Wave vector spectrum of the vibration modes in a finite latticet

Abstract

The vibration problem of an N × N finite lattice is solved up to N = 24. The calculations converge rapidly to give a theoretical result. The independent harmonic modes are standing wave phonons, which can only be classified according to species. The dynamics of molecules and crystals are unified and coincide with the continuous elastic media, but the theory also reflects the fundamental differences of microscopic and macroscopic motions. In a finite lattice, every type of phonon is related to a spectrum of wave vectors except the translation mode. Finite lattice dynamics yields a rotation mode together with many inner rotation modes, and leads to a physical picture of the surface wave phonon. Numerical computation shows that strengthening of the covalent property leads to the appearance of a boundary coupling soft mode. The mode patterns and wave vector spectra were drawn by computer.