188.8.131.52.2 Dynamical matrix and normal modes of vibration
This chapter discusses dynamical matrix and normal modes of vibration. The vibrations of the atoms in a crystal slab can be analyzed by expanding the potential energy in a power series in the atomic displacement components which specify the displacement of an atom from its equilibrium position. The harmonic coupling constants and cubic anharmonic coupling constants must satisfy the conditions of infinitesimal translational invariance and infinitesimal rotational invariance as well as conditions imposed by the point group symmetry of the crystal slab. The equations of motion for the displacements are complicated by two factors. First, there is no periodicity of the atomic sites in the direction normal to the surface. Second, the coupling constants are not necessarily the same as the corresponding coefficients for the infinite crystal. One finds that the normal modes can be classified according to the behavior of the amplitudes as the center of the slab is approached, provided that the slab is sufficiently thick. For surface modes, the amplitudes decrease dramatically (essentially exponentially) as the center of the slab is approached. For bulk modes, on the other hand, there is no dramatic decrease in the amplitudes as the center is approached.