Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications

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Abstract

Lattice dynamical considerations and a Born repulsive potential between atoms lead to equations for the elastic constants of the cubic NaCl, CsCl and ZnS lattices as a function of compression. The NaCl lattice is unstable for n < 4.6 and the CsCl lattice is unstable for n < 7.2, where n is the exponent of the repulsive power law. The pressure derivatives of the shear constants (c′, c44) show a strong dependence upon crystallographic structure; for the ZnS lattice both dc′/dP and dc44/dP are negative for all reasonable values of n. The theoretical values of dcij/dP compare favorably with experimental results. For the isotropic shear modulus (μ), dμ/dP is determined by Poisson's ratio and the coordination of the ions in the lattice. Low (and possibly negative) values of dμ/dP are likely for several materials of importance to geophysics; such values would make low-velocity zones possible and interpretation of shock-wave data difficult. Vanishing of a shear constant predicts phase transitions at compression (V/V0) of 0.95 for the ZnS lattice, 0.75 for the NaCl lattice, and 0.60 for the CsCl lattice. The CsCl transition is predicted in spite of the fact that none of the elastic constants have a negative pressure derivative at zero pressure. The equation of state parameters, K0and (dK/dP)0(where K is the bulk modulus), are almost independent of crystallographic structure. K0arises entirely from the Laplacian of the repulsive potential. (dK/dP)0= 1 3(n+7) so that the stability limits on n make it unlikely that (dK/dP)0< 3.5 for the NaCl lattice or less than 5 for the CsCl lattice. © 1970.

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Anderson, O. L., & Liebermann, R. C. (1970). Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications. Physics of the Earth and Planetary Interiors, 3(C), 61–85. https://doi.org/10.1016/0031-9201(70)90045-2

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