In this chapter we are concerned with the spectrum of characteristic vibrations of a crystalline solid. This subject leads to a consideration of the conditions for wave propagation in a periodic lattice, the energy content and specific heat of lattice waves, the particle aspects of quantized lattice vibrations (phonons), and the consequences of anharmonic coupling between atoms. These topics form a significant part of solid state physics, and their discussion additionally introduces us to the concepts of permitted and forbidden frequency ranges, concepts which will be encountered again in connection with electronic spectra of solids.

The zero-point energy and thermal energy of a solid are manifest in incessant complicated vibrations of the atoms. These vibrations have Fourier components at a variety of frequencies. Additional motion is superimposed if the solid is stimulated by some external source, and we usually assume that the principle of superposition applies to the sum of these motions, i.e., we assume that the effect of several disturbances is found by simply adding them together. This assumption sounds plausible, provided that we remain in the linear region (or region of elastic deformation) such that the restoring force on each atom is approximately proportional to its displacement (Hooke's Law). As we shall see in discussing thermal conductivity, there are some effects of nonlinearity or “anharmonicity” even for very modest atomic displacements.