Symmetry Properties of the Normal Vibrations of a Crystal

TLDR

The study applies group theory to analyze how crystal symmetry determines degeneracies and polarization vectors of vibrational modes. The paper proposes a criterion for identifying additional degeneracies arising from time‑reversal symmetry. The authors construct 3r‑dimensional symmetry matrices that commute with the dynamical matrix, use anti‑unitary operators to account for time‑reversal symmetry, and apply projection operators to derive eigenvectors, illustrating the method on diamond. The symmetry matrices form a multiplier representation of the wave‑vector point group, and their reduction yields the degeneracies, transformation properties, and eigenvector forms, enabling determination of lattice‑vibration symmetries and two‑phonon absorption selection rules in diamond.

Abstract

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.