Normal mode determination of perovskite crystal structures with octahedral rotations: Theory and applications

Nuclear site analysis methods are used to enumerate the normal modes of ABX₃ perovskite polymorphs with octahedral rotations. We provide the modes of the 14 subgroups of the cubic aristotype describing the Glazer octahedral tilt patterns, which are obtained from rotations of the BX ₆ octahedra with different sense and amplitude about high-symmetry axes. We tabulate all normal modes of each tilt system and specify the contribution of each atomic species to the mode displacement pattern, elucidating the physical meaning of the symmetry unique modes. We have systematically generated 705 schematic atomic displacement patterns for the normal modes of all 15 (14 rotated + 1 unrotated) Glazer tilt systems. We show through some illustrative examples how to use these tables to identify the octahedral rotations, symmetric breathing, and first-order Jahn-Teller anti-symmetric breathing distortions of the BX₆ octahedra, and the associated Raman selection rules. We anticipate that these tables and schematics will be useful in understanding the lattice dynamics of bulk perovskites and could serve as a reference point in elucidating the atomic origins of a wide range of physical properties of synthetic perovskite thin films and superlattices.