Octahedral tilts, symmetry-adapted displacive modes and polyhedral volume ratios in perovskite structures

The structures of tilted perovskites in each of the 15 tilt systems have been decomposed into the amplitudes of symmetry-adapted modes in order to provide a clear and unambiguous definition of the tilt angles. A full expression in terms of the mode amplitudes for the ratio of the volumes of the two polyhedra within the perovskite structure for each of the 15 tilt systems is derived, along with more general expressions in terms of either mode amplitudes or tilt angles that can be used to estimate this ratio when the distortions of the octahedra are small.