InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 334
## Section 2.5.3.3.5. Symmetry elements observed by CBED M. Tanaka
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In the above sections, point-group and space-group determination methods were described following the theory of Buxton *et al*. (1976). They assumed that the observable symmetry elements are those of an infinitely extended parallel-sided specimen or of diperiodic plane figures. CBED patterns determine diffraction groups. Crystal point groups are identified by consulting Fig. 2.5.3.4, which gives the relations between diffraction groups and crystal point groups. When the assumption made by Buxton *et al*. (1976) is accepted in a strict sense, CBED symmetry *m*_{2} caused by a twofold rotation axis oblique to the specimen surface, which is not a symmetry element of a diperiodic plane figure, ought not to be observed. However, the symmetry *m*_{2} due to a twofold rotation axis in the direction of an Si film with [100] surface normal has been clearly observed at [111] electron incidence (Tanaka *et al.*, 1988, p. 33). This indicates that crystal symmetry elements oblique to the specimen surface are observable when the specimen is tilted. An important condition for CBED is that the top and bottom surfaces be parallel over the specimen area illuminated by the incident beam. CBED observes the symmetry elements of a crystal to the extent that the boundary conditions at the specimen surface do not break the symmetries of the CBED patterns. Gjønnes & Gjønnes (1985) reported that the breaking of CBED symmetry due to a surface oblique to the incident beam is practically negligible.

In Section 2.5.3.3 on space-group determination, space-group symmetry elements of crystals which have glide and screw components parallel to the specimen surface were considered to act as space-group symmetry elements of diperiodic plane figures by mitigating the strict application of the assumption of diperiodic plane figures. In fact, vertical glide planes with a glide vector not parallel to the specimen surface, which were dealt in Section 2.5.3.3.4, are not the symmetry elements of diperiodic plane figures. Ishizuka (1982) showed theoretically that a vertical glide plane with a vertical glide vector produces dynamical extinction lines in HOLZ discs if the Laue zones are well separated. Tanaka *et al.* (1988, pp. 214–225) tabulated the extinction lines appearing in HOLZ discs caused by the vertical glide planes whose glide vectors are not only parallel but also not parallel to the specimen surface. Dynamical extinction lines caused by the glide planes with a glide vector not parallel to the surface have been demonstrated using FeS_{2} and MgAl_{2}O_{4} (Tanaka *et al.*, 1988, pp. 51–61).

Vertical 2_{1}, 3_{1}, 3_{2}, …, 6_{5} screw axes, which are not symmetry elements of diperiodic plane figures, are expected to form dynamical extinction lines in kinematically forbidden reflections that are located in the direction of the screw axes or of the surface normal. The extinction lines, however, are difficult to observe in ordinary CBED. Thus, CBED does not observe all the symmetry elements of the crystal space groups but observes many more symmetry elements than those of the diperiodic plane figures. It is clear now that it makes no sense to construct space groups using actually observable symmetry elements because they do not form a complete set of groups. It is of no importance to give the relation between the 230 space groups of crystals and the 80 space groups of diperiodic plane figures. Buxton *et al.*'s theory, which determines crystal point groups with the help of diperiodic plane figures, is very beautiful and successful. However, it is not correct to state that CBED observes the symmetry elements of the diperiodic plane figures. The use of the groups of diperiodic plane figures should be recognized as a convention for the sake of convenience. As a further example, horizontal screw axes and horizontal glide planes must be located at the middle of a specimen to form symmetry elements of the diperiodic plane figures. However, those screw axes and glide planes which are not located at the middle of a specimen do produce CBED symmetries. Since we now know that CBED does not observe the symmetries of the diperiodic plane figures but observes those of a physical crystalline specimen, we can determine the corresponding infinite crystal symmetries more freely, by using our knowledge of the symmetries of the sample concerned, guided but not restricted by the beautiful theory of Buxton *et al.* (1976).

One point to note, for symmetry determination, is that one has to be aware of spurious symmetries that appear for crystals of certain structure types (Tanaka *et al.*, 1988, pp. 20–32 and 42–45) and destroy the correct determination of the point and space groups. Another point for precise symmetry determination is that one has to be aware of how CBED symmetry is destroyed by a small breakdown of crystal symmetry (Tanaka *et al.*, 1988 pp. 46–47).

### References

Buxton, B., Eades, J. A., Steeds, J. W. & Rackham, G. M. (1976).*The symmetry of electron diffraction zone axis patterns. Philos. Trans. R. Soc. London Ser. A*,

**181**, 171–193.

Gjønnes, J. & Gjønnes, K. (1985).

*Bloch wave symmetries and inclined surfaces*.

*Ultramicroscopy*,

**18**, 77–82.

Ishizuka, K. (1982).

*Translation symmetries in convergent-beam electron-diffraction*.

*Ultramicroscopy*,

**9**, 255–258.

Tanaka, M., Terauchi, M. & Kaneyama, T. (1988).

*Convergent-Beam Electron Diffraction II*. Tokyo: JEOL Ltd.