International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 2.5, pp. 285-306   | 1 | 2 |

Section 2.5.3. Space-group determination by convergent-beam electron diffraction2

P. Goodmanb

2.5.3. Space-group determination by convergent-beam electron diffraction2

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2.5.3.1. Introduction

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2.5.3.1.1. CBED

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Convergent-beam electron diffraction, originating in the experiments of Kossel and Möllenstedt (Kossel & Möllenstedt, 1938[link]) has been established over the past two decades as a powerful technique for the determination of space group in inorganic materials, with particular application when only microscopic samples are available. Relatively recently, with the introduction of the analytical electron microscope, this technique – abbreviated as CBED – has become available as a routine, so that there is now a considerable accumulation of data from a wide range of materials. A significant extension of the technique in recent times has been the introduction of LACBED (large-angle CBED) by Tanaka & Terauchi (1985[link]). This technique allows an extensive angular range of single diffraction orders to be recorded and, although this method cannot be used for microdiffraction (since it requires an extensive single-crystal area), new LACBED applications appear regularly, particularly in the field of semiconductor research (see Section 2.5.3.6[link]).

The CBED method relies essentially on two basic properties of transmission electron diffraction, namely the radical departure from Friedel's law and the formation of characteristic extinction bands within space-group-forbidden reflections. Departure from Friedel's law in electron diffraction was first noted experimentally by Miyake & Uyeda (1950)[link]. The prediction of space-group-forbidden bands (within space-group-forbidden reflections) by Cowley & Moodie (1959)[link], on the other hand, was one of the first successes of N-beam theory. A detailed explanation was later given by Gjønnes & Moodie (1965)[link]. These are known variously as `GM' bands (Tanaka et al., 1983[link]), or more simply and definitively as `GS' (glide–screw) bands (this section). These extinctions have a close parallel with space-group extinctions in X-ray diffraction, with the reservation that only screw axes of order two are accurately extinctive under N-beam conditions. This arises from the property that only those operations which lead to identical projections of the asymmetric unit can have N-beam dynamical symmetries (Cowley et al., 1961[link]).

Additionally, CBED from perfect crystals produces high-order defect lines in the zero-order pattern, analogous to the defect Kikuchi lines of inelastic scattering, which provide a sensitive measurement of unit-cell parameters (Jones et al., 1977[link]; Fraser et al., 1985[link]; Tanaka & Terauchi, 1985[link]).

The significant differences between X-ray and electron diffraction, which may be exploited in analysis, arise as a consequence of a much stronger interaction in the case of electrons (Section 2.5.2[link]). Hence, thin, approximately parallel-sided crystal regions must be used in high-energy (100 kV–1 MV) electron transmission work, so that diffraction is produced from crystals effectively infinitely periodic in only two dimensions, leading to the relaxation of three-dimensional diffraction conditions known as `excitation error' (Chapter 5.2[link] ). Also, there is the ability in CBED to obtain data from microscopic crystal regions of around 50 Å in diameter, with corresponding exposure times of several seconds, allowing a survey of a material to be carried out in a relatively short time.

In contrast, single-crystal X-ray diffraction provides much more limited symmetry information in a direct fashion [although statistical analysis of intensities (Wilson, 1949[link]) will considerably supplement this information], but correspondingly gives much more direct three-dimensional geometric data, including the determination of unit-cell parameters and three-dimensional extinctions.

The relative strengths and weaknesses of the two techniques make it useful where possible to collect both convergent-beam and X-ray single-crystal data in a combined study. However, all parameters can be obtained from convergent-beam and electron-diffraction data, even if in a somewhat less direct form, making possible space-group determination from microscopic crystals and microscopic regions of polygranular material. Several reviews of the subject are available (Tanaka, 1994[link]; Steeds & Vincent, 1983[link]; Steeds, 1979[link]). In addition, an atlas of characteristic CBED patterns for direct phase identification of metal alloys has been published (Mansfield, 1984[link]), and it is likely that this type of procedure, allowing N-beam analysis by comparison with standard simulations, will be expanded in the near future.

