International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 2.2, p. 17

Section 2.2.2. Space groups with more than one description

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.2. Space groups with more than one description

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For several space groups, more than one description is available. Three cases occur:

  • (i) Two choices of origin (cf. Section 2.2.7[link])

    For all centrosymmetric space groups, the tables contain a description with a centre of symmetry as origin. Some centrosymmetric space groups, however, contain points of high site symmetry that do not coincide with a centre of symmetry. For these 24 cases, a further description (including diagrams) with a high-symmetry point as origin is provided. Neither of the two origin choices is considered standard. Noncentrosymmetric space groups and all plane groups are described with only one choice of origin.

    Examples

    • (1) Pnnn (48)

      Origin choice 1 at a point with site symmetry 222

      Origin choice 2 at a centre with site symmetry [\bar{1}].

    • (2) [Fd\bar{3}m] (227)

      Origin choice 1 at a point with site symmetry [\bar{4}3m]

      Origin choice 2 at a centre with site symmetry [\bar{3}m].

  • (ii) Monoclinic space groups

    Two complete descriptions are given for each of the 13 monoclinic space groups, one for the setting with `unique axis b', followed by one for the setting with `unique axis c'.

    Additional descriptions in synoptic form are provided for the following eight monoclinic space groups with centred lattices or glide planes:[\eqalign{&C2\ (5), Pc\ (7), Cm\ (8), Cc\ (9), C2/m\ (12), P2/c\ (13),\hfill\cr &P2_{1}/c\ (14), C2/c\ (15)\hfill}]These synoptic descriptions consist of abbreviated treatments for three `cell choices', here called `cell choices 1, 2 and 3'. Cell choice 1 corresponds to the complete treatment, mentioned above; for comparative purposes, it is repeated among the synoptic descriptions which, for each setting, are printed on two facing pages. The cell choices and their relations are explained in Section 2.2.16[link].

  • (iii) Rhombohedral space groups

    The seven rhombohedral space groups R3 (146), [R\bar{3}] (148), R32 (155), R3m (160), R3c (161), [R\bar{3}m] (166), and [R\bar{3}c] (167) are described with two coordinate systems, first with hexagonal axes (triple hexagonal cell) and second with rhombohedral axes (primitive rhombohedral cell). For both descriptions, the same space-group symbol is used. The relations between the cell parameters of the two cells are listed in Chapter 2.1[link] .

    The hexagonal triple cell is given in the obverse setting (centring points [{2 \over 3}, {1 \over 3}, {1 \over 3};\; {1 \over 3}, {2 \over 3}, {2 \over 3}]). In IT (1935)[link], the reverse setting (centring points [{1 \over 3}, {2 \over 3}, {1 \over 3}; \;{2 \over 3}, {1 \over 3}, {2 \over 3}]) was employed; cf. Chapter 1.2[link] .

References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]








































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