A neutron diffraction study of anharmonic thermal vibrations in cubic CsPbX₃

A neutron diffraction study of cubic CsPbX₃ (X = Cl or Br) has been carried out over the temperature ranges 325-623 K for CsPbCl₃ and 408-673 K for CsPbBr₃. The temperature factors for the perovskite structure were derived following the method of Matsubara [Prog. Theor. Phys. (1975), 53, 1210-1211] which includes the use of cumulant coefficients to characterize anharmonic components for an Einstein model. The potential parameters were then obtained using a numerical integration method to analyse the temperature dependence of the temperature factors. It was found that the anharmonic components in the potential were very large for the Cs and X atoms which undergo displacements on passing through the phase transitions at lower temperatures (321 K for CsPbCl₃ and 403 K for CsPbBr₃). On the other hand, a harmonic potential is quite adequate to describe the thermal vibration of the Pb atoms, which are not displaced at the phase transitions. Thus, the existence of the anharmonicity in the cubic phase seems to be anticipating the atomic displacement through the successive phase transitions for these substances. In addition to this anharmonicity, the temperature factors of the X atoms parallel to the (100) plane show an anomalous behaviour near the cubic to tetragonal phase-transition temperature, which should be connected with the softening of phonon mode at this phase transition.