Surface Vibrational Modes in Crystal Lattices with Complex Interatomic Interactions

The investigation of surface vibrational modes in crystal lattices is complicated by the necessity for satisfying the free boundary conditions at the surface if the interatomic interactions include next-nearest neighbor interactions, next-next-nearest neighbor interactions, etc. The number of equations specifying the free boundary conditions may then become rather large. A general method is presented in this paper for the investigation of the normal modes of one-, two-, and three-dimensional lattices which are finite or semi-infinite in one dimension and have free boundary surfaces, assuming interatomic interactions of various ranges. The method is illustrated by calculations of the normal modes for the finite and semi-infinite diatomic linear chains with nearest and next-nearest neighbor interactions. Some remarks are made regarding the applications of the mathematical techniques to the evaluation of certain continuant determinants of large order.