The atoms of a crystal execute small, oscillatory motions about their equilibrium positions called lattice vibrations. These vibrations are stimulated by thermal energy or by external agents such as electromagnetic and mechanical forces. As with molecular vibrations, the atomic motions of the lattice can be expressed as linear combinations of the normal modes of motion. Classically, the energy contained in a given normal mode is unrestricted. In quantum theory the energy in a normal mode is quantized in discrete units of ħω. A quantum (ħω)ofenergyin a normal mode of vibration is called a phonon. More loosely, the lattice vibration wave in a crystal is also called a phonon.

Because of the translation symmetry of an (infinite) crystal the normal modes are characterized by a wavevector, k. In the case of lattice vibrations we associate a vector with the physical displacement of each atom from its equilibrium position. The Cartesian components of displacements transform in the same way as the p-orbitals and therefore the application of space-group theory to lattice vibrations is analogous to finding the tight-binding energy bands of a crystal with only p-orbitals on each atom. The method of analysis of lattice vibrations is the same as that employed in Chapter 10 for tight-binding energy bands. Instead of energy bands we obtain “phonon branches”.