International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 309-310   | 1 | 2 |

## Section 2.5.3.2.2. Identification of three-dimensional symmetry elements

M. Tanakaf

#### 2.5.3.2.2. Identification of three-dimensional symmetry elements

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It is difficult to imagine the symmetries in CBED patterns generated by the three-dimensional symmetry elements of the sample. The reason is that if a three-dimensional symmetry element is applied to a specimen, it turns it upside down, which is impractical in most experiments. The reciprocity theorem of scattering theory (Pogany & Turner, 1968 ) enables us to clarify the symmetries of CBED patterns expected from these three-dimensional symmetry elements. A graphical method for obtaining CBED symmetries due to sample symmetry elements is described in the papers of Goodman (1975 ), Buxton et al. (1976 ) and Tanaka (1989 ). The CBED symmetries of the three-dimensional symmetries do not appear in the zone-axis patterns, but do in a diffraction disc set at the Bragg condition, each of which we call a dark-field pattern (DP). The CBED symmetries obtained are illustrated in Fig. 2.5.3.2 . A horizontal twofold rotation axis 2′, a horizontal mirror plane m′, an inversion centre i and a fourfold rotary inversion produce symmetries mR (m2), 1R, 2R and 4R in DPs, respectively. Figure 2.5.3.2 | top | pdf |Illustration of symmetries appearing in dark-field patterns (DPs). (a) mR and m2; (b) 1R; (c) 2R; (d) 4R, originating from 2′, m′, i and , respectively.

Next we explain the symbols of the CBED symmetries. (1) Operation mR is shown in the left-hand part of Fig. 2.5.3.2 (a), which implies successive operations of (a) a mirror m with respect to a twofold rotation axis, transforming an open circle beam (○) in reflection G into a beam (+) in reflection G′ and (b) rotation R of this beam by π about the centre point of disc G′ (or the exact Bragg position of reflection G′), resulting in position ○ in reflection G′. The combination of the two operations is written as mR. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the left-hand part of the figure become one reflection G, and a mirror symmetry, whose mirror line is perpendicular to vector G and passes through the centre of disc G, appears between the two beams (the right-hand side figure of Fig. 2.5.3.2 a). The mirror symmetry is labelled m2 after the twofold rotation axis. (2) Operation 1R (Fig. 2.5.3.2 b) for a horizontal mirror plane is a combination of a rotation by 2π of a beam (○) about a zone axis O (symbol 1), which is equivalent to no rotation, and a rotation by π of the beam about the exact Bragg position or the centre of disc G. (3) Operation 2R is a rotation by π of a beam (○) in reflection G about a zone axis (symbol 2), which transforms the beam into a beam (+) in reflection −G, followed by a rotation by π of the beam (+) about the centre of disc −G, resulting in the beam (○) in disc −G (Fig. 2.5.3.2 c). The symmetry is called translational symmetry after Goodman (1975 ) because the pattern of disc +G coincides with that of disc −G by a translation. It is emphasized that an inversion centre is identified by the test of translational symmetry about a pair of ±G dark-field patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4R (Fig. 2.5.3.2 d) can be understood in a similar manner. It is noted that regular letters are symmetries about a zone axis, while subscripts R represent symmetries about the exact Bragg position. We call a pattern that contains an exact Bragg position (if possible at the disc centre) a dark-field pattern. As far as CBED symmetries are concerned, we do not use the term dark-field pattern if a disc does not contain the exact Bragg position.

The four three-dimensional symmetry elements are found to produce different symmetries in the DPs. These facts imply that these symmetry elements can be identified unambiguously from the symmetries of CBED patterns.

### 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.
Goodman, P. (1975). A practical method of three-dimensional space-group analysis using convergent-beam electron diffraction. Acta Cryst. A31, 804–810.
Pogany, A. P. & Turner, P. S. (1968). Reciprocity in electron diffraction and microscopy. Acta Cryst. A24, 103–109.
Tanaka, M. (1989). Symmetry analysis. J. Electron Microsc. Tech. 13, 27–39.