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. 285306

2.5.3. Spacegroup determination by convergentbeam electron diffraction^{2}
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.
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.
The following guidelines, the result of accumulated experience from several laboratories, are given in an experimentally based sequence, and approximately in order of value and reliability.
These results are illustrated in Table 2.5.3.2 and by actual examples in Section 2.5.3.5.
Space groups may very well be identified using CBED patterns from an understanding of the diffraction properties of realspace symmetry elements, displayed for example in Table 2.5.3.2. It is, however, of great assistance to have the symmetries tabulated in reciprocal space, to allow direct comparison with the pattern symmetries.
There are three generally useful ways in which this can be done, and these are set out in Tables 2.5.3.3 to 2.5.3.5. The simplest of these is by means of point group, following the procedures of Buxton et al. (1976). Next, the CBED pattern symmetries can be listed as diperiodic groups which are space groups in two dimensions, allowing identification with a restricted set of threedimensional space groups (Goodman, 1984b). Finally, the dynamic extinctions (GS bands and zerolayer absences) can be listed for each nonsymmorphic space group, together with the diffraction conditions for their observation (Tanaka et al., 1983; Tanaka & Terauchi, 1985). Descriptions for these tables are given below.




The list of examples given here must necessarily be regarded as unsatisfactory considering the vastness of the subject, although some attempt has been made to choose a diverse range of problems which will illustrate the principles involved. Some particular aspects, however, need further mention.
One of these concerns the problem of examining largeunitcell materials with a high diffractionpattern density. This limits the possible convergence angle, if overlap is to be avoided, and leaves numerous but featureless discs [for example Goodman (1984b)]. Technical advances which have been made to overcome this problem include the beamrocking technique (Eades, 1980) and LACBED (Tanaka et al., 1980), both of which are reviewed by Tanaka & Terauchi (1985) and Eades et al. (1983). The disadvantage of these latter methods is that they both require a significantly larger area of specimen than does the conventional technique, and it may be that more sophisticated methods of handling the crowded conventional patterns are still needed.
Next, the matter of accuracy must be considered. There are two aspects of the subject where this is of concern. Firstly, there is a very definite limit to the sensitivity with which symmetry can be detected. In a simple structure of mediumlight atoms, displacements of say 0.1 Å or less from a pseudomirror plane could easily be overlooked. An important aspect of CBED analysis, not mentioned above, is the Nbeam computation of patterns which is required when something approaching a refinement (in the context of electron diffraction) is being attempted. Although this quantitative aspect has a long history [for example see Johnson (1972)], it has only recently been incorporated into symmetry studies as a routine (Creek & Spargo, 1985; Tanaka, 1994). Multislice programs which have been developed to produce computersimulated pattern output are available (Section 2.5.3.8).
Next there is concern as to the allocation of a space group to structures which microscopically have a much lower symmetry (Goodman et al., 1984). This arises because the volume sampled by the electron probe necessarily contains a large number of unit cells. Reliable microscopic interpretation of certain nonstoichiometric materials requires that investigations be accompanied by highresolution microscopy. Frequently (especially in mineralogical samples), nonstoichiometry implies that a space group exists only on average, and that the concept of absolute symmetry elements is inapplicable.
From earlier and concluding remarks it will be clear that combined Xray/CBED and CBED/electronmicroscopy studies of inorganic materials represents the standard ideal approach to spacegroup analysis at present; given this approach, all the spacegroup problems of classical crystallography appear soluble. As has been noted earlier, it is important that HREM be considered jointly with CBED in determining space group by electron crystallography, and that only by this joint study can the socalled `phase problem' be completely overcome. The example of the spacegroup pairs and has already been cited. Using CBED, it might be expected that FOLZ lines would show a break from twofold symmetry with the incident beam aligned with a axis. However, a direct distinction should be made apparent from highresolution electron micrographs. Other less clearcut cases occur where the HREM images allow a spacegroup distinction to be made between possible space groups of the same arithmetic class, especially when only one morphology is readily obtained (e.g. , , ).
The slightly more subtle problem of distinguishing enantiomorphic spacegroup pairs can be solved by one of two approaches: either the crystal must be rotated around an axis by a known amount to obtain two projections, or the required threedimensional phase information can be deduced from specific threebeaminteraction data. This problem is part of the more general problem of solving handedness in an asymmetric structure, and is discussed in detail by Johnson & Preston (1994).

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