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. 856-857

Section 3.6.2.2.5. Symbol of the subgroup [{\cal D}] of index 2 of [{\cal F}({\cal D})]

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

3.6.2.2.5. Symbol of the subgroup [{\cal D}] of index 2 of [{\cal F}({\cal D})]

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For magnetic group types [{\cal F}({\cal D})], the magnetic group type symbol of the subgroup [{\cal D}] is given in the third column of the survey of magnetic groups, see e.g. Table 3.6.2.2[link]. If [{\cal F}({\cal D})] is a group [{\cal M}_T], then the subgroup [{\cal D}] is defined by the translational group of [{\cal F}({\cal D})] and the unprimed coset representatives of [{\cal F}({\cal D})].

Example

Consider the three-dimensional magnetic space-group type 16.3.101 [P2'2'2]. The representative group [P2'2'2] is defined by the translational subgroup [\cal T] denoted by the letter P generated by the translations[\{1|1,0,0)\}\quad \{1|0,1,0\}\quad \{1|0,0,1\}]and the standard set of coset representatives[\{1|0\}\quad \{2_{100}|0\}'\quad \{2_{010}|0\}'\quad \{2_{001}|0\}.]The subgroup [{\cal D}] of index 2 of the representative group [{\cal F}({\cal D}) = P2'2'2] is defined by the translational group [\cal T] denoted by the letter P and the cosets {1∣0} and {2001∣0}, and is a group of type P2.

If [{\cal F}({\cal D})] is a group [{\cal M}_R], then the subgroup [{\cal D}] is defined by the non-primed translational group of [{\cal F}({\cal D})] and all the cosets of the standard set of coset representatives of the group [{\cal F}({\cal D})].

Example

Consider the three-dimensional magnetic space-group type 16.4.102 P2a222. The representative group P2a222 is defined by the translational group [\cal T] denoted by the symbol P2a generated by the translations[\{1|1,0,0\}'\quad \{1|0,1,0\}\quad \{1|0,0,1\}]and the standard set of coset representatives[\{1|0\}\quad \{2_{100}|0\}\quad \{2_{010}|0\}\quad \{2_{001}|0\}.]The subgroup [{\cal D}] of index 2 of the representative group [{\cal F}({\cal D}) = P_{2a}222] is defined by the translational subgroup [\cal T] denoted by the symbol P2a, i.e. the translations generated by[\{1|2,0,0\}\quad \{1|0,1,0\}\quad \{1|0,0,1\}]and the standard set of cosets of P2a222. The group [{\cal D}] is a group of type P222.

While the group type symbol of [{\cal D}] is given, the coset representatives of the subgroup [{\cal D}] of [{\cal F}({\cal D})] derived from the standard set of coset representatives of [{\cal F}({\cal D})] may not be identical with the standard set of coset representatives of the representative group of type [{\cal D}] found in the survey of magnetic group types. Consequently, to show the relationship between this subgroup [{\cal D}] and the listed representative group of groups of type [{\cal D}] additional information is provided: a new coordinate system is defined in which the coset representatives of this subgroup [{\cal D}] are identical with the standard set of coset representatives listed for the representative group of groups of type [{\cal D}]: Let (O; a, b, c) be the coordinate system in which the group [{\cal F}({\cal D})] is defined. O is the origin of the coordinate system, and a, b and c are the basis vectors of the coordinate system. a, b and c represent a set of basis vectors of a primitive cell for primitive lattices and of a conventional cell for centred lattices. A second coordinate system, defined by (O + p; a′, b′, c′), is given in which the coset representatives of this subgroup [{\cal D}] are identical with the standard set of coset representatives listed for the representative group of groups of type [\cal D]. O + p is referred to as the location of the subgroup [{\cal D}] in the coordinate system of the group [{\cal F}({\cal D})] (Kopský, 2011[link]). The origin is first translated from O to O + p. On translating the origin from O to O + p, a coset representative {Rτ} becomes {Rτ + Rpp} (Litvin, 2005[link], 2008b[link]; see also Section 1.5.2.3[link] ). This is followed by changing the basis vectors a, b and c to a′, b′ and c′, respectively. The basis vectors a′, b′, c′ define the conventional unit cell of the non-primed subgroup [{\cal D}] of [{\cal F}({\cal D})] in the coordinate system (O; a, b, c) in which [{\cal F}({\cal D})] is defined. (O + p; a′, b′, c′) is given immediately following the group type symbol for the subgroup [{\cal D}] of [{\cal F}({\cal D})]. [In Litvin (2013[link]), for typographical simplicity, the symbols `O +' are omitted.]

