Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 292-295

Section Use of CBED in study of crystal defects, twins and non-classical crystallography

P. Goodmanb Use of CBED in study of crystal defects, twins and non-classical crystallography

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  • (i) Certain crystal defects lend themselves to analysis by CBED and LACBED. In earlier work, use was made of the high sensitivity of HOLZ line geometry to unit-cell parameters (Jones et al., 1977[link]). A computer program (Tanaka & Terauchi, 1985[link]) is available for simulating relative line positions from lattice geometry, assuming kinematical scattering, which at least provides a valid starting point since these spacings are mainly determined from geometric considerations. Fraser et al. (1985)[link], for example, obtained a sensitivity of 0.03% in measurements of cubic-to-tetragonal distortions in this way, although the absolute accuracy was not established.

  • (ii) By contrast, techniques have been devised for evaluating Bragg-line splitting caused by the action of a strain field within the single crystal. One method depends upon the observation of splitting in HOLZ lines (Carpenter & Spence, 1982[link]). More recently, the use of LACBED has allowed quantitative evaluation of lattice distortions in semiconductor heterostructures (e.g. containing GaAs–InGaAs interfaces). This technique has been reviewed by Chou et al. (1994)[link].

  • (iii) Quite distinct from this is the analysis of stacking faults between undistorted crystal domains (Johnson, 1972[link]). Coherent twin boundaries with at least a two-dimensional coincidence site lattice can be considered in a similar fashion (Schapink et al., 1983[link]). In marked contrast to electron-microscopy image analysis these boundaries need to be parallel (or nearly so) to the crystal surfaces rather than inclined or perpendicular to them for analysis by CBED or LACBED.

    The term `rigid-body displacement' (RBD) is used when it is assumed that no strain field develops at the boundary. A classification of the corresponding bi-crystal symmetries was developed by Schapink et al. (1983)[link] for these cases. Since experimental characterization of grain boundaries is of interest in metallurgy, this represents a new area for the application of LACBED and algorithms invoking reciprocity now make routine N-beam analysis feasible.

    The original investigations, of a mid-plane stacking fault in graphite (Johnson, 1972[link]) and of a mid-plane twin boundary in gold (Schapink et al., 1983[link]), represent classic examples of the influence of bi-crystal symmetry on CBED zone-axis patterns, whereby the changed central-plane symmetry is transformed through reciprocity into an exact diffraction symmetry. (a) In the graphite [(P6_{3}/mmc)] example, the hexagonal pattern of the unfaulted graphite is replaced by a trigonal pattern with mid-plane faulting. Here a mirror plane at the centre of the perfect crystal (A–B–A stacking) is replaced by an inversion centre at the midpoint of the single rhombohedral cell A–B–C; the projected symmetry is also reduced from hexagonal to trigonal: both whole pattern and central beam then have the symmetry of 3m1. The 2H polytype of TaS2 [(P6_{3}/mmc)] (Tanaka & Terauchi, 1985[link]) gives a second clear example. (b) In the case of a [111] gold crystal, sectioning the f.c.c. structure parallel to [111] preparatory to producing the twin already reduces the finite crystal symmetry to [R\bar{3}m], i.e. a trigonal space group for which the central beam, and the HOLZ reflections in particular, exhibit the trigonal symmetry of 31m (rather than the 3m1 of trigonal graphite). A central-plane twin boundary with no associated translation introduces a central horizontal mirror plane into the crystal. For the zone-axis pattern the only symmetry change will be in the central beam, which will become centrosymmetric, increasing its symmetry to 6mm. Using diffraction-group terminology these cases are seen to be relative inverses. Unfaulted graphite has the BESR group [6mm1_{R}] (central beam and whole pattern hexagonal); central-plane faulting results in a change to the group [6_{R}mm_{R}]. Unfaulted [111] gold correspondingly has the BESR group symmetry [6_{R}mm_{R}]; central-plane twinning results in the addition of the element [1_{R}] (for a central mirror plane), leading to the group [6mm1_{R}].