2.5.3.1.2. Zone-axis patterns from CBED

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Symmetry analysis is necessarily tied to examination of patterns near relevant zone axes, since the most intense N-beam interaction occurs amongst the zero-layer zone-axis reflections, with in addition a limited degree of upper-layer (higher-order Laue zone) interaction. There will generally be several useful zone axes accessible for a given parallel-sided single crystal, with the regions between axes being of little use for symmetry analysis. Only one such zone axis can be parallel to a crystal surface normal, and a microcrystal is usually chosen at least initially to have this as the principal symmetry axis. Other zone axes from that crystal may suffer mild symmetry degradation because the N-beam lattice component (`excitation error' extension) will not have the symmetry of the structure (Goodman, 1974[link]; Eades et al., 1983[link]).

Upper-layer interactions, responsible for imparting three-dimensional information to the zero layer, are of two types: the first arising from `overlap' of dynamic shape transforms and causing smoothly varying modulations of the zero-layer reflections, and the second, caused by direct interactions with the upper-layer, or higher-order Laue zone lines, leading to a sharply defined fine-line structure. These latter interactions are especially useful in increasing the accuracy of space-group determination (Tanaka et al., 1983[link]), and may be enhanced by the use of low-temperature specimen stages. The presence of these defect lines in convergent-beam discs, occurring especially in low-symmetry zone-axis patterns, allows symmetry elements to be related to the three-dimensional structure (Section 2.5.3.5[link]; Fig. 2.5.3.4c[link]).

To the extent that such three-dimensional effects can be ignored or are absent in the zero-layer pattern the projection approximation (Chapter 5.2[link] ) can be applied. This situation most commonly occurs in zone-axis patterns taken from relatively thin crystals and provides a useful starting point for many analyses, by identifying the projected symmetry.

2.5.3.2. Background theory and analytical approach

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2.5.3.2.1. Direct and reciprocity symmetries: types I and II

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Convergent-beam diffraction symmetries are those of Schrödinger's equation, i.e. of crystal potential, plus the diffracting electron. The appropriate equation is given in Section 2.5.2[link] [equation (2.5.2.6)[link]] and Chapter 5.2 [equation (5.2.2.1)][link] in terms of the real-space wavefunction ψ. The symmetry elements of the crystal responsible for generating pattern symmetries may be conveniently classified as of two types (I and II) as follows.

  • I. The direct (type I: Table 2.5.3.1[link]) symmetries imposed by this equation on the transmitted wavefunction given z-axis illumination ([{\bf k}_{0}], the incident wavevector parallel to Z, the surface normal) are just the symmetries of ϕ whose operation leaves both crystal and z axis unchanged. These are also called `vertical' symmetry elements, since they contain Z. These symmetries apply equally in real and reciprocal space, since the operator [\nabla^{2}] has circular symmetry in both spaces and does nothing to degrade the symmetry in transmission. Hence, for high-symmetry crystals (zone axis parallel to z axis), and to a greater or lesser degree for crystals of a more general morphology, these zone-axis symmetries apply both to electron-microscope lattice images and to convergent-beam patterns under z-axis-symmetrical illumination, and so impact also on space-group determination by means of high-resolution electron microscopy (HREM). In CBED, these elements lead to whole pattern symmetries, to which every point in the pattern contributes, regardless of diffraction order and Laue zone (encompassing ZOLZ and HOLZ reflections).

