The cubic-tetragonal phase transition of CH 3 NH 3 PbI 3 was investigated by single crystal X-Ray diffractometry. The crystal structure was refined at five temperatures in the teragonal phase. The PbI 6 octahedron rorates around the c-axis alternatively to construct the SrTiO 3 -type tetragonal structure. A methylammonium ion is partially orderd; 24 disordered states in the cubic phase are reduced to 8. With decreasing temperature, the rotation angle of the octahedron increases monotonically, which indicates it is an order parameter of the cubictetragonal transition.

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In order to understand the mechanism of the structural phase transition in CH 3 NH 3 PbBr 3 , neutron diffraction data from the single crystal is analyzed by the least-squares method and the maximum entropy method. The disordered configuration of a methylammonium ion in a cage surrounded by eight PbBr 6 octahedrons is identified unambiguously. With comparing the low-temperature structure, the transition is characterized by the condensation of the rotational motion of the PbBr 6 octahedron around the Pb-Br axis, which accompanies the partial ordering of the methylammonium ion to reduce the site symmetry.

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In order to describe a structural phase transition at low temperature, a quantum particle within a local potential is considered. According to the general formalism presented by Salje et al. ͓Z. Phys. B 82, 399 ͑1991͔͒, a quasiharmonic approximation is applied to the local potential and the interaction is replaced by the mean field one. The rigorous effective potential is reduced from a double Morse-type potential. The order parameter, the variance, and the effective soft frequency are given by analytic equations with potential parameters. As an application, a molecular motion of H 2 PO 4 in KDP is considered and the disappearance of the ordered phase at low temperature under high pressure is discussed. The difference between KDP and DKDP crystals are described successfully by the presented model.