Example

For the three-dimensional magnetic space-group type 10.4.52, [{\cal F}({\cal D}) = P2/m'], one finds in Litvin (2013[link])2

Serial No.SymbolNon-primed subgroup of index 2Standard set of coset representatives
10.4.52 P2/m P2 (0, 0, 0; a, b, c) {1∣0} {2010∣0} {1∣0}′ {m010∣0}′

The translational subgroup of the subgroup [{\cal D}] = P2 of [{\cal F}({\cal D}) = P2/m'] is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the coset representatives of this group are {1∣0} and {2010∣0}, the unprimed coset representatives on the right. This subgroup [{\cal D}] is of type P2. In Litvin (2013[link]), listed for the group type 3.1.8, P2, one finds the identical two coset representatives. Consequently, there is no change in the coordinate system, i.e. [{\bf p} = (0,0,0)] and a′ = a, b′ = b and c′ = c. In the coordinate system of the magnetic group [P2/m'], the coset representatives of its subgroup [{\cal D} = P2] are identical with the standard set of coset representatives of the group type P2.

Example

For the three-dimensional magnetic space-group type 16.7.105, [{\cal F}({\cal D}) = P_{2c} 22'2'] one has

Serial No.SymbolNon-primed subgroup of index 2Standard set of coset representatives
16.7.105 P2c22′2′ P2221 (0, 0, 0; a, b, 2c) {1∣0} {2100∣0} {2010∣0, 0, 1} {2001∣0, 0, 1}

The translational subgroup of the subgroup [{\cal D} = P222_1] of [{\cal F}({\cal D}) = P_{2c} 22'2'] is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 2}, and the coset representatives of this group are all those coset representatives on the right. This subgroup [{\cal D}] is of type P2221. Listed for the group type 17.1.106 P2221, one finds a different set of coset representatives:[\{1|0\}\quad \{2_{100}|0\}\quad \{2_{010}|0,0,{\textstyle{1\over 2}}\}\quad \{2_{001}|0,0,{\textstyle{1\over 2}}\}.]Consequently, to show the relationship between this subgroup [{\cal D}] of [{\cal F}({\cal D})] and the listed representative group of the group type P2221 we change the coordinate system in which [{\cal D}] is defined to (0, 0, 0; a, b, 2c). In this new coordinate system the coset representatives of the subgroup [{\cal D}] are identical with the coset representatives of the representative group of the group type P2221.

Example

For the three-dimensional magnetic space-group type 18.4.116, P2121′2′, one has

Serial No.SymbolNon-primed subgroup of index 2Standard set of coset representatives
18.4.116 P2121′2′ P21 (0, ¼, 0; c, a, b) {1∣0} {2100∣½, ½, 0} {2010∣½, ½, 0}′ {2001∣0}′

The translational subgroup of [{\cal D}] is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the coset representatives of this group are {1∣0} and {2100∣½, ½, 0}, the unprimed coset representatives on the right. The group [{\cal D}] is of type P21. For the magnetic group type 4.1.15 P21 one finds a different set of coset representatives: {1∣0} and {2010∣0, ½, 0}. Consequently, to show the relationship between the subgroup [{\cal D}] of [{\cal F}({\cal D})] and the listed representative group of the group type P21, we change the coordinate system in which the subgroup [{\cal D}] is defined to (0, ¼, 0; c, a, b). The origin is first translated from O to O + p, where p = (0, ¼, 0), and then a new set of basis vectors, a′ = c, b′ = a and c′ = b, is defined. In this new coordinate system the coset representatives of the subgroup [{\cal D}] are identical with the standard set of coset representatives of the representative group of the group type P21.

References

Kopský, V. (2011). Private communication.
Litvin, D. B. (2005). Tables of properties of magnetic subperiodic groups. Acta Cryst. A61, 382–385.
Litvin, D. B. (2008b). Tables of crystallographic properties of magnetic space groups. Acta Cryst. A64, 419–424.
Litvin, D. B. (2013). Magnetic Group Tables, 1-, 2- and 3-Dimensional Magnetic Subperiodic Groups and Space Groups. Chester: International Union of Crystallography. Freely available from http://www.iucr.org/publ/978–0–9553602–2–0 .








































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