Second order elasticity theory: An improved formulation of the Grüneisen parameter at high pressure

The second order theory of elasticity, in which terms to second order in strain are retained in calculating atomic bond length changes and elastic moduli, is extended to describe thermal vibration of a face-centred cubic crystal. Coupled with equations relating the pressure dependences of elastic constants, this yields a new formulation of the thermal Grüneisen parameter γ in terms of pressure P, incompressibility K and rigidity, μ

$\text{λ}\text{1}\text{2}\text{d}\text{K}\text{d}\text{P}\text{?}\text{1}\text{2}\text{+}\text{1}\text{9}\text{P}\text{K}\text{?}\text{1}\text{3}\text{?}\text{1}\text{9}\text{P}\text{K}\text{f}\text{1?}\text{2}\text{3}\text{P}\text{K}\text{?}\text{2}\text{3}\text{P}\text{K}\text{f}$

where f = 24 (3 K ? 2 P)/(3 K + 115 μ + 90 P). The factor f arises from bond interactions and the case f = 1, representing independent bonds (no interactions), yields the free-volume γ- Since we have shown earlier that the second order elasticity theory provides a convincing explanation of the elasticity of the inner core, we believe that the new formula is appropriate for the inner core. It is, however, inadequate to describe the lower mantle γ, in which atomic bond angle rigidity, not considered here, may be appreciable.