Elastic soft mode and charge ordering of (formula presented)

The elastic constant of (Formula presented) in (Formula presented) relating to the transverse ultrasonic mode shows a pronounced softening around the structural phase transition point at (Formula presented) K associated with the charge ordering of (Formula presented) and (Formula presented) states. The group theoretical analysis leads to the charge fluctuation mode with (Formula presented) symmetry coupled to the elastic strain (Formula presented) of the soft (Formula presented) mode. In a framework of the Landau phenomenological theory for the phase transition, the order parameter (Formula presented) below (Formula presented) gives rise to the linear chain of (Formula presented) ions among the body diagonal [111] direction as well as the trigonal distortion of (Formula presented) Furthermore, this theory explains experimental observations of the elastic softening (Formula presented) above (Formula presented) the structural change from cubic phase of (Formula presented) to trigonal phase of (Formula presented) and the first-order nature of the transition.