Simple group-theoretical method to determine the symmetry of soft modes in general second order phase transitions

It is shown how one can obtain, extending Birman's method, the symmetry of the soft modes in general second order phase transitions which occurs at $\text{k}\text{≠0}$ points in Brillouin zone accompanying the primitive cell volume enlargement. This method is much simpler than the usual Landau method which utilizes the free energy expansion and its minimization. Gd₂(MoO₄)₃ is taken as an example. The case in which several arms of star-k contribute to the transition is also discussed using the example of C⁶_4h→C⁶_2h.