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. 310
## Section 2.5.3.2.3. Identification of two-dimensional symmetry elements M. Tanaka
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Two-dimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zone-axis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a bright-field pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the bright-field disc [the central or `direct' (000) beam]. The WP is composed of the BP and the pattern formed by the surrounding diffraction discs, which are not exactly excited. The two-dimensional symmetry elements *m*, 1, 2, 3, 4 and 6 yield symmetry *m*_{v} and one-, two-, three-, four- and sixfold rotation symmetries, respectively, in WPs, where the suffix *v* for *m*_{v} is assigned to distinguish it from mirror symmetry *m*_{2} caused by a horizontal twofold rotation axis.

It should be noted that a BP shows not only the zone-axis symmetry but also three-dimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to three-dimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the three-dimensional symmetry elements *m*′, *i*, 2′ and produce, respectively, symmetries 1_{R}, 1, *m*_{2} and 4 in the BP, instead of 1_{R}, 2_{R}, *m*_{2} and 4_{R} in the DPs (Fig. 2.5.3.2). We mention that the BP cannot distinguish whether a specimen crystal has an inversion centre or not, because an inversion centre forms the lowest symmetry 1 in the BP.

In conclusion, all the two-dimensional symmetry elements can be identified from the WP symmetries.