Dynamics of a Semi-Infinite Crystal Lattice in a Quasiharmonic Approximation. II. The Normal-Mode Analysis of a Semi-Infinite Lattice
This is the second of two papers dealing with the dynamics of semi-infinite crystal lattices. The present paper is concerned with the small-amplitude vibrations of the atoms in a semi-infinite crystal lattice, with a free boundary, about their actual static equilibrium positions. The static equilibrium configuration (at zero temperature) of such a lattice was discussed in the first paper. The normal modes of the semi-infinite lattice are derived in a harmonic approximation. These modes are represented as linear combinations of elementary bulk modes and elementary surface waves, the latter having a wave vector whose component normal to the surface is complex. All the elementary modes which specify a given normal mode have the same frequency, and the same two-dimensional reduced wave vector kρ. The normal modes are again classified into "bulk modes" and "surface modes." The latter represent a vibrational state, consistent with a free boundary, in which the displacements decrease essentially exponentially with the distance from the free boundary. The dispersion relation of the surface modes is discussed in some detail. The theory is developed in terms of the complex vibrational energy-band structure. The approach is related to Heine's analysis of generalized Bloch functions in terms of the complex electronic energy-band structure.