Vibrational Dynamics within the Embedded-Atom-Method Formalism and the Relationship to Born-von-K\'arm\'an Force Constants.

We derive expressions for the dynamical matrix of a crystalline solid with total potential energy described by an embedded-atom-method (EAM) potential. We make no assumptions regarding the number of atoms per unit cell. These equations can be used for calculating both bulk phonon modes as well the modes of a slab of material, which is useful for the study of surface phonons. We further discuss simplifications that occur in cubic lattices with one atom per unit cell. The relationship of Born-von-K\'arm\'an (BvK) force constants - which are readily extracted from experimental vibrational dispersion curves - to the EAM potential energy is discussed. In particular, we derive equations for BvK force constants for bcc and fcc lattices in terms of the functions that define an EAM model. The EAM - BvK relationship is useful for assessing the suitability of a particular EAM potential for describing vibrational spectra, which we illustrate using vibrational data from the bcc metals K and Fe and the fcc metal Au.