Non-symbolized symmetry elements

Burzlaff, H.; Zimmermann, H.

doi:10.1107/97809553602060000526

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. 12.3, pp. 831-832

Section 12.3.3. Non-symbolized symmetry elements

H. Burzlaff^a and H. Zimmermann^b ^*

^a Universität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and ^bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail: helmuth.zimmermann@krist.uni-erlangen.de

12.3.3. Non-symbolized symmetry elements

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Certain symmetry elements are not given explicitly in the full symbol because they can easily be derived. They are:

(i) Rotoinversion axes that are not used to indicate the lattice symmetry directions.
(ii) Rotation axes 2 included in the axes 4, $[\overline{4}]$ and 6 and rotation axes 3 included in the axes $[\overline{3}]$ , 6 and $[\overline{6}]$ .
(iii) Additional symmetry elements occurring in space groups with centred unit cells. These types of operation can be deduced from the product of the centring translation (I, g) with a symmetry operation (W, w). The new symmetry operation $[({\bi W},\; {\bi g} + {\bi w})]$ again has W as rotation part but a different glide/screw part if the component of g parallel to the symmetry element corresponding to W is not a lattice vector; cf. Chapter 4.1 .

Example

Space group $[C2/c\ (15)]$ has a twofold axis along b with screw part $[{\bi w}_{g} = (0,0,0)]$ . The translational part of the centring operation is $[{\bi g} = (\textstyle{1\over 2},\textstyle{1\over 2},0)]$ .

An additional axis parallel to b thus has a translation part $[{\bi g} + {\bi w}_{g} = (\textstyle{1\over 2},\textstyle{1\over 2},0)]$ . The component $[(0,\textstyle{1\over 2},0)]$ indicates a screw axis $[2_{1}]$ in the y direction, whereas the component $[(\textstyle{1\over 2},0,0)]$ indicates the location of this axis in $[(\textstyle{1\over 4},y,0)]$ . Similarly, it can be shown that glide plane c combined with the centring gives a glide plane n.

In the same way, in rhombohedral and cubic space groups, a rotation axis 3 is accompanied by screw axes $[3_{1}]$ and $[3_{2}]$ .

In space groups with centred unit cells, the location parts of different symmetry elements may coincide. In $[I\overline{4}2m]$ , for example, the mirror plane m contains simultaneously a non-symbolized glide plane n. The same applies to all mirror planes in Fmmm.
(iv) Symmetry elements with diagonal orientation always occur with different types of glide/screw parts simultaneously. In space group $[P\overline{4}2m]$ the translation vector along a can be decomposed as $[{\bi w} = (1,0,0) = (\textstyle{1\over 2},\textstyle{1\over 2},0) + (\textstyle{1\over 2}, -\textstyle{1\over 2},0) = {\bi w}_{g} + {\bi w}_{l}.]$ The diagonal mirror plane with normal along $[[1\overline{1}0]]$ passing through the origin is accompanied by a parallel glide plane with glide part $[(\textstyle{1\over 2},\textstyle{1\over 2},0)]$ shifted by $[(\textstyle{1\over 4},-\textstyle{1\over 4},0)]$ . The same arguments lead to the occurrence of screw axes $[2_{1}]$ , $[3_{1}]$ and $[3_{2}]$ connected with diagonal rotation axes 2 or 3.
(v) For some investigations connected with klassengleiche subgroups , it is convenient to introduce an extended full space-group symbol that comprises all symmetry elements indicated in (iii) and (iv). The basic concept may be found in papers by Hermann (1929) and in IT (1952). These concepts have been applied by Bertaut (1976) and Zimmermann (1976); cf. Part 4 .

Example

The full symbol of space group Imma (74) is $[I{2_{1} \over m} {2_{1} \over m} {2_{1} \over a}.]$

The I-centring operation introduces additional rotation axes and glide planes for all three sets of lattice symmetry directions. The extended full symbol is $[I{2,2_{1} \over m,n} {2,2_{1} \over m,n} {2,2_{1} \over a,b} \quad \hbox{or} \quad I\openup2pt\matrix{\displaystyle{2_{1} \over m} &\displaystyle{2_{1} \over m} &\displaystyle{2_{1} \over a}\cr \displaystyle{2 \over n} &\displaystyle{2 \over n} &\displaystyle{2 \over b}\cr}.]$ This symbol shows immediately the eight subgroups with a P lattice corresponding to point group mmm: $[Pmma \sim Pmmb,\quad Pnma \sim Pmnb,\quad Pmna \sim Pnmb\quad \hbox{and}\quad Pnna \sim Pnnb.\hfill]$

References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]Google Scholar

Bertaut, E. F. (1976). Study of principal subgroups and their general positions in C and I groups of class mmm – D_2h. Acta Cryst. A32, 380–387.Google Scholar

Hermann, C. (1929). Zur systematischen Strukturtheorie IV. Untergruppen. Z. Kristallogr. 69, 533–555.Google Scholar

Zimmermann, H. (1976). Ableitung der Raumgruppen aus ihren klassengleichen Untergruppenbeziehungen. Z. Kristallogr. 143, 485–515.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 12.3, pp. 831-832