The wavevector substar group in reciprocal space and its representation

A new concept of the wavevector substar group is established which, in the study of translational symmetry breaking of a crystal, only considers the particular arms of the wavevector star taking part in the phase transition; this is in contrast with the traditional Landau theory which considers all of the arms of the wavevector star. It is shown that this new concept can be used effectively to investigate the interesting physical properties of crystals associated with translational symmetry breaking. It is shown that studies on complicated phase transitions related to reducible representations, such as those in perovskite KMnF₃ multiferroics and the high-temperature superconductor La_2/3Mg_1/2W_1/2O₃ (La₄Mg₃W₃O₁₈), are much simplified by the new concept. The theory of the wavevector substar group and its representation is a powerful mathematical tool for the study of various symmetry-breaking phenomena in solid-state crystals.