Group-theoretic analysis of long-wavelength vibrations of polar crystals

A group-theoretic method is described for analyzing the long-wavelength lattice vibrations of polar crystals made up of deformable and polarizable ions. A set of $3r$-dimensional matrices is constructed (where $r$ is the number of ions in a primitive unit cell of the crystal), each of which commutes with the dynamical matrix for such crystals, and which also provides a representation of the point group $G_0(\hat{k};-\hat{k})$, which is the subgroup of the point group of the space group of the crystal whose elements {$R$} have the property that $\mathrm{R}\hat{k}=\pm \hat{k}$. Here $\mathrm{R}$ is a 3 × 3 real, orthogonal matrix representative of the symmetry operation $R$, while $\hat{k}$ is a unit vector in the direction of the wave vector $\overrightarrow{\mathrm{k}}$ of the long-wavelength lattice vibrations being studied. Reduction of this matrix representation yields the symmetries of the long-wavelength normal modes of the crystal, and the forms of the corresponding eigenvectors can be obtained by projection-operator techniques. Additional degeneracies imposed by time-reversal symmetry are automatically taken into account in this treatment, which is illustrated by applying it to an analysis of the long-wavelength vibration modes of graphite.