Ferroic classifications extended to ferrotoroidic crystals

Aizu's characterization of the 773 species of phase transitions by magnetization, polarization and strain is extended to include characterization by the recently observed toroidal moment. The resulting distinction quadruplet characterization is then used to classify the species into sub-ensembles, extending Schmid's concept of ensembles of species to include classification by toroidal moments. Tables are given of the distinction quadruplet characterization of each species and the species in each ensemble and sub-ensemble. The form of primary and secondary ferroic property tensors invariant under the 122 magnetic point groups have also been tabulated for use in determining the characterization of species and possible domain switching. In both cases, physical property tensors related to the toroidal moment are included.