Monoclinic subgroups

Wondratschek, H.; Aroyo, M. I.

doi:10.1107/97809553602060000542

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. 2.1, p. 44 | 1 | 2 |

Section 2.1.2.5.1. Monoclinic subgroups

Hans Wondratschek^a ^* and Mois I. Aroyo^b

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and ^bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

2.1.2.5.1. Monoclinic subgroups

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Rules :

(a) If the monoclinic axis of $[{\cal H}]$ is the b or c axis of the basis of $[{\cal G}]$ , then the setting of $[{\cal H}]$ is also `unique axis b' or `unique axis c'. In particular, if $[{\cal G}]$ is monoclinic, then the settings of $[{\cal G}]$ and $[{\cal H}]$ agree.
(b) If the monoclinic axis of $[{\cal H}]$ is neither b nor c in the basis of $[{\cal G}]$ , then for $[{\cal H}]$ the setting `unique axis b' is chosen.
(c) The cell choice is always `cell choice 1' with the symbols C and c for unique axis b, and A and a for unique axis c.

Remarks (see also the following examples):

Rule (a) is valid for the many cases where the setting of $[{\cal H}]$ is `inherited' from $[{\cal G}]$ . In particular, this always holds for isomorphic subgroups.
Rule (b) is applied if $[{\cal G}]$ is orthorhombic and the monoclinic axis of $[{\cal H}]$ is the a axis of $[{\cal G}]$ and if $[{\cal H}]$ is a monoclinic subgroup of a trigonal group. Rule (b) is not natural, but specifies a preference for the setting `unique axis b'. This seems to be justified because the setting `unique axis b' is used more frequently in crystallographic papers and the standard short HM symbol is also referred to it.
Rule (c) implies a choice of that cell which is most explicitly described in the tables of IT A. By this choice, the centring type and the glide vector are fixed to the conventional values of `cell choice 1'.

The necessary adjustment is performed through a coordinate transformation, i.e. by a change of the basis and by an origin shift, see Section 2.1.3.3.

Example 2.1.2.5.1

$[{\cal G}=P12/m1]$ , No. 10; unique axis b.

II Maximal klassengleiche subgroups, Enlarged unit cell

$[[2]\, {\bf a}'=2{\bf a}]$ , both subgroups .

The monoclinic axis b is retained but the glide reflection a is converted into a glide reflection c ( is the conventional HM symbol for cell choice 1).
$[[2]\, {\bf b}'=2{\bf b},\, {\bf c}'=2{\bf c}]$ , all four subgroups .

The monoclinic axis b is retained but the A centring is converted into the conventional C centring ( is the conventional HM symbol for cell choice 1).
$[[2]\, {\bf a}'=2{\bf a},\, {\bf c}'=2{\bf c}]$ , both subgroups .

The monoclinic axis b is retained. The glide reflection is designated by `e' (simultaneous c- and a-glide reflection in the same plane perpendicular to b). The nonconventional B centring is converted into the conventional primitive setting P, by which the e-glide reflection also becomes a c-glide reflection.

Example 2.1.2.5.2

$[{\cal G}\,=\,P112/m]$ , No. 10; unique axis c.

II Maximal klassengleiche subgroups, Enlarged unit cell

$[[2]\, {\bf a}'=2{\bf a}]$ , both subgroups .

The monoclinic axis c and the glide reflection a are retained because is the conventional full HM symbol for unique axis c, cell choice 1.
$[[2]\, {\bf b}'=2{\bf b},\, {\bf c}'=2{\bf c}]$ , all four subgroups .

The monoclinic axis c and the A centring are retained because is the conventional full HM symbol for this setting.
$[[2]\, {\bf a}'=2{\bf a},\, {\bf b}'=2{\bf b}]$ , both subgroups .

The monoclinic axis c is retained. The glide reflection is designated by `e' (simultaneous a- and b-glide reflection in the same plane perpendicular to c). The nonconventional C centring is converted into the conventional primitive setting P, by which the e-glide reflection also becomes an a-glide reflection.

Example 2.1.2.5.3

$[{\cal G}\,=\,Pban]$ , No. 50; origin choice 1.

I Maximal (monoclinic) translationengleiche subgroups

$[[2]\, P112/n]$ : conventional unique axis c; nonconventional glide reflection n. The monoclinic axis c is retained but the glide reflection n is adjusted to a glide reflection a in order to conform to the conventional symbol of cell choice 1.
$[[2]\, P12/a1]$ : conventional unique axis b; nonconventional glide reflection a. The monoclinic axis b is retained but the glide reflection a is adjusted to a glide reflection c of the conventional symbol , cell choice 1.
$[[2]\, P2/b11]$ : nonconventional monoclinic unique axis a; nonconventional glide reflection b. The monoclinic axis a is transformed to the conventional unique axis b; the glide reflection b is adjusted to the conventional symbol of the setting unique axis b, cell choice 1.

References

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, p. 44