International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 285286

Convergentbeam electron diffraction, originating in the experiments of Kossel and Möllenstedt (Kossel & Möllenstedt, 1938) has been established over the past two decades as a powerful technique for the determination of space group in inorganic materials, with particular application when only microscopic samples are available. Relatively recently, with the introduction of the analytical electron microscope, this technique – abbreviated as CBED – has become available as a routine, so that there is now a considerable accumulation of data from a wide range of materials. A significant extension of the technique in recent times has been the introduction of LACBED (largeangle CBED) by Tanaka & Terauchi (1985). This technique allows an extensive angular range of single diffraction orders to be recorded and, although this method cannot be used for microdiffraction (since it requires an extensive singlecrystal area), new LACBED applications appear regularly, particularly in the field of semiconductor research (see Section 2.5.3.6).
The CBED method relies essentially on two basic properties of transmission electron diffraction, namely the radical departure from Friedel's law and the formation of characteristic extinction bands within spacegroupforbidden reflections. Departure from Friedel's law in electron diffraction was first noted experimentally by Miyake & Uyeda (1950). The prediction of spacegroupforbidden bands (within spacegroupforbidden reflections) by Cowley & Moodie (1959), on the other hand, was one of the first successes of Nbeam theory. A detailed explanation was later given by Gjønnes & Moodie (1965). These are known variously as `GM' bands (Tanaka et al., 1983), or more simply and definitively as `GS' (glide–screw) bands (this section). These extinctions have a close parallel with spacegroup extinctions in Xray diffraction, with the reservation that only screw axes of order two are accurately extinctive under Nbeam conditions. This arises from the property that only those operations which lead to identical projections of the asymmetric unit can have Nbeam dynamical symmetries (Cowley et al., 1961).
Additionally, CBED from perfect crystals produces highorder defect lines in the zeroorder pattern, analogous to the defect Kikuchi lines of inelastic scattering, which provide a sensitive measurement of unitcell parameters (Jones et al., 1977; Fraser et al., 1985; Tanaka & Terauchi, 1985).
The significant differences between Xray and electron diffraction, which may be exploited in analysis, arise as a consequence of a much stronger interaction in the case of electrons (Section 2.5.2). Hence, thin, approximately parallelsided crystal regions must be used in highenergy (100 kV–1 MV) electron transmission work, so that diffraction is produced from crystals effectively infinitely periodic in only two dimensions, leading to the relaxation of threedimensional diffraction conditions known as `excitation error' (Chapter 5.2 ). Also, there is the ability in CBED to obtain data from microscopic crystal regions of around 50 Å in diameter, with corresponding exposure times of several seconds, allowing a survey of a material to be carried out in a relatively short time.
In contrast, singlecrystal Xray diffraction provides much more limited symmetry information in a direct fashion [although statistical analysis of intensities (Wilson, 1949) will considerably supplement this information], but correspondingly gives much more direct threedimensional geometric data, including the determination of unitcell parameters and threedimensional extinctions.
The relative strengths and weaknesses of the two techniques make it useful where possible to collect both convergentbeam and Xray singlecrystal data in a combined study. However, all parameters can be obtained from convergentbeam and electrondiffraction data, even if in a somewhat less direct form, making possible spacegroup determination from microscopic crystals and microscopic regions of polygranular material. Several reviews of the subject are available (Tanaka, 1994; Steeds & Vincent, 1983; Steeds, 1979). In addition, an atlas of characteristic CBED patterns for direct phase identification of metal alloys has been published (Mansfield, 1984), and it is likely that this type of procedure, allowing Nbeam analysis by comparison with standard simulations, will be expanded in the near future.
Symmetry analysis is necessarily tied to examination of patterns near relevant zone axes, since the most intense Nbeam interaction occurs amongst the zerolayer zoneaxis reflections, with in addition a limited degree of upperlayer (higherorder Laue zone) interaction. There will generally be several useful zone axes accessible for a given parallelsided single crystal, with the regions between axes being of little use for symmetry analysis. Only one such zone axis can be parallel to a crystal surface normal, and a microcrystal is usually chosen at least initially to have this as the principal symmetry axis. Other zone axes from that crystal may suffer mild symmetry degradation because the Nbeam lattice component (`excitation error' extension) will not have the symmetry of the structure (Goodman, 1974; Eades et al., 1983).
