Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, p. 292   | 1 | 2 |

Section Examples

P. Goodmanb Examples

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  • (1) Determination of centrosymmetry; examples from the hexagonal system. Fig.[link](a) illustrates the allocation of planar point groups from [0001] zone-axis patterns of [\beta \hbox{-Si}_{3}\hbox{N}_{4}] (left-hand side) and β-GaS (right-hand side); the patterns exhibit point symmetries of 6 and 6mm, respectively, as indicated by the accompanying geometric figures, permitting point groups 6 or [6/m], and 6mm or [6/mmm], in three dimensions. Alternative zone axes are required to distinguish these possibilities, the actual test used (testing for the element m′ or the centre of symmetry) being largely determined in practice by the type of crystal preparation.


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    (a) Zone-axis patterns from hexagonal structures β-[\hbox{Si}_{3}\hbox{N}_{4}] (left) and β-GaS (right) together with the appropriate planar figures for point symmetries 6 and 6mm, respectively. (b) [[12\bar{1}0]] zone-axis pattern from β-[\hbox{Si}_{3}\hbox{N}_{4}], showing Friedel's law breakdown in symmetry between 0002 and [000\bar{2}] reflections (Bando, 1981[link]). (c) Conjugate pair of [1\bar{1}01/\bar{1}101] patterns from β-GaS, taken near the [[110\bar{2}]] zone axis, showing a translational symmetry associated with structural centrosymmetry.

    Fig.[link](b) shows the CBED pattern from the [[11\bar{2}0]] zone axis of [\beta \hbox{-Si}_{3}\hbox{N}_{4}] (Bando, 1981[link]), using a crystal with the corresponding cleavage faces. The breakdown of Friedel's law between reflections 0002 and [000\bar{2}] rules out the point group [6/m] (the element m′ from the first setting is not present) and establishes 6 as the correct point group.

    Also, the GS bands in the 0001 and [000\bar{1}] reflections are consistent with the space group [P6_{3}]. [Note: screw axes [6_{1}], [6_{3}] and [6_{5}] are not distinguished from these data alone (Tanaka et al., 1983[link]).]

    Fig.[link](c) shows CBED patterns from the vicinity of the [[1\bar{1}02]] zone axis of β-GaS, only 11.2° rotated from the [0001] axis and accessible using the same crystal as for the previous [0001] pattern. This shows a positive test for centrosymmetry using a conjugate reflection pair [1\bar{1}01/\bar{1}10\bar{1}], and establishes the centrosymmetric point group [6/mmm], with possible space groups Nos. 191, 192, 193 and 194. Rotation of the crystal to test the extinction rule for [hh2\bar{h}l] reflections with l odd (Goodman & Whitfield, 1980[link]) establishes No. 194 [(P6_{3}/mmc)] as the space group.

    Comment: These examples show two different methods for testing for centrosymmetry. The ±H test places certain requirements on the specimen, namely that it be reasonably accurately parallel-sided – a condition usually met by easy-cleavage materials like GaS, though not necessarily by the wedge-shaped refractory Si3N4 crystals. On the other hand, the 90° setting, required for direct observation of a possible perpendicular mirror plane, is readily available in these fractured samples, but not for the natural cleavage samples.

  • (2) Point-group determination in the cubic system, using Table[link] Fig.[link] shows [001] (cyclic) zone-axis patterns from two cubic materials, which serve to illustrate the ability to distinguish cubic point groups from single zone-axis patterns displaying detailed central-beam structures. The left-hand pattern, from the mineral gahnite (Ishizuka & Taftø, 1982[link]) has 4mm symmetry in both the whole pattern and the central (bright-field) beam, permitting only the BESR group [4mm1_{R}] for the cubic system (column III, Table[link]; this same observation establishes the crystallographic point group as m3m (column V of Table[link]). The corresponding pattern for the χ-phase precipitate of stainless steel (Steeds & Evans, 1980[link]) has a whole-pattern symmetry of only 2mm, lower than the central-beam (bright-field) symmetry of 4mm (this lower symmetry is made clearest from the innermost reflections bordering the central beam). This combination leads to the BESR group [4_{R}mm_{R}] (column III, Table[link], and identifies the cubic point group as [\bar{4}3m].


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    Zone-axis patterns from cubic structures gahnite (left) (Ishizuka & Taftø, 1982[link]) and χ-phase precipitate (right) (Steeds & Evans, 1980[link]).

