The theory of elasticity and of wave-propagation in crystals from the atomistic standpoint

Abstract

The static method of obtaining the strain energy function of a crystal has been developed for the case where the potential energy of the entire lattice is a general quadratic in the nuclear displacements of the atoms of the crystal. It is shown that for heterogeneous strains, the deformation energy of the crystal is a quadratic in all the nine strain components. For small homogeneous deformations, the strain energy function reduces to the form of the corresponding function in the elasticity theory. The wave equations obtained from this energy function by the variational procedure are identical with the wave equations of the elasticity theory. By comparing either the two forms of the strain energy function or the two sets of waveequations, expressions for the elastic constants can be obtained in terms of the atomic force constants. Starting from the most general expression for the strain energy function which is a quadratic in all the nine strain components and by assuming that the strain components are all linearly independent functions of the position vector of any point of the solid, the wave equations of Begbie and Born have been derived by means of Hamilton’s variational principle. But these equations are not reducible to the symmetric form of the wave equations of the elasticity theory without further assumption of additional relations among the force constants. Since there is no justification for such new relations which restrict the generality of the force scheme used, the expressions of Begbie-Born and of Kun Huang for the elastic constants of crystals are not valid in a general force scheme. The expressions for the elastic constants which follow the different theoretical procedures are derived for the case of diamond and are compared with the experimental results. Finally, a cubic equation whose roots determine the limiting group velocities of the long acoustic waves travelling in any direction of the crystal, has been derived; this replaces the expression for the velocities of the long acoustic waves given in an earlier paper by the author.