The structural evolution of selected perovskites containing Jahn–Teller cations has been investigated in the light of a formal analysis of symmetry hierarchies for phase transitions driven by octahedral tilting and Jahn–Teller cooperative distortions. General expressions derived from the strain/order-parameter coupling relationships allowed by symmetry are combined with observed changes in lattice parameters to reveal details of order-parameter evolution and coupling. LuVO3, YbVO3, YVO3 and CeVO3 are representative of systems which develop Jahn–Teller ordering schemes associated with irreducible representations M^+_2 and R^+_3 of the space group Pm\bar 3m. Tilting of their octahedra is associated with M^+_3 and R^+_4. The Pnma (M^+_3+R^+_4 tilting) ↔ P21/a (M^+_3+R^+_4 tilting, R^+_3 Jahn–Teller order) transition below room temperature is close to second order in character. Shear strains which depend primarily on tilt angles show little variation, implying that there is only weak coupling between the tilting and Jahn–Teller order parameters. The subsequent P21/a ↔ Pnma (M^+_3+R^+_4 tilting, M^+_2 Jahn–Teller order) is first order in character, and involves either a reduction in the R^+_4 tilt angle or a change in the strength of tilt/Jahn–Teller order-parameter coupling. In LaMnO3, the isosymmetric Pnma (M^+_3+R^+_4 tilting) ↔ Pnma (M^+_3+R^+_4 tilting, M^+_2 Jahn–Teller order) transition can be described in terms of a classical first-order transition conforming to a 246 Landau expansion with negative fourth-order coefficients. Strain evolution in Ba-doped samples suggests that the transition becomes second order in character and reveals a new strain relaxation mechanism in LaMnO3 which might be understood in terms of local strain heterogeneities due to the disordering of distorted MnO6 octahedra. Transitions in PrAlO3 and La0.5Ba0.5CoO3 illustrate the transformation behaviour of systems in which the Jahn–Teller ordering scheme is associated with the irreducible representation \Gamma^+_3. Overall, coupled tilting + Jahn–Teller phase transitions in perovskites conform to mean-field behaviour, consistent with the underlying role of strain in promoting long interaction lengths.
Neural network-based order parameter for phase transitions and its applications in high-entropy alloys