  • (3) Analysis of data from FeS2 illustrating use of Tables[link] and[link]. FeS2 has a cubic structure for which a complete set of data has been obtained by Tanaka et al. (1983)[link]; the quality of the data makes it a textbook example (Tanaka & Terauchi, 1985[link]) for demonstrating the interpretation of extinction bands.

    Figs.[link](a) and (b) show the [001] (cyclic) exact zone-axis pattern and the pattern with symmetrical excitation of the 100 reflection, respectively (Tanaka et al., 1983[link]).


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    (a) CBED pattern from the exact [001] (cyclic) zone-axis orientation of FeS2. (b) Pattern from the [001] zone axis oriented for symmetrical excitation of the 100 reflection (central in the printed pattern) [from the collection of patterns presented in Tanaka et al. (1983)[link]; originals kindly supplied by M. Tanaka].

    • (i) Using Table[link], since there are GS bands, the pattern group must be listed in column II(ii); since a horizontal `b' glide plane is present (odd rows are absent in the [b^{*}] direction), the symbol must contain a `b′' (or `a′') (cf. Fig.[link]). The only possible cubic group from Table[link] is No. 205.

    • (ii) Again, a complete GS cross, with both G and S arms, is present in the 100 reflection (central in Fig.[link]), confirmed by mirror symmetries across the G and S lines. From Table[link] only space group No. 205 has the corresponding entry in the column for `[100] cyclic' with GS in the cubic system (space groups Nos. 198–230). Additional patterns for the [110] setting, appearing in the original paper (Tanaka et al., 1983[link]), confirm the cubic system, and also give additional extinction characteristics for 001 and 1[\bar{1}]0 reflections (Tanaka et al., 1983[link]; Tanaka & Terauchi, 1985[link]).

  • (4) Determination of centrosymmetry and space group from extinction characteristics. Especially in working with thin crystals used in conjunction with high-resolution lattice imaging, it is sometimes most practical to determine the point group (i.e. space-group class) from the dynamic extinction data. This is exemplified in the Moodie & Whitfield (1984)[link] studies of orthorhombic materials. Observations on the zero-layer pattern for [\hbox{Ge}_{3}\hbox{SbSe}_{3}] with a point symmetry of 2mm, and with GS extinction bands along odd-order h00 reflections, together with missing reflection rows in the 0k0 direction, permit identification from Table[link]. This zone-axis pattern has the characteristics illustrated in Figs.[link] and hence (having both missing rows and GS bands) should be listed in both II(ii) and II(iii). Hence the diffraction group must be either No. 40 or 41. Here, the class mmm, and hence centrosymmetry, has been identified through non-symmorphic elements.

    This identification leaves seven possible space groups, Nos. 52, 54, 56, 57, 60, 61 and 62, to be distinguished by hkl extinctions.

    The same groups are identified from Table[link] by seeking the entry GS `−' in one of the [001] (cyclic) entries for the orthorhombic systems. With the assumption that no other principal zone axis is readily available from the same sample (which will generally be true), Table[link], in the last three columns, indicates which minor zone axes should be sought in order to identify the space group, from the glide-plane extinctions of `G' bands. For example, space group 62 has no h0l extinctions, but will give 0kl extinction bands `G' according to the rules for an `n' glide, i.e. in reflections for which [k + l = 2n + 1]. Again, if the alternative principal settings are available (from the alternative cleavages of the sample) the correct space group can be found from the first three columns of Table[link].

    From the above discussions it will be clear that Tables[link] and[link] present information in a complementary way: in Table[link] the specific pattern group is indexed first with the possible space groups following, while in Table[link] the space group is indexed first, and the possible pattern symmetries are then given, in terms of the standard International Tables setting.


First citation Bando, Y. (1981). Weak asymmetry in β-Si3N4 as revealed by convergent beam electron diffraction. Acta Cryst. B39, 185–189.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 Ishizuka, K. & Taftø, J. (1982). Kinematically allowed reflections caused by scattering via HOLZ. Proc. Electron Microsc. Soc. Am. pp. 688–689.Google Scholar
First citation Moodie, A. F. & Whitfield, H. J. (1984). CBED and HREM in the electron microscope. Ultramicroscopy, 13, 265–278.Google Scholar
First citation Steeds, J. W. & Evans, N. S. (1980). Practical examples of point and space group determination in convergent beam diffraction. Proc. Electron Microsc. Soc. Am. pp. 188–191.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. (1985). Convergent-beam electron diffraction. Tokyo: JEOL Ltd.Google Scholar

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