Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 288-289   | 1 | 2 |

Section Pattern observation of individual symmetry elements

P. Goodmanb Pattern observation of individual symmetry elements

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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.

  • (i) The value of X in an X-fold rotation axis is made immediately obvious in a zone-axis pattern, although a screw component is not detected in the pattern symmetry.

    Roto-inversionary axes require special attention: [\bar{6}] and [\bar{3}] may be factorized, as in Tables[link],[link] and[link], to show better the additional CBED symmetries ([3/m'] and [3 \times \bar{1}'], respectively). [\bar{4}] cannot be decomposed further (Table[link] and generates its own diffraction characteristics in non-projective patterns (see Section[link]). This specific problem of observing the fourfold roto-inversion symmetry has been resolved recently by Tanaka et al. (1994)[link] using both CBED and LACBED techniques.

  • (ii) Vertical mirror plane determination may be the most accurate crystal point-symmetry test, given that it is possible to follow the symmetry through large crystal rotations (say 5 to 15°) about the mirror normal. It is also relatively unaffected by crystal surface steps as compared to (v)[link] below.

  • (iii) Horizontal glide planes are determined unequivocally from zero-layer absences when the first Laue zone is recorded, either with the main pattern or by further crystal rotation; i.e. a section of this zone is needed to determine the lateral unit-cell parameters. This observation is illustrated diagrammatically in Fig.[link].

  • (iv) An extinction (GS) line or band through odd-order reflections of a zone-axis pattern indicates only a projected glide line . This is true because both [P2_{1}] (No. 4) and Pa (No. 7) symmetries project into `pg' in two dimensions. However, the projection approximation has only limited validity in CBED. For all crystal rotations around the [2_{1}] axis, or alternatively about the glide-plane `a' normal, dynamic extinction conditions are retained. This is summarized by saying that the diffraction vector [{\bf K}_{0{g}}] should be either normal to a screw axis or contained within a glide plane for the generation of the S or G bands, respectively. Hence [P2_{1}] and Pa may be distinguished by these types of rotations away from the zone axis with the consequence that the element [2_{1}] in particular is characterized by extinctions close to the Laue circle for the tilted ZOLZ pattern (Goodman, 1984b[link]), and that the glide a will generate extinction bands through both ZOLZ and HOLZ reflections for all orientations maintaining Laue-circle symmetry about the S band (Steeds et al., 1978[link]).

    As a supplement to this, in a refined technique not universally applicable, Tanaka et al. (1983)[link] have shown that fine-line detail from HOLZ interaction can be observed which will separately identify S- (21) and G-band symmetry from a single pattern (see Fig.[link]).

  • (v) The centre-of-symmetry (or ±H) test can be made very sensitive by suitable choice of diffraction conditions but requires a reasonably flat crystal since it involves a pair of patterns (the angular beam shift involved is very likely to be associated with some lateral probe shift on the specimen). This test is best carried out at a low-symmetry zone axis, free from other symmetries, and preferably incorporating some fine-line HOLZ detail, in the following way. The hkl and [\bar{h}\bar{k}\bar{l}] reflections are successively illuminated by accurately exchanging the central-beam aperture with the diffracted-beam apertures, having first brought the zone axis on to the electron-microscope optic axis. This produces the symmetrical ±H condition.

  • (vi) In seeking internal [m_{R}] symmetry as a test for a horizontal diad axis it is as well to involve some distinctive detail in the mirror symmetry (i.e. simple two-beam-like fringes should be avoided), and also to rotate the crystal about the supposed diad axis, to avoid an [m_{R}] symmetry due to projection [for examples see Fraser et al. (1985)[link] and Goodman & Whitfield (1980)[link]].

  • (vii) The presence or absence of the in-disc centrosymmetry element [1_{R}] formally indicates the presence or absence of a horizontal mirror element m′, either as a true mirror or as the mirror component of a horizontal glide plane g′. In this case the absence of symmetry provides more positive evidence than its presence, since absence is sufficient evidence for a lack of central-mirror crystal symmetry but an observed symmetry could arise from the operation of the projection approximation. If some evidence of the three-dimensional interaction is included in the observation or if three-dimensional interaction (from a large c axis parallel to the zone axis) is evident in the rest of the pattern, this latter possibility can be excluded. Interpretation is also made more positive by extending the angular aperture, especially by the use of LACBED.

These results are illustrated in Table[link] and by actual examples in Section[link].


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. (1984b). A retabulation of the 80 layer groups for electron diffraction usage. Acta Cryst. A40, 633–642.Google Scholar
First citation Goodman, P. & Whitfield, H. J. (1980). The space group determination of GaS and Cu3As2S3I by convergent beam electron diffraction. Acta Cryst. A36, 219–228.Google Scholar
First citation Steeds, J. W., Rackham, G. M. & Shannon, M. D. (1978). On the observation of dynamically forbidden lines in two and three dimensional electron diffraction. In Electron diffraction 1927–1977. Inst. Phys. Conf. Ser. No. 41, pp. 135–139.Google Scholar
First citation Tanaka, M., Sekii, H. & Nagasawa, T. (1983). Space group determination by dynamic extinction in convergent beam electron diffraction. Acta Cryst. A39, 825–837.Google Scholar
First citation Tanaka, M., Terauchi, M. & Tsuda, K. (1994). Convergent-beam electron diffraction III. Tokyo: JEOL–Maruzen.Google Scholar

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