Applications of liquid state physics to the Earth's core

By use of the modern theory of liquids and some guidance from the hard-sphere model of liquid structure, the following new results have been derived for application to the Earth's outer core. (1) d K/ dP ⋍ 5 - 5. 6P / K, where K is the incompressibility and P the pressure. This is valid for a high-pressure liquid near its melting point, provided that the pressure is derived primarily from a strongly repulsive pair potential φ. This result is consistent with seismic data, except possibly in the lowermost region of the outer core, and demonstrates the approximate universality of d K/d P proposed by Birch (1939) and Bullen (1949). (2) dln T_M/dln ρ = ( γC_V - 1)/( C _V - 3/2), where T _M is the melting point, ρ the density, γ the atomic thermodynamic Grüneisen parameter and C_V the atomic contribution to the specific heat in units of Boltzmann's constant per atom. This reduces to Lindemann's law for C_V = 3 and provides further support for the approximate validity of this law. (3) It follows that the "core paradox" of Higgins and Kennedy can only occur if γ < 2/3. However, it is shown that γ < 2/3 ⇔ ∫ ^∞₀ (∂g/∂T) _ρ r( d/ dr)(r ² φ) dr > 0 , which cannot be achieved for any strongly repulsive pair potential φ and the corresponding pair distribution function g. It is concluded that γ > 2/3 and that the core paradox is almost certainly impossible for any conceivable core composition. Approximate calculations suggest that γ ∼ 1.3-1.5 in the core. Further work on the thermodynamics of the liquid core must await development of a physically realistic pair potential, since existing pair potentials may be unsatisfactory.