International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 309310
Section 2.5.3.2.2. Identification of threedimensional symmetry elements
M. Tanaka^{f}

It is difficult to imagine the symmetries in CBED patterns generated by the threedimensional symmetry elements of the sample. The reason is that if a threedimensional 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 threedimensional 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 threedimensional symmetries do not appear in the zoneaxis patterns, but do in a diffraction disc set at the Bragg condition, each of which we call a darkfield 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 m_{R} (m_{2}), 1_{R}, 2_{R} and 4_{R} in DPs, respectively.

Illustration of symmetries appearing in darkfield patterns (DPs). (a) m_{R} and m_{2}; (b) 1_{R}; (c) 2_{R}; (d) 4_{R}, originating from 2′, m′, i and , respectively. 
Next we explain the symbols of the CBED symmetries. (1) Operation m_{R} is shown in the lefthand 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 m_{R}. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the lefthand 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 righthand side figure of Fig. 2.5.3.2a). The mirror symmetry is labelled m_{2} after the twofold rotation axis. (2) Operation 1_{R} (Fig. 2.5.3.2b) 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 2_{R} 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. 2.5.3.2c). 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 darkfield patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4_{R} (Fig. 2.5.3.2d) 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 darkfield pattern. As far as CBED symmetries are concerned, we do not use the term darkfield pattern if a disc does not contain the exact Bragg position.
The four threedimensional 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 threedimensional spacegroup analysis using convergentbeam 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.