The Green Function and Its Use in the Derivation of Additional Boundary Conditions for a Semi-Infinite System

Abstract

A new method is proposed for the calculation of the Green function for semi-infinite systems of coupled harmonic oscillators. For a model of the one-dimensional chain the expression of the Green function is derived, which is exact for arbitrarily long-range interactions between oscillators. It demonstrates clearly the absence of surface modes irrespective of the nature of the coupling. The Green function, used as the dielectric susceptibility of the model system, is shown to reproduce the additional boundary conditions (ABC) of Mahan and Obermair. The ABC for the q²-spatial dispersion are also discussed and the importance of the exact Green function is emphasized.