Lattice models based upon empirical two-body potential functionsare used to predict the elastic constants of “mantle-candidate”mineralsat 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 rutilestructures. Calculations for NaCl (rock salt), MgO (rock salt), Al_2MgO_4(spinel), and TiO_2 (rutile) are compared with recent high-precisionultrasonic data. The effect of van der Waals forces and second-neighboranion-anion interactions is shown to be small. The NaCl and MgO dataare best fit with an exponential cation-anion repulsive potential. Theelastic constants of MgO cannot be well fit unless the ionicity (valenceproduct) is lowered to 0.7 of its full ionic value. For NaCl this is notrequired. The shear instability (C_(44) = 0) is predicted for both NaCl andMgO, but the exact pressure is sensitive to the details of the potential.
Using the Mg-O two-body potential found for periclase, Al_2MgO_4spinel was investigated using only two pieces of input datum, K and ρ.Although the predicted elastic constants were in good agreement with thedata, the pressure derivatives were not. The discrepancy is caused bya large contribution from the internal deformations which occur in allnon-centro symmetric structures. The same result was found for TiO_2.Arelaxation of the rigid-ion and central-force approximations may correctthis discrepancy.
Using the Mg-O bond parameters found for periclase and theSi-Q bond parameters found from K and ρ of stishovite, the elasticproperties of the high-pressure polymorph δ–Mg_2SiO_4 spinel werepredicted. The predicted equilibrium density was in agreement withprevious experimental extrapolations; the predicted μ parameter wasin agreement with prior estimates based on bond-length arguments, andthe predicted bulk modulus was in agreement with prior systematicsestimates. However, the internal deformation contribution againdominated the pressure derivatives and caused both the predicted V_pand V_s to be lower than the corresponding seismic velocities in the"spinel region" of the mantle. A comparison of MgO (rock salt) andSiO_2 (stishovite) with the seismic profiles for the "post-spinel" lowermantle shows a discrepancy in both absolute value and gradient. Unlikethe silicate spinel, this is not obviously caused by the internal deformations.The lattice models predict that both TiO_2 and stishovite willbecome unstable in shear (1/2 (C_(11) – C_(12) = 0) at high pressure.
Other methods of using laboratory data to interpret seismicprofiles are reviewed. Birch's formulation of isotropic finite straintheory is corrected and used to test the homogeneity and adiabaticityof the lower mantle of recent earth-inversion models. Systematics areshown to be insufficient to treat the shear properties. Although latticemodels are limited by empirical approximations to the complex bondingforces, the empiricism is on a more basic level than that of velocitydensity systematics previously used to interpret seismic profiles. Byusing lattice models, one gains the natural dependence of both the compressionaland shear properties on the crystal structure.
