities present, equality of the chemical potentials for each impurity element imposes a relation between the mol fractions in the liquid and the solid, so that with S, O and Si we have three such relations. But these three relations must be consistent with the accurate values of the mass densities in the inner and outer core deduced from seismic and free-oscillation data [7^9]. We shall show that ab initio results for the densities and chemical potentials in the liquid and solid Fe/S, Fe/O and Fe/Si alloys determine with useful accuracy the mol fraction of O and the sum of the S and Si mol fractions in the outer and inner core, as well as giving the temperature at the ICB. The chemical potential WX of a solute X in a solid or liquid solution is conventionally expressed as WX =W 2 X+kBT ln aX, where W 2 X is a constant and aX is the activity [10]. To re£ect the fact that aX becomes equal to the mol fraction cX in the dilute limit cXC0, it is common practice to write aX = QX cX, where the activity coe¤cient QX has the property QXC1 as cXC0. The chemical potential can therefore be expressed as: WX ˆ W 2X ‡ kBT ln…Q XcX† ˆ W 2X ‡ kBT ln Q X‡ kBT ln cX …1† which we rewrite as: WX ˆ ~ WX ‡ kBT ln cX …2† where ~ WX = ~ W 2 X+kBT ln QX. It is helpful to focus on the quantity ~ WX for two reasons: ¢rst, because it is a convenient quantity to obtain by ab initio calculations [6] ; second, because at low concentrations the activity coe¤cient QX will deviate only weakly from unity by an amount proportional to cX, and by the properties of the logarithm the same will be true of ~ WX. By Eq. 2, equality of the chemical potentials WX and WX in coexisting liquid and solid (superscripts l and s, respectively) then requires that: ~ W l X ‡ kBT ln cX ˆ ~ W X ‡ kBT ln cX …3†