Magnetic, electric and toroidal polarization modes describing the physical properties of crystals. NdFeO₃ case

The structure and the physical phenomena that occur in a crystal can be described by using a suitable set of symmetry-adapted modes. The classification of magnetic modes in crystals presented in Fabrykiewicz et al. [Acta Cryst. (2021), A77, 327–338] is extended to a classification of electric and toroidal (anapole) modes in crystals. These three classifications are based on magnetic point groups, which are used in two contexts: (i) the magnetic point group of the magnetic crystal class and (ii) the magnetic site-symmetry point group of the Wyckoff position of interest. The classifications for magnetic, electric and toroidal modes are based on the properties of the three generalized inversions: space inversion 1, time inversion 1′ and the space-and-time inversion 1′. It is emphasized that none of these three inversions is more important than the other two. A new notation for symmetry operation symbols and magnetic point group symbols is proposed; each operation is presented as a product of one proper rotation and one generalized inversion. For magnetic, electric and toroidal orderings there are 64 modes: three pure ferro(magnetic/electric/toroidal) modes, 13 mixed ferro(magnetic/electric/toroidal) and antiferro(magnetic/electric/toroidal) modes, and 48 pure antiferro(magnetic/electric/toroidal) modes. The proposed classification of modes leads to useful observations: the electric and toroidal modes have many symmetry limitations similar to those already known for the magnetic modes, e.g. a continuous reorientation of the magnetic or electric or toroidal moments is possible only in triclinic or monoclinic symmetry. An antiferro(magnetic/electric/toroidal) ordering with a weak perpendicular ferro(magnetic/electric/toroidal) component is possible only in monoclinic or orthorhombic symmetry. The general classifications of magnetic, electric and toroidal modes are presented for the case of NdFeO₃.