Abstract：
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.
摘要：
可以通过使用一组合适的对称适应模式来描述晶体中发生的结构和物理现象。Fabrykiewicz等人介绍了晶体中磁模的分类。[ActaCryst。(2021),A77，327-338]扩展到晶体中的电和环形（anapole）模式的分类。这三种分类是基于磁点组，在两种情况下使用：（i）磁性晶体类的磁性点组和（ii）感兴趣的Wyckoff位置的磁性位点对称点组。磁性的分类，电模式和环形模式基于三个广义反演的性质：空间反演1，时间反演1\'和时空反演1\'。要强调的是，这三个倒置中没有一个比其他两个更重要。提出了对称运算符号和磁点组符号的新表示法；每个操作都表示为一个适当旋转和一个广义反转的乘积。对于磁性，电和环形排序有64种模式：三种纯铁（磁/电/环形）模式，13混合铁（磁/电/环形）和反铁（磁/电/环形）模式，和48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：电模式和环形模式具有许多对称限制，类似于已知的磁模式。例如，磁矩或电矩或环形矩的连续重新定向只能在三斜对称或单斜对称的情况下进行。具有弱垂直铁（磁/电/环形）分量的反铁（磁/电/环形）排序仅在单斜或正交对称下是可能的。磁性的一般分类，对于NdFeO3，提出了电模式和环形模式。