  • (iv) Finally, no present-day discussion of electron-crystallographic investigations of symmetry could be complete without reference to two aspects of non-classical symmetries widely discussed in the literature in recent years. The recent discovery of noncrystallographic point symmetries in certain alloys (Shechtman et al., 1984[link]) has led to the study of quasi-crystallinity. An excellent record of the experimental side of this subject may be found in the book Convergent-beam electron diffraction III by Tanaka et al. (1994)[link], while the appropriate space-group theory has been developed by Mermin (1992)[link]. It would be inappropriate to comment further on this new subject here other than to state that this is clearly an area of study where combined HREM, CBED and selected-area diffraction (SAD) evidence is vital to structural elucidation.

    The other relatively new topic is that of modulated structures. From experimental evidence, two distinct structural phenomena can be distinguished for structures exhibiting incommensurate superlattice reflections. Firstly, there are `Vernier' phases, which exist within certain composition ranges of solid solutions and are composed of two extensive substructures, for which the superspace-group nomenclature developed by de Wolff et al. (1981)[link] is structurally valid (e.g. Withers et al., 1993[link]). Secondly, there are structures essentially composed of random mixtures of two or more substructures existing as microdomains within the whole crystal (e.g. Grzinic, 1985[link]). Here the SAD patterns will contain superlattice reflections with characteristic profiles and/or irregularities of spacings. A well illustrated review of incommensurate-structure analysis in general is given in the book by Tanaka et al. (1994)[link], while specific discussions of this topic are given by Goodman et al. (1992)[link], and Goodman & Miller (1993)[link].


First citation Carpenter, R. W. & Spence, J. C. H. (1982). Three-dimensional strain-field information in convergent-beam electron diffraction patterns. Acta Cryst. A38, 55–61.Google Scholar
First citation Chou, C. T., Anderson, S. C., Cockayne, D. J. H., Sikorski, A. Z. & Vaughan, M. R. (1994). Ultramicroscopy, 55, 334–347.Google Scholar
First citation 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
First citation Goodman, P. & Miller, P. (1993). Reassessment of the symmetry of the 221 PbBiSrCaCuO structure using LACBED and high-resolution SAD: the relevance of Cowley's theory of disorder scattering to a real-space structural analysis. Ultramicroscopy, 52, 549–556.Google Scholar
First citation Goodman, P., Miller, P., White, T. J. & Withers, R. L. (1992). Symmetry determination and Pb-site ordering analysis for the n = 1,2 PbxBi2 − xSr2Can − 1CunO4 + 2n + δ compounds by convergent-beam and selected-area electron diffraction. Acta Cryst. B48, 376–387.Google Scholar
First citation Grzinic, G. (1985). Calculation of incommensurate diffraction intensities from disordered crystals. Philos. Mag. A, 52, 161–187.Google Scholar
First citation Johnson, A. W. S. (1972). Stacking faults in graphite. Acta Cryst. A28, 89–93.Google Scholar
First citation 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
First citation Mermin, N. D. (1992). The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic and triclinic crystals. Rev. Mod. Phys. 64, 3–49.Google Scholar
First citation Schapink, F. W., Forgany, S. K. E. & Buxton, B. F. (1983). The symmetry of convergent-beam electron diffraction patterns from bicrystals. Acta Cryst. A39, 805–813.Google Scholar
First citation Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953.Google Scholar
First citation Tanaka, M. & Terauchi, M. (1985). Convergent-beam electron diffraction. Tokyo: JEOL Ltd.Google Scholar
First citation Tanaka, M., Terauchi, M. & Tsuda, K. (1994). Convergent-beam electron diffraction III. Tokyo: JEOL–Maruzen.Google Scholar
First citation Withers, R. L., Schmid, S. & Thompson, J. G. (1993). A composite modulated structure approach to the lanthanide oxide fluoride, uranium nitride fluoride and zirconium nitride fluoride solid-solution fields. Acta Cryst. B49, 941–951.Google Scholar
First citation Wolff, P. M. de, Janssen, T. & Janner, A. (1981). The superspace groups for incommensurate crystal structures with a one-dimensional modulation. Acta Cryst. A37, 625–636.Google Scholar

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