International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 781-782

Section 3.3.3.3. Non-symbolized symmetry elements

H. Burzlaffa and H. Zimmermannb*

aUniversität 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

3.3.3.3. 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, cf. Sections 1.4.2.4[link] and 1.5.4.1[link] . 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. Section 1.5.4.1[link] .

    Example

    Space group [C2/c\ (15)] has a twofold axis along b with screw part [{\bi w}_{g} = \pmatrix{0\cr 0\cr 0}]. The translational part of the centring operation is [{\bi g} = \let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0}].

    An additional axis parallel to b thus has a translation part [{\bi g} + {\bi w}_{g} = \let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0}]. The component [\pmatrix{0\cr \textstyle{1\over 2}\cr 0}] indicates a screw axis [2_{1}] in the b direction, whereas the component [\pmatrix{\textstyle{1\over 2}\cr 0\cr 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] (111) the translation vector along a can be decomposed as[{\bi w} = \pmatrix{1\cr 0\cr 0} = \let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0} + \pmatrix{\textstyle{1\over 2}\cr -\textstyle{1\over 2}\cr 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 [\let\normalbaselines\relax\openup2pt\pmatrix{\textstyle{1\over 2}\cr \textstyle{1\over 2}\cr 0}] passing through [\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 (for subgroups of space groups, cf. Section 1.7.1[link] ), it is convenient to introduce an extended Hermann–Mauguin symbol that comprises all symmetry elements indicated in (iii)[link] and (iv)[link]. The basic concept may be found in papers by Hermann (1929[link]) and in IT (1952[link]). These concepts have been applied by Bertaut (1976[link]) and Zimmermann (1976[link]); cf. Section 1.5.4.1[link] .

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 Hermann–Mauguin 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:[\displaylines{Pmma \sim Pmmb,\quad Pnma \sim Pmnb,\quad Pmna \sim Pnmb\quad \hbox{and}\cr \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).]
Bertaut, E. F. (1976). Study of principal subgroups and their general positions in C and I groups of class mmm – D2h. Acta Cryst. A32, 380–387.
Hermann, C. (1929). Zur systematischen Strukturtheorie IV. Untergruppen. Z. Kristallogr. 69, 533–555.
Zimmermann, H. (1976). Ableitung der Raumgruppen aus ihren klassengleichen Untergruppenbeziehungen. Z. Kristallogr. 143, 485–515.








































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