The symmetry of lattice vibrations in the zincblende and diamond structures

Abstract
A simple and practical method is presented for analysing the vibrational symmetries in any perfect crystal. It takes the space spanned by the complete set of normal modes of wave vector q, and factorizes it into a ‘cell space’ SC, and a ‘complex Euclidean space’ SE. Two simple equations describe the effect of the crystal symmetry operations in each of these factor spaces, so that the symmetry of all the normal modes can readily be determined. This formalism makes it easy to understand several qualitative features of vibrational spectra: for example, if the unit cell contains only two atoms and they are dissimilar, the irreducible representations inside the Brillouin zone always appear doubled. Similarly, it is shown that the number of different types of atom vibrating in any mode cannot exceed the number of times that modes of the same symmetry appear in the vibrational spectrum: so that if a mode has a unique symmetry, only one type of atom is vibrating in that mode, and all other atoms are stationary. Vibrations in the perovskite structure are discussed briefly by way of an example, and this is followed by a fuller treatment of the zincblende and diamond structures.

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