今天来学习一下 如何判断磁性结构的对称性

Magnetic space groups

共有四种磁空间群

• Type I (Colorless groups) 230: 纯空间群 无额外对称性
• Type II (Grey groups) 230: 空间群 + 每一个空间群的对称操作都会额外附带一个反对称操作 例如: SG166 + $\mathcal{T}$
• Type III(Colorless groups) 674: 空间群 + 一半数量的对称操作额外附带一个反对称操作
• Type IV (Colorless groups) 517: 空间群 + 额外的平移联合时间反演对称性

Type I

Type I magnetic space groups, $\mathcal{M}_I$ are identical to the ordinary space groups ${G}$:

$\mathcal{M}_I = {G}$

Type II

Type II magnetic space groups, $\mathcal{M}_II$, are made up of all the symmetry operations of the crystallographic space group, ${G}$, plus the product of those operations with time reversal operation, $\mathcal{T}$.

$\mathcal{M}_II = G + \mathcal{T}G$

Type III

Type III magnetic space groups, $\mathcal{M}_{III}$ are constructed using a group ${H}$, which is a subgroup of ${G}$ with index 2.

$\mathcal{M}_III = G + \mathcal{T}(G-H)$

Type IV

Type IV magnetic space groups, $\mathcal{M}_IV$ are are constructed with the use of a pure translation, {$E|t_0$}, which is Seitz notation for null rotation and a translation, $t_0$.

$$ \mathcal{M}{IV}=G+\mathcal{T}\left{E \mid t{0}\right}G $$

Magnetic stucture

To determine a micro magnetic structure, such techniques can be used:

• neutron diffraction

Determine a magnetic group via FINDSYM

Type of anti-ferromagnetic

[^1]Przystawa J, Suski W. Magnetic Ordering in Uranium Compounds with Crystal Structures of Anti‐Cu2Sb and PbFCl Types. Molecular Field Approximation[J]. physica status solidi (b), 1967, 20(2): 451-459.

A simple test

1. upload your cif file

2. modify the structure

Here I choose “Include magnetic moments. moments. Give components along the three vectors above which define a conventional unit cell.”

set the local mganetic momentum directing z-axis for each C atoms in graphene.

• then click OK

3. Checkout the result

 1Lattice parameters, a,b,c,alpha,beta,gamma:
 2   2.45600   2.45600  20.00000  90.00000  90.00000 120.00000
 3Centering: P
 4Number of atoms in unit cell:
 5    2
 6Type of each atom:
 72*C 
 8        Position                     Occ      Magnetic moment        
 9    1 C  0.33333  0.66667  0.25000    1.000    0.000  0.000  1.000  
10    2 C  0.66667  0.33333  0.25000    1.000    0.000  0.000  1.000 

这部分描述了我们上传的晶格结构信息, 该网站分析出了是我们这种设置是什么对称性

1 Space Group 191.240 P6/mm'm'  

同时，我们也可以下载这个cif文件，改后缀为cif，用vesta打开

当然也可以用该网站小组开发的 晶体结构软件 ISOVIZ

complex example

我们以 PtCl3 为例，判断其在不同磁结构下 属于什么磁群 有什么样的Mirror对称性