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. 286288

Convergentbeam diffraction symmetries are those of Schrödinger's equation, i.e. of crystal potential, plus the diffracting electron. The appropriate equation is given in Section 2.5.2 [equation (2.5.2.6)] and Chapter 5.2 [equation (5.2.2.1)] in terms of the realspace wavefunction ψ. The symmetry elements of the crystal responsible for generating pattern symmetries may be conveniently classified as of two types (I and II) as follows.

A minimal summary of basic theoretical points, otherwise found in Chapter 5.2 and numerous referenced articles, is given here.
For a specific zerolayer diffraction order the incident and diffracted vectors are and . Then the threedimensional vector has the patternspace projection, . The point gives the symmetrical Bragg condition for the associated diffraction disc, and is identifiable with the angular deviation of from the vertical z axis in threedimensional space (see Fig. 2.5.3.1). also defines the symmetry centre within the twodimensional disc diagram (Fig. 2.5.3.2); namely, the intersection of the lines S and G, given by the trace of excitation error, , and the perpendicular line directed towards the reciprocalspace origin, respectively. To be definitive it is necessary to index diffracted amplitudes relating to a fixed crystal thickness and wavelength, with both crystallographic and momentum coordinates, as , to handle the continuous variation of (for a particular diffraction order), with angles of incidence as determined by , and registered in the diffraction plane as the projection of .
Reciprocity was introduced into the subject of electron diffraction in stages, the essential theoretical basis, through Schrödinger's equation, being given by Bilhorn et al. (1964), and the Nbeam diffraction applications being derived successively by von Laue (1935), Cowley (1969), Pogany & Turner (1968), Moodie (1972), Buxton et al. (1976), and Gunning & Goodman (1992).
Reciprocity represents a reverseincidence configuration reached with the reversed wavevectors and , so that the scattering vector is unchanged, but is changed in sign and hence reversed (Moodie, 1972). The reciprocity equation, is valid independently of crystal symmetry, but cannot contribute symmetry to the pattern unless a crystalinverting symmetry element is present (since belongs to a reversed wavevector). The simplest case is centrosymmetry, which permits the righthand side of (2.5.3.1) to be complexconjugated giving the useful CBED pattern equation Since K is common to both sides there is a pointbypoint identity between the related distributions, separated by 2g (the distance between g and reflections). This invites an obvious analogy with Friedel's law, , with the reservation that (2.5.3.2) holds only for centrosymmetric crystals. This condition (2.5.3.2) constitutes what has become known as the ±H symmetry and, incidentally, is the only reciprocityinduced symmetry so general as to not depend upon a disc symmetrypoint or line, nor on a particular zone axis (i.e. it is not a point symmetry but a translational symmetry of the pattern intensity).

Horizontal glides, a′, n′ (diperiodic, primed notation), generate zerolayer absent rows, or centring, rather than GS bands (see Fig. 2.5.3.3). This is an example of the projection approximation in its most universally held form, i.e. in application to absences. Other examples of this are: (a) appearance of both G and S extinction bands near their intersection irrespective of whether glide or screw axes are involved; and (b) suppression of the influence of vertical, nonprimitive translations with respect to observations in the zero layer. It is generally assumed as a working rule that the zerolayer or ZOLZ pattern will have the rotational symmetry of the pointgroup component of the vertical screw axis (so that ). Elements included in Table 2.5.3.1 on this pretext are given in parentheses. However, the presence of rather than 2 ( rather than 3 etc.) should be detectable as a departure from accurate twofold symmetry in the firstorderLauezone (FOLZ) reflection circle (depicted in Fig. 2.5.3.3). This has been observed in the cubic structure of Ba_{2}Fe_{2}O_{5}Cl_{2}, permitting the space groups I23 and to be distinguished (Schwartzman et al., 1996). A summary of all the symmetry components described in this section is given diagrammatically in Table 2.5.3.2.
References
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