Seismological Applications of Lattice Theory

Citation

Sammis, Charles George (1971) Seismological Applications of Lattice Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/S6B1-6N30. https://resolver.caltech.edu/CaltechTHESIS:12192012-162926776

Abstract

Lattice models based upon empirical two-body potential functions are used to predict the elastic constants of “mantle-candidate” minerals at high pressures for direct comparison with seismic velocity profiles. The method of long waves, originally formulate d by Born and his coworkers, has been applied to solids in the rock salt, spinel, and rutile structures. Calculations for NaCl (rock salt), MgO (rock salt), Al_2MgO_4 (spinel), and TiO_2 (rutile) are compared with recent high-precision ultrasonic data. The effect of van der Waals forces and second-neighbor anion-anion interactions is shown to be small. The NaCl and MgO data are best fit with an exponential cation-anion repulsive potential. The elastic constants of MgO cannot be well fit unless the ionicity (valence product) is lowered to 0.7 of its full ionic value. For NaCl this is not required. The shear instability (C_(44) = 0) is predicted for both NaCl and MgO, but the exact pressure is sensitive to the details of the potential.

Using the Mg-O two-body potential found for periclase, Al_2MgO_4 spinel was investigated using only two pieces of input datum, K and ρ. Although the predicted elastic constants were in good agreement with the data, the pressure derivatives were not. The discrepancy is caused by a large contribution from the internal deformations which occur in all non-centro symmetric structures. The same result was found for TiO_2. A relaxation of the rigid-ion and central-force approximations may correct this discrepancy.

Using the Mg-O bond parameters found for periclase and the Si-Q bond parameters found from K and ρ of stishovite, the elastic properties of the high-pressure polymorph δ–Mg_2SiO_4 spinel were predicted. The predicted equilibrium density was in agreement with previous experimental extrapolations; the predicted μ parameter was in agreement with prior estimates based on bond-length arguments, and the predicted bulk modulus was in agreement with prior systematics estimates. However, the internal deformation contribution again dominated the pressure derivatives and caused both the predicted V_p and V_s to be lower than the corresponding seismic velocities in the "spinel region" of the mantle. A comparison of MgO (rock salt) and SiO_2 (stishovite) with the seismic profiles for the "post-spinel" lower mantle shows a discrepancy in both absolute value and gradient. Unlike the silicate spinel, this is not obviously caused by the internal deformations. The lattice models predict that both TiO_2 and stishovite will become unstable in shear (1/2 (C_(11) – C_(12) = 0) at high pressure.

Other methods of using laboratory data to interpret seismic profiles are reviewed. Birch's formulation of isotropic finite strain theory is corrected and used to test the homogeneity and adiabaticity of the lower mantle of recent earth-inversion models. Systematics are shown to be insufficient to treat the shear properties. Although lattice models are limited by empirical approximations to the complex bonding forces, the empiricism is on a more basic level than that of velocity density systematics previously used to interpret seismic profiles. By using lattice models, one gains the natural dependence of both the compressional and shear properties on the crystal structure.

Thesis Files