Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 310   | 1 | 2 |

Section Identification of two-dimensional symmetry elements

M. Tanakaf Identification of two-dimensional symmetry elements

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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 mv and one-, two-, three-, four- and sixfold rotation symmetries, respectively, in WPs, where the suffix v for mv is assigned to distinguish it from mirror symmetry m2 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 [\bar 4] produce, respectively, symmetries 1R, 1, m2 and 4 in the BP, instead of 1R, 2R, m2 and 4R in the DPs (Fig.[link]). 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.

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