On the Strain Dependence of the Vibrational Frequency Distributions of Simple Lattices

Expressions are derived for the first even moments ⟨ω 2 ⟩ and ⟨ω 4 ⟩ of the normal frequency distribution of a rigid system of particles interacting with central forces. These are used to obtain the strain dependence of ⟨ω 2 ⟩ and ⟨ω 4 ⟩ for any cubic lattice whose atoms all occupy centres of symmetry, up to the second order in the homogeneous strain parameters. By a different method the strain dependence of ⟨ω 6 ⟩ is derived for the special case of a face-centred cubic lattice with a potential ϕ( r ) between nearest neighbours, under the conditions of static equilibrium. The results are applied first to a face-centred cubic lattice with a 6-12 potential between all neighbours, and used to derive values of ω D ( n ) = [1/3( n + 3) ⟨ω n ⟩] 1/ n for n = 2 and n = 4. In earlier approximations the strain dependence of ln ω D ( n ) has been assumed not to vary with n , but the present results show that this is invalid for shear strain, and hence for calculating the elastic constants c 44 and c 11 - c 12 . Neglect of the interaction between all except nearest neighbours only slightly alters the strain dependence of ln ω D ( n ), and calculations with a more general potential ϕ( r ) are therefore made only with nearest neighbour interaction. Strong variation with n (indicating strong distortion of the shape of the frequency distribution under strain) is found to be always present for shear strain unless the cubic term in the pair potential is small. Results for n = 1 and n = 0, required for calculations of thermoelastic properties at T = 0 and at very high temperatures respectively, can be estimated by extrapolation from n = 6, 4 and 2. In particular, the contribution of the zero-point energy to the elastic constants can be determined to within about 10%, provided that the dominant forces are central forces between nearest neighbours.