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 G₀(^k;-^k), which is the subgroup of the point group of the space group of the crystal whose elements {R} have the property that R^k=±^k. Here R is a 3 ×{} 3 real, orthogonal matrix representative of the symmetry operation R, while ^k is a unit vector in the direction of the wave vector →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.