Conjugate subgroups

Müller, U.

doi:10.1107/97809553602060000547

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 3.1, pp. 432-433 | 1 | 2 |

Section 3.1.5. Conjugate subgroups

Ulrich Müller^a ^*

^a Fachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: mueller@chemie.uni-marburg.de

3.1.5. Conjugate subgroups

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Conjugate subgroups are different subgroups belonging to the same space-group type (they have the same Hermann–Mauguin symbol) and they have the same unit-cell size and the same shape for the conventional cell. They can be mapped onto one another by a symmetry operation of the starting group, i.e. they are symmetry-equivalent in this space group. They can occur only if the index of symmetry reduction is $[\geq 3]$ . The relations of the Wyckoff positions of a space group with the Wyckoff positions of any representative of a set of conjugate subgroups are always the same. Therefore, in principle it is sufficient to list the relations for only one representative.

Two kinds of conjugation of maximal subgroups can be distinguished, translational conjugation and orientational conjugation. Non-maximal subgroups can involve both kinds of conjugation, so the situation is more complicated in chains of group–subgroup relations, cf. Koch (1984) and Müller (1992). Since the present tables only list maximal subgroups, we will not discuss this here.

3.1.5.1. Translational conjugation

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Translational conjugation occurs when the group–subgroup relation involves a loss of translational symmetry. This happens when the conventional cell has been enlarged or when centring translations have been lost; this means that the primitive unit cell of the subgroup is larger (by a factor $[\geq 3]$ ). Translationally conjugate subgroups of a space group are symmetry-equivalent by a translation of the lattice of this space group. This way, isomorphic subgroups of index $[p \geq 3]$ have p conjugate subgroups (unless the cell enlargement occurs in a direction in which the origin may float). The existence of conjugate subgroups of this kind is not specifically mentioned in the tables. However, they can be recognized by looking in the column `Coordinates'. If a semicolon appears after the coordinate triplet, followed by values in parentheses to be added, and if, in addition, the index of symmetry reduction is $[\geq 3]$ , then conjugate subgroups usually exist. They differ in the locations of their origins by values corresponding to the values given in the parentheses.

Example 3.1.5.1.1

$[\textstyle{x,y,{{1}\over{3}} z;\;\pm(0,0,{{1}\over{3}})}]$ gives the positional coordinates in the subgroup originating from the coordinates of one unit cell of the starting group, namely $[\textstyle{\;x,y,{{1}\over{3}} z;\quad x, y, {{1}\over{3}} z+{{1}\over{3}};\quad x,y,{{1}\over{3}} z-{{1}\over{3}}.}]$ In addition, this also means that there are three conjugate subgroups. They differ in the locations of their origins referred to the origin of the starting space group by [0,0,0] , $[0,0,{{1}\over{3}}]$ and $[0,0,-{{1}\over{3}}]$ , expressed in terms of the coordinate system of the subgroup, which is equivalent to [0,0,0] , [0,0,-1] and [0,0,1] in the coordinate system of the starting group.

Primitive subgroups of face-centred cubic space groups have four conjugate subgroups. Because in this case no values have to be added to the coordinates, the existence of conjugate subgroups is expressed by the entry `4 conjugate subgroups'. They differ in their origin locations corresponding to the centring vectors of the face-centred cell.

Cell enlargements do not always produce conjugate subgroups. If the cell is being enlarged in a direction in which the origin may float, i.e. is not fixed by symmetry, no conjugate subgroups result. This applies to the following crystal classes:

1, enlargement in any direction;
2, mm2, 3, 3m, 4, 4mm, 6 and 6mm, enlargement in the direction of the unique axis;
m , enlargement parallel to the plane of symmetry.

Example 3.1.5.1.2

The cell enlargement $[{\bf a},\,{\bf b},\,5{\bf c}]$ of space group [Cmc2_1] , No. 36, (crystal class mm2) does not produce conjugate subgroups.

If one is unsure whether conjugate subgroups exist, this can be looked up in the tables of Chapter 2.3 of this volume, where all conjugate subgroups are always mentioned and joined by a left brace.

Example 3.1.5.1.3

For space group $[Pm\overline{3}m]$ , No. 221, two subgroups $[Im\overline{3}m]$ ( $[2{\bf a},\,2{\bf b},\,2{\bf c}]$ ) with index 4 are listed. Each of them belongs to a set of four conjugate subgroups which differ in their origin locations ( [0,0,0] ; [-1,0,0;] [0, -1,0] ; [0,0, -1] for the first listed subgroup, referred to the coordinate system of $[Pm\overline{3}m]$ ). This can be seen by the coordinate values to be added ( [0,0,0] ; $[{{1}\over{2}},0,0]$ ; $[0,{{1}\over{2}},0]$ ; $[0,0,{{1}\over{2}}]$ ; coordinate system of $[I\,m\overline{3}m]$ ). In Chapter 2.3 , all four conjugate subgroups and their origin shifts are listed and joined by a brace.

3.1.5.2. Orientational conjugation

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In this case, the conjugate subgroups have differently oriented unit cells that are equivalent by a symmetry operation other than a translation of the space group. This occurs in the following cases: orthorhombic subgroups of hexagonal space groups; monoclinic subgroups of trigonal (including rhombohedral) space groups; rhombohedral and tetragonal subgroups of cubic space groups. In these cases, the corresponding cell and coordinate transformations are listed for all conjugate subgroups after the word `conjugate'. Their Wyckoff symbols, being the same for all conjugate subgroups, are not repeated.

Example 3.1.5.2.1

The cubic space group $[P\, \overline{4}3m]$ , No. 215, has three tetragonal conjugate subgroups $[P\, \overline{4}2m]$ . Their tetragonal c axes correspond to the cubic a, b or c axes, respectively. In $[P\, \overline{4}3m]$ , a, b and c are symmetry-equivalent by the threefold rotation axes.

References

Koch, E. (1984). The implications of normalizers on group–subgroup relations between space groups. Acta Cryst. A40, 593–600.Google Scholar

Müller, U. (1992). Berechnung der Anzahl möglicher Strukturtypen für Verbindungen mit dichtest gepackter Anionenteilstruktur. I. Das Rechenverfahren. Acta Cryst. B48, 172–178.Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 3.1, pp. 432-433