Upperlayer interactions, responsible for imparting threedimensional information to the zero layer, are of two types: the first arising from `overlap' of dynamic shape transforms and causing smoothly varying modulations of the zerolayer reflections, and the second, caused by direct interactions with the upperlayer, or higherorder Laue zone lines, leading to a sharply defined fineline structure. These latter interactions are especially useful in increasing the accuracy of spacegroup determination (Tanaka et al., 1983), and may be enhanced by the use of lowtemperature specimen stages. The presence of these defect lines in convergentbeam discs, occurring especially in lowsymmetry zoneaxis patterns, allows symmetry elements to be related to the threedimensional structure (Section 2.5.3.5; Fig. 2.5.3.4c).
To the extent that such threedimensional effects can be ignored or are absent in the zerolayer pattern the projection approximation (Chapter 5.2 ) can be applied. This situation most commonly occurs in zoneaxis patterns taken from relatively thin crystals and provides a useful starting point for many analyses, by identifying the projected symmetry.
References
Cowley, J. M. & Moodie, A. F. (1959). The scattering of electrons by atoms and crystals. III. Singlecrystal diffraction patterns. Acta Cryst. 12, 360–367.Google ScholarCowley, J. M., Moodie, A. F., Miyake, S., Takagi, S. & Fujimoto, F. (1961). The extinction rules for reflections in symmetrical electron diffraction spot patterns. Acta Cryst. 14, 87–88.Google Scholar
Eades, J. A., Shannon, M. D. & Buxton, B. F. (1983). Crystal symmetry from electron diffraction. In Scanning electron microscopy, 1983/III, pp. 1051–1060. Chicago: SEM Inc.Google Scholar
Fraser, H. L., Maher, D. M., Humphreys, C. J., Hetherington, C. J. D., Knoell, R. V. & Bean, J. C. (1985). The detection of local strains in strained superlattices. In Microscopy of semiconducting materials, pp. 1–5. London: Institute of Physics.Google Scholar
Gjønnes, J. & Moodie, A. F. (1965). Extinction conditions in dynamic theory of electron diffraction patterns. Acta Cryst. 19, 65–67.Google Scholar
Goodman, P. (1974). The role of upper layer interactions in electron diffraction. Nature (London), 251, 698–701.Google Scholar
Jones, P. M., Rackham, G. M. & Steeds, J. W. (1977). Higher order Laue zone effects in electron diffraction and their use in lattice parameter determination. Proc. R. Soc. London Ser. A, 354, 197–222.Google Scholar
Kossel, W. & Möllenstedt, G. (1938). Electron interference in a convergent beam. Nature (London), 26, 660.Google Scholar
Mansfield, J. (1984). Convergent beam electron diffraction of alloy phases by the Bristol Group under the direction of John Steeds. Bristol: Adam Hilger.Google Scholar
Miyake, S. & Uyeda, R. (1950). An exception to Friedel's law in electron diffraction. Acta Cryst. 3, 314.Google Scholar
Steeds, J. W. (1979). Convergent beam electron diffraction. In Introduction to analytical electron microscopy, edited by J. J. Hren, J. I. Goldstein & D. C. Joy, pp. 387–422. New York: Plenum.Google Scholar
Steeds, J. W. & Vincent, R. (1983). Use of high symmetry zone axes in electron diffraction in determining crystal point and space groups. J. Appl. Cryst. 16, 317–324.Google Scholar
Tanaka, M. (1994). Convergentbeam electron diffraction. Acta Cryst. A50, 261–286.Google Scholar
Tanaka, M., Sekii, H. & Nagasawa, T. (1983). Space group determination by dynamic extinction in convergent beam electron diffraction. Acta Cryst. A39, 825–837.Google Scholar
Tanaka, M. & Terauchi, M. (1985). Convergentbeam electron diffraction. Tokyo: JEOL Ltd.Google Scholar
Wilson, A. J. C. (1949). The probability distribution of Xray intensities. Acta Cryst. 2, 318–321.Google Scholar