Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland.
Acta Crystallogr A Found Adv. 2023 Jan 1;79(Pt 1):80-94. doi: 10.1107/S2053273322009858.
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 NdFeO.
晶体中的结构和物理现象可以用一组合适的对称适应模式来描述。Fabrykiewicz 等人在[Acta Cryst. (2021), A77, 327-338]中提出的晶体中磁模的分类被扩展到晶体中电模和环形（反磁）模的分类。这三种分类基于磁点群，这些点群在两个方面使用：(i) 磁晶体类的磁点群，和 (ii) 感兴趣的维格纳位置的磁位对称点群。磁、电和环形模式的分类基于三个广义反转的性质：空间反转 1、时间反转 1' 和空间-时间反转 1'。强调的是，这三个反转中没有一个比其他两个更重要。提出了一种新的对称操作符号和磁点群符号表示法；每个操作都表示为一个纯旋转和一个广义反转的乘积。对于磁、电和环形有序，有 64 种模式：三种纯铁磁/电/环形模式，13 种混合铁磁/电/环形和反铁磁/电/环形模式，和 48 种纯反铁磁/电/环形模式。所提出的模式分类导致了有用的观察结果：电和环形模式具有许多与已经知道的磁模式类似的对称限制，例如，磁或电或环形矩的连续重定向仅在三斜或单斜对称中是可能的。仅在单斜或正交对称中才可能存在具有弱垂直铁磁/电/环形分量的反铁磁/电/环形有序。给出了 NdFeO 情况下的磁、电和环形模式的一般分类。
