Elastic Constants of Alkali Halide Crystals

Abstract

ADOPTING the simple Bom model for an alkali halide crystal, in which the ions are regarded as held in their respective positions by the electrostatic and the repulsion interactions between the ions, one may readily calculate the velocities of propagation v of long acoustic waves of wave-length λ along the principal directions [100], [110] and [111], and hence along any direction, in the crystal: where m₁ and m₂ are the masses of the two ions, 2πΔ_1n/λ denotes the difference in phase of the acoustic wave at any two ions 1 and n, and σ_n denotes summation over all the ions, except 1, in the crystal. (If D is the distance up to which the electrostatic interactions remain significant, λ is regarded as sufficiently large in comparison with D that cos(2πD/λ) can be put equal to 1 − ½(2πD/λ)².) where ψ(R_1n) is the interaction energy between the two ions 1 and n, separated by a distance R_1n, and ψ′ and ψ″ are its differential coefficients with respect to R_1n at its equilibrium value R_1n⁰. Q is the cosine of the angle between R_1n⁰ and the direction of displacement of the ions under the acoustic wave. Since the values of v² for different directions of propagation and corresponding directions of displacement are also known in terms of the elastic constants c₁₁, c₁₂ and c₄₄, we obtain, by comparing them with the values obtained from (1), expressions for the elastic constants in terms of the interactions between the ions. We shall denote by ɛ₁₁, ɛ₁₂, ɛ₄₄ the contributions to C₁₁, c₁₂, c₄₄ respectively from the electrostatic interactions, which are of long range, and by ρ₁₁, ρ₁₂, ρ₄₄ the contributions from repulsion interactions, which are assumed to fall exponentially with the increase in the distance of separation of the interacting ions. In view of the short range of these repulsion interactions, we regard them as confined to the immediate neighbours only.