Group-theoretical methods for obtaining distortions in crystals: Applications to vibrational modes and phase transitions

We describe group-theoretical methods for projecting out global distortions in a crystal from distortions that are localized at a single site. We present a simple method for determining which sets of distortions are independent. We discuss and resolve the difficulties that arise when considering physically irreducible representations, which are reducible on the complex numbers. We illustrate these methods with examples. Applications of these methods to crystal vibrations and phase transitions are presented and illustrated with the ${\mathrm{BaTiO}}_3$ structure.