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

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

Section 3.1.6. Monoclinic and triclinic subgroups

Ulrich Müllera*

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

3.1.6. Monoclinic and triclinic subgroups

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Aside from the two choices for the unique axis and the three possible cell choices given in Volume A, the unit cell of a monoclinic space group allows many more settingsthat can be interconverted by transformations such as [{\bf a}\pm q{\bf c},\,{\bf b},\,{\bf c}] with an integer value for q (Sayari & Billiet, 1977[link]). The most commonly chosen cell is the one with the shortest basis vectors a and c and a non-acute angle β. For triclinic space groups the `reduced' cell is preferred, which depends on the metric values of the lattice (Billiet & Rolley Le Coz, 1980[link]).

Some relations always require a cell transformation, for example rhombohedral to monoclinic relations. A group–subgroup relation in which the subgroup is monoclinic or triclinic can always be chosen together with a cell transformation that produces one of the cells mentioned. The transformation to be chosen depends on the cell metrics of the starting space group. For general tables we therefore cannot specify a priori the kind of cell transformation that will be needed.

The settings listed for monoclinic and triclinic subgroups were chosen in such a way that axes transformations are avoided or kept to a minimum. Depending on the cell metrics, this may result in cells that do not have the shortest possible basis vectors. Unfortunately, transformation of a monoclinic or triclinic cell setting to another one may cause an interchange of Wyckoff labels(within the Wyckoff sets). Frequently, several possible cell settings of the same monoclinic subgroup have been listed; the entry for the subgroup then is followed by the word `or' or `alternative', plus another entry.

Example 3.1.6.1

Space group Cmcm, No. 63, has the subgroup [P\,1\,1\,2_1/m], No. 11. It requires a cell transformation which is given as [{\bf a},\,{{1}\over{2}}(-{\bf a}+{\bf b}),\,{\bf c}]. The following two lines list two other possible cell transformations for the same subgroup after the words `or': [{{1}\over{2}}({\bf a}-{\bf b}),\,{\bf b},\,{\bf c}] and [{{1}\over{2}}({\bf a}-{\bf b}),\,{{1}\over{2}}({\bf a}+{\bf b}),\,{\bf c}]. These three options cause different relations for the Wyckoff positions 4b and 8d of Cmcm.

Caution should also be exerted when different cell choices of monoclinic cells are involved. Monoclinic subgroups may refer to any of the three cell choices listed in Volume A. As long as these cell choices are used as listed in Volume A, no problems should arise. However, interconversions from one setting to another and especially nonconventional settings require special attention.

Example 3.1.6.2

The common setting of space group No. 15 is [C12/c1], which means unique axis b and cell choice 1; the glide plane c is located at [y=0] (and [y={{1}\over{2}}]). By interchanging the axes b and [-{\bf c}], [C12/c1] becomes [B112/b] with the b glide plane at [z=0]. This was the setting listed in International Tables for X-ray Crystallography (1952, 1965, 1969[link]) for unique axis c. However, since the 1983 edition of Volume A, [B112/b] does not correspond to one of the listed cell choices. Instead, they are now [A112/a] (cell choice 1) or [B112/n] (cell choice 2) or [I112/b] (cell choice 3). Note that for all three cell choices the glide plane mentioned in the symbol is at [z=0]. [B112/n] also has a glide plane in the b direction, but unlike [B112/b] it is at [z={{1}\over{4}}]. [B112/n] and [B112/b] can be set up with the same unit-cell dimensions, but with origins shifted by [{{1}\over{4}}, 0, {{1}\over{4}}]. The full Hermann–Mauguin symbol always shows uniquely which is the setting.

References

Billiet, Y. & Rolley Le Coz, M. (1980). Le groupe P1 et ses subgroupes. II. Tables des sous-groupes. Acta Cryst. A36, 242–248.Google Scholar
International Tables for X-ray Crystallography (1952, 1965, 1969). Vol. I, Symmetry groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.Google Scholar
Sayari, A. & Billiet, Y. (1977). Subgroups and changes of standard settings of triclinic and monoclinic space groups. Acta Cryst. A33, 985–986.Google Scholar








































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