International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.6, pp. 852-853

## Section 3.6.2.1. Reduced magnetic superfamilies of magnetic groups

D. B. Litvina*

aDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: u3c@psu.edu

#### 3.6.2.1. Reduced magnetic superfamilies of magnetic groups

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Let denote a crystallographic group. The magnetic superfamily of consists of the following set of groups:

 (1) The group . (2) The group , the direct product of the group and the time-inversion group , the latter consisting of the identity 1 and time inversion . (3) All groups , subdirect products of the groups and . is a subgroup of index 2 of . Groups of this kind will also be denoted by .

The third subset is divided into two subdivisions:

 (3a) Groups , where is an equi-translational (translatio­nen­gleiche) subgroup of . (3b) Groups , where is an equi-class (klassengleiche) subgroup of .1

Two magnetic groups and are called equivalent if there exists an affine transformation that maps onto and onto (Opechowski, 1986). If only non-equivalent groups are included, then the above set of groups is referred to as the reduced magnetic superfamily of .

#### Example

We consider the crystallographic point group . The magnetic superfamily of the group consists of five groups: , the group , and the three groups , and . Since the latter three groups are all equivalent, the reduced magnetic superfamily of the group consists of only three groups: , , and one of the three groups , and .

#### Example

In the reduced magnetic space group superfamily of there are five groups: , , and three groups , and . The groups and are equi-translational magnetic space groups and is an equi-class magnetic space group .

A magnetic group has been defined as a symmetry group of a spin arrangement (Opechowski, 1986). With this definition, since , a group is then not a magnetic group. However, there is not universal agreement on the definition or usage of the term magnetic group. Two definitions (Opechowski, 1986) have magnetic groups as symmetry groups of spin arrangements, with one having only groups , of the three types of groups , and , defined as magnetic groups, while a second having both group and defined as magnetic groups. Here we shall refer to all groups in a magnetic superfamily of a group as magnetic groups, while cognizant of the fact that groups cannot be a symmetry group of a spin arrangement.

### References

Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North Holland.