International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 7-8   | 1 | 2 |

Section 1.2.4.  International (Hermann–Mauguin) symbols for subperiodic groups

V. Kopskýa and D. B. Litvinb*

a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA
Correspondence e-mail:  u3c@psu.edu

1.2.4. International (Hermann–Mauguin) symbols for subperiodic groups

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Both the short and the full Hermann–Mauguin symbols consist of two parts: (i) a letter indicating the centring type of the conventional cell, and (ii) a set of characters indicating symmetry elements of the subperiodic group.

  • (i) The letters for the two centring types for layer groups are the lower-case italic letter p for a primitive cell and the lower-case italic letter c for a centred cell. For rod and frieze groups there is only one centring type, the one-dimensional primitive cell, which is denoted by the lower-case script letter [{\scr p}].

  • (ii) The one or three entries after the centring letter refer to the one or three kinds of symmetry directions of the conventional crystallographic basis. Symmetry directions occur either as singular directions or as sets of symmetrically equivalent symmetry directions. Only one representative of each set is given. The sets of symmetry directions and their sequence in the Hermann–Mauguin symbol are summarized in Table 1.2.4.1[link].

    Table 1.2.4.1 | top | pdf |
    Sets of symmetry directions and their positions in the Hermann–Mauguin symbol

    In the standard setting, periodic directions are [100] and [010] for the layer groups, [001] for the rod groups, and [10] for the frieze groups.

    (a) Layer groups and rod groups.

      Symmetry direction (position in Hermann–Mauguin symbol)
    Primary Secondary Tertiary
    Triclinic None
    Monoclinic [100] [010] [001]
    Orthorhombic
    Tetragonal [001] [[100]] [[1\bar{1}0]]
    [[010]] [[110]]
    Trigonal [001] [[100]] [[1\bar{1}0]]
    Hexagonal [[010]] [[120]]
    [[\bar{1}\bar{1}0]] [[\bar{2}\bar{1}0]]

    (b) Frieze groups.

      Symmetry direction (position in Hermann–Mauguin symbol)
    Primary Secondary Tertiary
    Oblique Rotation point in plane    
    Rectangular [10] [01]

Each position in the Hermann–Mauguin symbol contains one or two characters designating symmetry elements, axes and planes that occur for the corresponding crystallographic symmetry direction. Symmetry planes are represented by their normals; if a symmetry axis and a normal to a symmetry plane are parallel, the two characters are separated by a slash, e.g. the 4/m in [{\scr p}4/mcc] (R40). Crystallographic symmetry directions that carry no symmetry elements are denoted by the symbol `1', e.g. p3m1 (L69) and p112 (L2). If no misinterpretation is possible, entries `1' at the end of the symbol are omitted, as in p4 (L49) instead of p411. Subperiodic groups that have in addition to translations no symmetry directions or only centres of symmetry have only one entry after the centring letter. These are the layer-group types p1 (L1) and [p\bar{1}] (L2), the rod-group types [{\scr p}1] (R1) and [{\scr p}\bar{1}] (R2), and the frieze group [{\scr p}1] (F1).








































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