Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, p. 295

Section Present limitations and general conclusions

P. Goodmanb Present limitations and general conclusions

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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 large-unit-cell materials with a high diffraction-pattern density. This limits the possible convergence angle, if overlap is to be avoided, and leaves numerous but featureless discs [for example Goodman (1984b)[link]]. Technical advances which have been made to overcome this problem include the beam-rocking technique (Eades, 1980[link]) and LACBED (Tanaka et al., 1980[link]), both of which are reviewed by Tanaka & Terauchi (1985)[link] and Eades et al. (1983)[link]. 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 medium-light 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 N-beam 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)[link]], it has only recently been incorporated into symmetry studies as a routine (Creek & Spargo, 1985[link]; Tanaka, 1994[link]). Multi-slice programs which have been developed to produce computer-simulated pattern output are available (Section[link]).

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[link]). 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 high-resolution 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 X-ray/CBED and CBED/electron-microscopy studies of inorganic materials represents the standard ideal approach to space-group analysis at present; given this approach, all the space-group 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 so-called `phase problem' be completely overcome. The example of the space-group pairs [I222/I2_{1}2_{1}2_{1}] and [I23/I2_{1}3] 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 [2_{1}] axis. However, a direct distinction should be made apparent from high-resolution electron micrographs. Other less clear-cut cases occur where the HREM images allow a space-group distinction to be made between possible space groups of the same arithmetic class, especially when only one morphology is readily obtained (e.g. [P222_{1}], [P22_{1}2_{1}], [P2_{1}2_{1}2_{1}]).

The slightly more subtle problem of distinguishing enantiomorphic space-group 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 three-dimensional phase information can be deduced from specific three-beam-interaction 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)[link].


First citation Creek, R. C. & Spargo, A. E. C. (1985). Electron optical study of rutile. J. Appl. Cryst. 18, 197–204.Google Scholar
First citation Eades, J. A. (1980). Another way to form zone axis patterns. Inst. Phys. Conf. Ser. 52, 9–12.Google Scholar
First citation 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
First citation Goodman, P. (1984b). A retabulation of the 80 layer groups for electron diffraction usage. Acta Cryst. A40, 633–642.Google Scholar
First citation Goodman, P., McLean, J. D., Wilson, I. J. & Olsen, A. (1984). Optical microdiffraction and image analysis of subsymmetries in Nb2O5 tunnel structures. In Analytical electron microscopy–1984, pp. 130–134. San Francisco Press.Google Scholar
First citation Johnson, A. W. S. (1972). Stacking faults in graphite. Acta Cryst. A28, 89–93.Google Scholar
First citation Johnson, A. W. S. & Preston, A. R. (1994). Some notes on the selection of structural chirality by CBED. Ultramicroscopy, 55, 348–355.Google Scholar
First citation Tanaka, M. (1994). Convergent-beam electron diffraction. Acta Cryst. A50, 261–286.Google Scholar
First citation Tanaka, M., Saito, P., Ueno, K. & Harada, Y. (1980). Large angle convergent-beam electron diffraction. J. Electron. Microsc. 29, 408–412.Google Scholar
First citation Tanaka, M. & Terauchi, M. (1985). Convergent-beam electron diffraction. Tokyo: JEOL Ltd.Google Scholar

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