Subperiodic group types

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000647

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

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International Tables for Crystallography (2006). Vol. E. ch. 1.2, p. 5 | 1 | 2 |

Section 1.2.1.1. Subperiodic group types

V. Kopský^a and D. B. Litvin^b ^*

^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

1.2.1.1. Subperiodic group types

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The subperiodic groups are classified into affine subperiodic group types, i.e. affine equivalence classes of subperiodic groups. There are 80 affine layer-group types and seven affine frieze-group types. There are 67 crystallographic and an infinity of noncrystallographic affine rod-group types. We shall consider here only rod groups of the 67 crystallographic rod-group types and refer to these crystallographic affine rod-group types simply as affine rod-group types.

The subperiodic groups are also classified into proper affine subperiodic group types, i.e. proper affine classes of subperiodic groups. For layer and frieze groups, the two classifications are identical. For rod groups, each of eight affine rod-group types splits into a pair of enantiomorphic crystallographic rod-group types. Consequently, there are 75 proper affine rod-group types. The eight pairs of enantiomorphic rod-group types are $[{\scr p}4_1]$ (R24), $[{\scr p}4_3]$ (R26); $[{\scr p}4_122]$ (R31), $[{\scr p}4_322]$ (R33); $[{\scr p}3_1]$ (R43), $[{\scr p}3_2]$ (R44); $[{\scr p}3_112]$ (R47), $[{\scr p}3_212]$ (R48); $[{\scr p}6_1]$ (R54), $[{\scr p}6_5]$ (R58); $[{\scr p}6_2]$ (R55), $[{\scr p}6_4]$ (R57); $[{\scr p}6_122]$ (R63), $[{\scr p}6_522]$ (R67); and $[{\scr p}6_222]$ (R64), $[{\scr p}6_422]$ (R66). (Each subperiodic group is given in the text by its Hermann–Mauguin symbol followed in parenthesis by a letter L, R or F to denote it, respectively, as a layer, rod or frieze group, and its sequential numbering from Parts 2 , 3 or 4 .) We shall refer to the proper affine subperiodic group types simply as subperiodic group types.

References

International Tables for Crystallography (2006). Vol. E. ch. 1.2, p. 5