International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 9.7, pp. 897-899
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Wilson (1993d) classified the space groups by degree of symmorphism. A fully symmorphic space group contains only the `syntropic' symmetry elements
and a fully antimorphic space group contains only the `antitropic' elements
The remaining symmetry elements
are `atropic'. The two triclinic space groups, P1 and
, contain only `atropic' elements, and are thus not classified by these criteria. The rest are divided into five groups, in accordance with the balance of symmetry elements within the unit cell. For the 71 non-triclinic space groups symmorphic in the strict sense (Wilson, 1993d
; Subsection 1.4.2.1
), the classification gives:
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The distribution of the 230 space groups (11 enantiomorphic couples merged) by arithmetic crystal class and degree of symmorphism is given in Table 9.7.1.2.
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A few points about the symmorphic groups are worth noting. The 14 `fully symmorphic' space groups are those that (i) have primitive cells and (ii) have no secondary or tertiary axes (three each monoclinic, orthorhombic, tetragonal, hexagonal; two trigonal; no cubic). Secondary axes, even though syntropic in the conventional space-group notation, generate additional antitropic axes in accordance with the principles set out by Bertaut (2005, Chap. 4.1
). These additional axes are not indicated in the `full' Hermann–Mauguin space-group symbol, but should appear in the `extended' symbol (Bertaut, 2005
, Table 4.3.2.1
4). As a result of the additional axes, 21 symmorphic space groups with primitive cells are shifted to the `tending to symmorphism' column (five tetragonal, six trigonal, five hexagonal, five cubic). Lattice centring has a similar or greater effect; the 36 centred symmorphic space groups are spread over the three columns `tending to symmorphism' (seven), `equally balanced' (20) and `tending to antimorphism' (nine).
From the nature of the definitions, no symmorphic space group can be `fully antimorphic'. The 14 groups under the latter heading consist of (i) 12 space groups with no special positions (Subsection 9.7.4.1), and (ii) two space groups whose only special positions have symmetry
(Subsection 9.7.4.2
). On these criteria, the two triclinic groups excluded from discussion would fall naturally into the column `fully antimorphic'. The remaining space groups have no obvious outstanding characteristics. Most of them fall under the heading `tending to antimorphism', though there are some in each of the columns `tending to symmorphism' and `equally balanced'.
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