The perturbation treatment of the deformation of a crystal

Abstract

A static deformation of a crystal can be treated as a quantum mechanical perturbation. It is shown here how the Hamiltonian of the deformed system may be regarded as how the Hamiltonian of the deformed system may be the perturbed form of the harmonic Hamiltonian, anharmonic terms being included in the perturbation. The usual second quantisation and diagram techniques may then be used to develop expressions for the stress tensor, elastic constants, internal strain, electric polarisation and piezoelectric coefficients. Comparison is made with an earlier standard lattice treatment, which does not obtain the correct index symmetries of the above physical quantities when they are described as cartesian tensors, because the treatment includes inconsistencies with the rotational invariance property of the interaction potential function. The present treatment is correct is this respect and so correction terms to the coefficients of the earlier treatment are obtained. The momentum contribution of the perturbation term is also included directly for the first time, and it can be easily evaluated. The perturbation term is rigorously expanded WRT the elements of the Lagrangian finite strain tensor eta . This expansion automatically displays all the required index symmetries. It also vanishes trivially for rotation since eta then vanishes, a property that cannot easily be displayed when the perturbation is expanded WRT the finite displacement gradient u. The theory also includes the evaluation of the static lattice contribution to the stress tensor in a natural way, it being well known that this term cannot be expressed in terms of the usual lattice potential coefficients. Rotational invariance conditions on the lattice potential coefficients are discussed and general sets of these are derived.