The quantum theory of crystal deformation: the harmonic approximation and rotational invariance

Abstract

In order that such thermodynamic quantities as the stress tensor, elastic constants and piezoelectric coefficients have the correct index symmetries when evaluated theoretically, it is necessary that the theory be consistent with the rotational invariance property of the particle interaction potential function. In this work, the conditions under which the harmonic lattice model is consistent with this property are found and it is shown that the harmonic lattice potential coefficients must then have a certain simple form. In general, the usual harmonic model does not satisfy these conditions. The dynamical properties of a rotationally invariant harmonic model are described. It is shown rigorously for such a model that on deformation the phonon spectrum is unchanged and that there is no internal strain, so that the stress tensor and elastic constants are temperature independent. The rotationally invariant harmonic model is used to provide a zero-order Hamiltonian for the treatment of the deformation of an anharmonic crystal. This approach, besides being completely consistent with the rotational invariance property, has the advantage that the momentum perturbation term, recently introduced into the theory of deformation, is completely absorbed into the zero-order Hamiltonian. The perturbation diagram treatment is thus simplified. By summing over an infinite number of a certain type of diagram, a new formula is obtained for the temperature-independent internal strain contribution to the second-order elastic constants of the ordinary (that is, not necessarily rotationally invariant) harmonic model of common use. The consequences of crystal symmetry for this model are discussed.