Temperature and Pressure Derivatives of Elastic Constants With Application to the Mantle

Abstract

The temperature and pressure derivatives of the elastic moduli M of solids can be cast into the form of dimensionless logarithmic anharmonic {DLA} parameters, ∂ ln M/∂ In p = {M} at constant temperature, pressure, or entropy (T, P, S), where p is the density. These parameters show little variation from material to material and are expected to show little variation with temperature at high temperature. Most of the available derivative data for ionic solids has been renormalized and analyzed for dependency on ion type, crystal structure, and other parameters. The {DLA} parameters exhibit little variation and little correlation with crystal structure for most close-packed halides and oxides. There are small systematic variations with ionic radius, Gruneisen's γ, and the bulk modulus-rigidity ratio (K/G). Temperature and pressure derivatives are correlated because of the importance of the volume-dependent, or extrinsic, terms. The intrinsic terms {K}_v and {G}_v are also highly correlated, even for open-packed structures, where ∂ ln M/α∂T = {M}v. These correlations make it possible to estimate the derivatives of high-pressure phases. The spinel forms of olivine are predicted to have "normal" derivatives, and therefore the magnitude of the modulus or velocity jump associated with the olivine-spinel transition near 400 km should be similar to that measured in the laboratory. The actual size of the 400-km discontinuity is much less, indicating the presence of substantial quantities of minerals other than olivine in the upper mantle or transition region. Recent calculations in apparent support of a homogeneous olivine-rich (>60%) mantle are based on choices for the derivatives of β- and γ-Mg_(2)SiO_4, which are unlike other ionic crystals. There is no evidence that these phases should be anomalous in their physical properties. The temperature and pressure derivatives of ionic crystals depend on the nature of the ions and their coordination.

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