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 [{\cal F}] denote a crystallographic group. The magnetic superfamily of [{\cal F}] consists of the following set of groups:

  • (1) The group [{\cal F}].

  • (2) The group [\ispecialfonts{\cal F}\!{\sfi 1}'\equiv {\cal F} \times {\sfi 1}'], the direct product of the group [{\cal F}] and the time-inversion group [\ispecialfonts{\sfi 1}'], the latter consisting of the identity 1 and time inversion [1'].

  • (3) All groups [\ispecialfonts{\cal F}({\cal D})\equiv{\cal D} \,\cup({\cal F} - {\cal D})1'\equiv {\cal F}\ \underline{\times} \ {\sfi 1}'], subdirect products of the groups [{\cal F}] and [\ispecialfonts{\sfi 1}']. [{\cal D}] is a subgroup of index 2 of [{\cal F}]. Groups of this kind will also be denoted by [{\cal M}].

The third subset is divided into two subdivisions:

  • (3a) Groups [{\cal M}_{T}], where [{\cal D}] is an equi-translational (translatio­nen­gleiche) subgroup of [{\cal F}].

  • (3b) Groups [{\cal M}_R], where [{\cal D}] is an equi-class (klassengleiche) subgroup of [{\cal F}].1

Two magnetic groups [{\cal F}_1({\cal D}_1)] and [{\cal F}_2({\cal D}_2)] are called equivalent if there exists an affine transformation that maps [{\cal F}_1] onto [{\cal F}_2] and [{\cal D}_1] onto [{\cal D}_2] (Opechowski, 1986[link]). If only non-equivalent groups [{\cal F}({\cal D})] are included, then the above set of groups is referred to as the reduced magnetic superfamily of [{\cal F}].

Example

We consider the crystallographic point group [{\cal F} = 2_x2_y2_z]. The magnetic superfamily of the group [2_x2_y2_z] consists of five groups: [{\cal F} = 2_x2_y2_z], the group [\ispecialfonts{\cal F}\!{\sfi 1}' = 2_x2_y2_z1'], and the three groups [{\cal F}({\cal D}) = 2_x2_y2_z(2_x)], [2_x2_y2_z(2_y)] and [2_x2_y2_z(2_z)]. Since the latter three groups are all equivalent, the reduced magnetic superfamily of the group [{\cal F} = 2_x2_y2_z] consists of only three groups: [2_x2_y2_z], [2_x2_y2_z1'], and one of the three groups [2_x2_y2_z(2_x)], [2_x2_y2_z(2_y)] and [2_x2_y2_z(2_z)].

Example

In the reduced magnetic space group superfamily of [{\cal F} = Pnn2] there are five groups: [{\cal F} = Pnn2], [\ispecialfonts{\cal F}\!{\sfi 1}' = Pnn21'], and three groups [{\cal F}({\cal D}) = Pnn2(Pc)], [Pnn2(P2)] and [Pnn2(Fdd2)]. The groups [Pnn2(Pc)] and [Pnn2(P2)] are equi-translational magnetic space groups [{\cal M}_T] and [Pnn2(Fdd2)] is an equi-class magnetic space group [{\cal M}_R].

A magnetic group has been defined as a symmetry group of a spin arrangement [{\bf S}(r)] (Opechowski, 1986[link]). With this definition, since [1'{\bf S}(r) = - {\bf S}(r)], a group [\ispecialfonts{\cal F}\!{\sfi 1}'] 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[link]) have magnetic groups as symmetry groups of spin arrangements, with one having only groups [{\cal F}({\cal D})], of the three types of groups [\cal F], [\ispecialfonts{\cal F}\!{\sfi 1}'] and [{\cal F}({\cal D})], defined as magnetic groups, while a second having both group [{\cal F}] and [{\cal F}({\cal D})] defined as magnetic groups. Here we shall refer to all groups in a magnetic superfamily of a group [{\cal F}] as magnetic groups, while cognizant of the fact that groups [\ispecialfonts{\cal F}\!{\sfi 1}'] cannot be a symmetry group of a spin arrangement.

References

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








































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