    Table 2.5.3.1| top | pdf |
    Listing of the symmetry elements relating to CBED patterns under the classifications of `vertical' (I), `horizontal' (II) and combined or roto-inversionary axes

    I. Vertical symmetry elements
    International symbols
    [\eqalign{&2, 3, 4, 6 \quad (2_{1}, 3_{1}, \ldots)\cr &m \;\phantom{2, 3, 4, 6} (c)\cr &a, b \!\!\phantom{2, 3, 4, 6} (n)}]
    II. Horizontal symmetry elements
      Diperiodic symbols BESR symbols
    2′ m
    [2'_{1}]  
    m [1_{R}]
    a′, b′, n  
    [\bar{1}'] [2_{R}]
    [\hbox{I} + \hbox{II}] [\bar{4}'] [4_{R}]
    [\hbox{I} \times \hbox{II}] [\bar{3}' = 3 \times \bar{1}'] [6_{R} = 3 \cdot 2_{R}]
      [\bar{6}' = 3 \times m'] [31_{R}]
  • II. Reciprocity-induced symmetries, on the other hand, depend upon ray paths and path reversal, and in the present context have relevance only to the diffraction pattern. Crystal-inverting or horizontal crystal symmetry elements combine with reciprocity to yield indirect pattern symmetries lacking a one-to-one real-space correspondence, within individual diffraction discs or between disc pairs. Type II elements are assumed to lie on the central plane of the crystal, midway between surfaces, as symmetry operators; this assumption amounts to a `central plane' approximation, which has a very general validity in space-group-determination work (Goodman, 1984a[link]).

A minimal summary of basic theoretical points, otherwise found in Chapter 5.2[link] and numerous referenced articles, is given here.

For a specific zero-layer diffraction order [g\ (=h, k)] the incident and diffracted vectors are [{\bf k}_{0}] and [{\bf k}_{g}]. Then the three-dimensional vector [{\bf K}_{0g} = {1 \over 2} ({\bf k}_{0} + {\bf k}_{g})] has the pattern-space projection, [{\bf K}_{g} = {}^{p}[{\bf K}_{0g}]]. The point [{\bf K}_{g} = {\bf 0}] gives the symmetrical Bragg condition for the associated diffraction disc, and [{\bf K}_{g} \neq {\bf 0}] is identifiable with the angular deviation of [{\bf K}_{0g}] from the vertical z axis in three-dimensional space (see Fig. 2.5.3.1[link]). [{\bf K}_{g} = {\bf 0}] also defines the symmetry centre within the two-dimensional disc diagram (Fig. 2.5.3.2[link]); namely, the intersection of the lines S and G, given by the trace of excitation error, [{\bf K}_{g} = {\bf 0}], and the perpendicular line directed towards the reciprocal-space origin, respectively. To be definitive it is necessary to index diffracted amplitudes relating to a fixed crystal thickness and wavelength, with both crystallographic and momentum coordinates, as [{\bf u}_{g, \, K}], to handle the continuous variation of [{\bf u}_{g}] (for a particular diffraction order), with angles of incidence as determined by [{\bf k}_{0}], and registered in the diffraction plane as the projection of [{\bf K}_{0g}].

[Figure 2.5.3.1]

Figure 2.5.3.1 | top | pdf |

Vector diagram in semi-reciprocal space, using Ewald-sphere constructions to show the `incident', `reciprocity' and `reciprocity × centrosymmetry' sets of vectors. Dashed lines connect the full vectors [{\bf K}_{0g}] to their projections [{\bf K}_{g}] in the plane of observation.

[Figure 2.5.3.2]

Figure 2.5.3.2 | top | pdf |

Diagrammatic representation of a CBED disc with symmetry lines m, mR (alternate labels G, S) and the central point [{\bf K}_{g} = 0].

2.5.3.2.2. Reciprocity and Friedel's law

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Reciprocity was introduced into the subject of electron diffraction in stages, the essential theoretical basis, through Schrödinger's equation, being given by Bilhorn et al. (1964)[link], and the N-beam diffraction applications being derived successively by von Laue (1935)[link], Cowley (1969)[link], Pogany & Turner (1968)[link], Moodie (1972)[link], Buxton et al. (1976)[link], and Gunning & Goodman (1992)[link].

