Symmetry Properties of the Normal Vibrations of a Crystal

A group-theoretic study is made of the degeneracies of the normal modes of vibration of a crystal and of the manner in which the polarization vectors describing these modes transform under the operations of the space group of the crystal. To describe the effects of the spatial symmetry operations a set of $3r$-dimensional matrices is constructed, where $r$ is the number of atoms in a primitive unit cell of the crystal, each of which commutes with the Fourier-transformed dynamical matrix for each value of the wave vector labeling the modes. These matrices are shown to provide a multiplier representation of the point group of the wave vector. The reduction of this representation yields the degeneracies (due to spatial symmetry) and transformation properties of the polarization vectors corresponding to a given wave vector, while the forms of the eigenvectors are obtained by projection operator techniques. For appropriate wave vectors, the consequences of time-reversal symmetry on the degeneracies and polarization vectors are investigated by introducing an anti-unitary matrix operator which commutes with the Fourier-transformed dynamical matrix. A criterion for the existence of extra degeneracies due to time-reversal symmetry is presented. The symmetries of lattice vibrations and selection rules for two-phonon absorption processes corresponding to several values of k in the first Brillouin zone of diamond are determined to illustrate the methods developed in this paper.