Refining perovskite structures to pair distribution function data using collective Glazer modes as a basis

Structural modelling of octahedral tilts in perovskites is typically done using the symmetry constraints of the resulting space group. In most cases, this introduces more degrees of freedom than those strictly necessary to describe only the octahedral tilts. It can therefore be a challenge to disentangle the octahedral tilts from other structural distortions such as cation displacements and octahedral distortions. This paper reports on the development of constraints for modelling pure octahedral tilts and implemented the constraints in diffpy-CMI, a powerful package to analyse pair distribution function (PDF) data. The program allows features in the PDF that come from rigid tilts to be separated from non-rigid relaxations, provides an intuitive picture of the tilting, and as it has many fewer refinable variables than the unconstrained space-group fits, provides robust and stable refinements of the tilt components. It further demonstrates the use of the model on the canonical tilted perovskite CaTiO$_3$ which has a known Glazer tilt system $\alpha^+ \beta^- \beta^-$. The Glazer model fits comparably to the corresponding space group model $Pnma$ below $r$ = 14 A and becomes progressively worse than the space group model at higher $r$ due to non-rigid distortions in the real material.