Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 128-129

Section 1.6.6. Space groups for nanocrystals by electron microscopy

J. C. H. Spencec

1.6.6. Space groups for nanocrystals by electron microscopy

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The determination of crystal space groups may be achieved by the method of convergent-beam electron microdiffraction (CBED) using a modern transmission electron microscope (TEM). A detailed description of the CBED technique is given by Tanaka (2008[link]) in Section 2.5.3[link] of Volume B; here we give a brief overview of the capabilities of the method for space-group determination, for completeness. A TEM beam focused to nanometre dimensions allows study of nanocrystals, while identification of noncentrosymmetric crystals is straightforward, as a result of the strong multiple scattering normally present in electron diffraction. (Unlike single scattering, this does not impose inversion symmetry on diffraction patterns, but preserves the symmetry of the sample and its boundaries.) CBED patterns also allow direct determination of screw and glide space-group elements, which produce characteristic absences, despite the presence of multiple scattering, in certain orientations. These absences, which remain for all sample thicknesses and beam energies, may be shown to occur as a result of an elegant cancellation theorem along symmetry-related multiple-scattering paths (Gjønnes & Moodie, 1965[link]). Using all of the above information, most of the 230 space groups can be distinguished by CBED. The remaining more difficult cases (such as space groups that differ only in the location of their symmetry elements) are discussed in Spence & Lynch (1982[link]), Eades (1988[link]), and Saitoh et al. (2001[link]). Enantiomorphic pairs require detailed atomistic simulations based on a model, as in the case of quartz (Goodman & Secomb, 1977[link]). Multiple scattering renders Bragg intensities sensitive to structure-factor phases in noncentrosymmetric structures, allowing these to be measured with a tenth of a degree accuracy (Zuo et al., 1993[link]). Unlike X-ray diffraction, electron diffraction is very sensitive to ionicity and bonding effects, especially at low angles, allowing extinction-free charge-density mapping with high accuracy (Zuo, 2004[link]; Zuo et al., 1999[link]). Because of its sensitivity to strain, CBED may also be used to map out local phase transformations which cause space-group changes on the nanoscale (Zuo, 1993[link]; Zhang et al., 2006[link]).

In simplest terms, a CBED pattern is formed by enlarging the incident beam divergence in the transmission diffraction geometry, as first demonstrated G. Mollenstedt in 1937 (Kossel & Mollenstedt, 1942[link]). Bragg spots are then enlarged into discs, and the intensity variation within these discs is studied, in addition to that of the entire pattern, in the CBED method. The intensity variation within a disc displays a complete rocking curve in each of the many diffracted orders, which are simultaneously excited and recorded. The entire pattern thus consists of many independent `point' diffraction patterns (each for a slightly different incident beam direction) laid beside each other. Fig.[link] shows a CBED pattern from the wurtzite structure of ZnO, with the beam normal to the c axis (Wang et al., 2003[link]). The intensity variation along a line running through the centres of these discs (along the c axis) is not an even function, strongly violating Friedel's law for this elastic scattering. At higher scattering angles, curvature of the Ewald sphere allows three-dimensional symmetry elements to be determinated by taking account of `out-of-zone' intensities in the outer higher-order Laue zone (HOLZ) rings near the edge of the detector. Since sub-ångstrom-diameter electron probes and nanometre X-ray laser probes (Spence et al., 2012[link]) are now being used, the effect of the inevitable coherent interference between overlapping convergent-beam orders on space-group determination must be considered (Spence & Zuo, 1992[link]).


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Polarity determination by convergent-beam electron diffraction. A CBED pattern from ZnO with the beam normal to the c axis is shown. The intensity distribution along c does not have inversion symmetry, reflecting the noncentrocentrosymmetric nature of the structure. Reproduced with permission from Wang et al. (2003[link]). Copyright (2003) by The American Physical Society.

A systematic approach to space-group determination by CBED has been developed by several groups. In general, one would determine the symmetry of the projection diffraction group first (ignoring diffraction components along the beam direction z), then add the z-dependent information seen in HOLZ lines, allowing one to finally identify the point group from tables, by combining all this information. After indexing the pattern, in order to determine a unit cell the Bravais lattice is next determined. The form of the three-dimensional reciprocal lattice and its centring can usually be determined by noting the registry of Bragg spots in a HOLZ ring against those in the zero-order (ZOLZ) ring. Finally, by setting up certain special orientations, tests are applied for the presence of screw and glide elements, which are revealed by a characteristic dark line or cross within the CBED discs. Tables can again then be used to combine these translational symmetry elements with the previously determined point group, to find the space group. As a general experimental strategy, one first seeks mirror lines (perhaps seen in Kikuchi patterns), then follows these around using the two-axis goniometer fitted to modern TEM instruments in a systematic search for other symmetry elements. Reviews of the CBED method can be found in Steeds & Vincent (1983[link]), in Goodman (1975[link]), and in the texts by Tanaka et al. (1988[link]). A textbook-level worked example of space-group determination by CBED can be found in Spence & Zuo (1992[link]) and in the chapter by A. Eades in Williams & Carter (2009[link]).


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Goodman, P. & Secomb, T. W. (1977). Identification of enantiomorphously related space groups by electron diffraction. Acta Cryst. A33, 126–133.
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Saitoh, K., Tsuda, K., Terauchi, M. & Tanaka, M. (2001). Distinction between space groups having principal rotation and screw axes, which are combined with twofold rotation axes, using the coherent convergent-beam electron diffraction method. Acta Cryst. A57, 219–230.
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Williams, D. & Carter, C. B. (2009). Transmission Electron Microscopy, ch. 6. New York: Springer.
Zhang, P., Kisielowski, C., Istratov, A., He, H., Nelson, C., Mardinly, J., Weber, E. & Spence, J. C. H. (2006). Direct strain measurement in a 65 nm node strained silicon transistor by convergent-beam electron diffraction. Appl. Phys. Lett. 89, 161907.
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Zuo, J. M., Spence, J. C. H., Downs, J. & Mayer, J. (1993). Measurement of individual structure-factor phases with tenth-degree accuracy: the 00.2 reflection in BeO studied by electron and X-ray diffraction. Acta Cryst. A49, 422–429.

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