Theoretical properties of cubic crystals at arbitrary pressure—II. Shear moduli

Abstract

G eneral expressions are derived for calculating the shear moduli μ and μ' at arbitrary pressure for cubic crystals in which the interatomic interaction energies are modelled by pairwise functions Ø. Under hydrostatic loading, k−2/3μ = μ' + 2P (where k is the bulk modulus and P is the pressure); this relation is, in effect, a generalization of the Cauchy condition ( C 12 = C 44), applicable for such crystals at non-zero pressure. Calculations of the shear moduli are carried out with the aid of a particular family of functions Ø, and a systematic investigation is made of the influence of the crystal structure and the characteristics of the potential upon the relative elasticity at arbitrary pressure. In particular, detailed computations are made (i) for the three cubic Bravais lattices, (ii) for lattices with varying degrees of anharmonicity (as characterized by a “potential range parameter” β, and (iii) over a wide range of all-round stretch λ. For large values of β, the elastic behaviour of the lattice approaches that of a collection of “hard spheres” with nearest neighbour interactions only; there exists a lower limit to the value of β for which the behaviour of the crystal, in a sense, approaches that of a continuum. Calculations are also made of derivatives d N μ(λ)/ dλ N and d N μ'(λ)/ dλ N, at zero pressure, and these “higher order moduli” are used in forming series expansion approximations to the functions μ(λ) and μ'(λ) at non-zero pressure. Finally, comparisons are made among the approximate values of μ(λ) and μ'(λ), determined from successively higher order series expansions (i.e. with the coefficients based on the zero pressure, higher order moduli), and the “exact” values evaluated in the current state.