Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section Identification of three-dimensional symmetry elements

M. Tanakaf 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[link]) 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[link]), Buxton et al. (1976[link]) and Tanaka (1989[link]). 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.[link]. A horizontal twofold rotation axis 2′, a horizontal mirror plane m′, an inversion centre i and a fourfold rotary inversion [\bar 4] produce symmetries mR (m2), 1R, 2R and 4R in DPs, respectively.


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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 [\bar 4], respectively.

Next we explain the symbols of the CBED symmetries. (1) Operation mR is shown in the left-hand part of Fig.[link](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.[link]a). The mirror symmetry is labelled m2 after the twofold rotation axis. (2) Operation 1R (Fig.[link]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.[link]c). The symmetry is called translational symmetry after Goodman (1975[link]) 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.[link]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.


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.

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