Wave vector star channel and star channel group in the reciprocal lattice space of crystal

In the paper, a new method determining the wave vector star channel in the reciprocal lattice space of crystal in the light of the translational symmetry breaking is proposed, and, in order to consider the phase transitions according to the wave vector star channel, the conception of wave vector star channel group is adopted. By this method, it is revealed that the phase transitions in crystal are induced not by any arbitrary combinations of arms of the given star, but by the selected combinations of arms which are satisfied by symmetry of the parent phase. The wave vector star channel group is defined as the set of elements of space group leaving the wave vector star channel invariant. We show that the conception of wave vector star channel group can be efficiently used in studying the translational symmetry breaking related to all the Lifshitz wave vector stars of 230 space groups.