Unified Perturbation Treatment for Phonons in Metallic Covalent and Ionic Crystals

An interpolation formula which interpolates the exact Hartree expression for the electronic part of the dynamical matrix between the limits of strong and weak crystal potentials and which becomes rigorous in the zero wave vector limit is used to avoid to the problems connected with the inversion of the screening tensor. In the ionic part of the dynamical matrix the Gaussian charge densities of the usual Ewald technique are replaced by the true charge densities around each ion in the unperturbed crystal. The arising expressions are transformed in such a way that the compensation with the corresponding terms in the electronic part can be done explicitely. As a result one gets an alternative formula for the dynamical matrix of simple metals, a practicable formula for transition and noble metals and covalent crystals, and a rather simple expression for ionic crystals which has some link to the physical picture of a shell model. It is shown that the given theory describes the qualitatively correct pressure dependence of the phonon frequencies.