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Fabrykiewicz, Piotr; Przeniosło, Radosław; Sosnowska, Izabela
Acta crystallographica. Section A, Foundations and advances, January 2023, 20230101, Volume: 79, Issue: 1Journal Article
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. A classification of the modes for magnetic, electric and toroidal polarization in crystals is proposed. The classification relies on magnetic point groups. A new notation of magnetic point groups based on the decomposition on proper rotation and generalized inversions is used.
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