Lattice Vibrations in Finite Systems with Boundary Conditions Given as the Extrapolation Lengths: I. Continuum Model

Lattice vibrations in finite systems with the boundary conditions given as the extrapolation lengths are studied on the basis of the continuum model. The system may be symmetric or asymmetric regarding the extrapolation lengths, which can be positive, negative, or of mixed signs. In general, even cosine modes and odd sine modes appear alternately from the low-frequency side, but the lowest cosine mode changes to the cosh mode and the second-lowest sine mode may change to the sinh mode when the extrapolation lengths at both boundaries are negative. On the other hand, it is not possible to assign the sin (sinh) mode or cos (cosh) mode to the lowest-frequency mode in the case of mixed-sign extrapolation lengths. Orthogonality between the normal modes can be proved using the boundary conditions in spite of the fact that the wavenumbers K-i of the normal modes are mutually incommensurate.