Tabulation of irreducible representations of the crystallographic space groups and their superspace extensions

Acta Cryst. (2013). A69, 388–395 (http://doi.org/mt6)

This A₂BB′O₆ double-perovskite example (an idealization of La₂CoRuO₆) simultaneously exhibits a variety of order parameters, all of which can be projected using IRs of the parent cubic symmetry Pm−3m. Here, the IRs M₃⁺ and R₄⁺ generate ferrorotational and antiferrorotational octahedral rotations, respectively; R₁⁺ generates the B–B′ site ordering; mΛ₃ generates the antiferromagnetic structure; and Γ₃⁺ and Γ₅⁺ generate the strains.

The irreducible representations (IRs) of the parent symmetry of a system provide a symmetry-motivated parameter set for describing any periodic or aperiodic system distortion. We present complete-group IRs for all crystallographic space groups and their extensions to (3+d)-dimensional superspace, including all special and non-special k vectors, commensurate and incommensurate. A new matrix form makes their exhaustive tabulation possible for the first time. They can now be incorporated into diverse software packages and applied to the solution and refinement of low-symmetry structures that arise in phase transitions.