Relations between symmetry elements

Bertaut, E. F.

doi:10.1107/97809553602060000509

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

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International Tables for Crystallography (2006). Vol. A. ch. 4.3, p. 71

Section 4.3.4.2. Relations between symmetry elements

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

4.3.4.2. Relations between symmetry elements

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In the crystal classes 42(2), 4m(m), $[\bar{4}2\hbox{(}m\hbox{)}]$ or $[\bar{4}m\hbox{(}2\hbox{)}]$ , $[4/m\;2/m\;\hbox{(}2/m\hbox{)}]$ , where the tertiary symmetry elements are between parentheses, one finds $[4 \times m = (m) = \bar{4} \times 2; \ 4 \times 2 = (2) = \bar{4} \times m.]$ Analogous relations hold for the space groups. In order to have the symmetry direction of the tertiary symmetry elements along [ $[1\bar{1}0]$ ] (cf. Table 2.2.4.1 ), one has to choose the primary and secondary symmetry elements in the product rule along [001] and [010].

Example

In $[P4_{1}2(2)\ (91)]$ , one has $[4_{1} \times 2 = (2)]$ so that $[P4_{1}2]$ would be the short symbol. In fact, in IT (1935), the tertiary symmetry element was suppressed for all groups of class 422, but re-established in IT (1952), the main reason being the generation of the fourfold rotation as the product of the secondary and tertiary symmetry operations: $[4 = (m) \times m]$ etc.

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).]Google Scholar

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 4.3, p. 71