Reciprocity represents a reverse-incidence configuration reached with the reversed wavevectors [\bar{{\bf k}}_{0} = - {\bf k}_{g}] and [\bar{{\bf k}}_{g} = - {\bf k}_{0}], so that the scattering vector [\Delta{\bf k} = {\bf k}_{g} - {\bf k}_{0} = \bar{{\bf k}}_{0} - \bar{{\bf k}}_{g}] is unchanged, but [\bar{{\bf K}}_{0g} = {1 \over 2} (\bar{{\bf k}}_{0} + \bar{{\bf k}}_{g})] is changed in sign and hence reversed (Moodie, 1972[link]). The reciprocity equation, [{\bf u}_{g, \, {\bf K}} = {\bf u}_{g, \, \bar{{\bf K}}}^{*}, \eqno(2.5.3.1)] is valid independently of crystal symmetry, but cannot contribute symmetry to the pattern unless a crystal-inverting symmetry element is present (since [\bar{{\bf K}}] belongs to a reversed wavevector). The simplest case is centrosymmetry, which permits the right-hand side of (2.5.3.1)[link] to be complex-conjugated giving the useful CBED pattern equation [{\bf u}_{g, \, {\bf K}} = {\bf u}_{\bar{g}, \, {\bf K}}. \eqno(2.5.3.2)] Since K is common to both sides there is a point-by-point identity between the related distributions, separated by 2g (the distance between g and [\bar{g}] reflections). This invites an obvious analogy with Friedel's law, [F_{g} = F_{\bar{g}}^{*}], with the reservation that (2.5.3.2)[link] holds only for centrosymmetric crystals. This condition (2.5.3.2)[link] constitutes what has become known as the ±H symmetry and, incidentally, is the only reciprocity-induced symmetry so general as to not depend upon a disc symmetry-point or line, nor on a particular zone axis (i.e. it is not a point symmetry but a translational symmetry of the pattern intensity).

2.5.3.2.3. In-disc symmetries

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  • (a) Dark-field (diffracted-beam) discs . Other reciprocity-generated symmetries which are available for experimental observation relate to a single (zero-layer) disc and its origin [{\bf K}_{g} = {\bf 0}], and are summarized here by reference to Fig. 2.5.3.2[link], and given in operational detail in Table 2.5.3.2[link]. The notation subscript R, for reciprocity-induced symmetries, introduced by Buxton et al. (1976)[link] is now adopted (and referred to as BESR notation). Fig. 2.5.3.2[link] shows a disc crossed by reference lines m and [m_{R}]. These will be mirror lines of intensity if: (a) g is parallel to a vertical mirror plane; and (b) g is parallel to a horizontal diad axis, respectively. The third possible point symmetry, that of disc centrosymmetry ([1_{R}] in BESR notation) will arise from the presence of a horizontal mirror plane. Lines m and [m_{R}] become the GS extinction lines G and S when glide planes and screw axes are present instead of mirror planes and diad axes.

    Table 2.5.3.2| top | pdf |
    Diagrammatic illustrations of the actions of five types of symmetry elements (given in the last column in Volume A diagrammatic symbols) on an asymmetric pattern component, in relation to the centre of the pattern at [{\bf K}_{00} = {\bf 0}], shown as `⊕', or in relation to the centre of a diffraction order at [{\bf K}_{0g} = {\bf 0}], shown as `+'

    TypeSymmetry elementObservation and actionIn combinationInterpretation
    Vertical 4 [Scheme scheme1]   [Scheme scheme2]
    m; a [Scheme scheme3] [Scheme scheme5] [Scheme scheme6]
    Horizontal 2′; [2'_{1}] [Scheme scheme4]
    [i(\bar{1}')] [Scheme scheme7]   [Scheme scheme8]
    m′; a [Scheme scheme9]   [Scheme scheme10]
  • (b) Bright-field (central-beam) disc . The central beam is a special case since the point [{\bf K}_{0} = {\bf 0}] is the centre of the whole pattern as well as of that particular disc. Therefore, both sets of rotational symmetry (types I[link] and II[link]) discussed above apply (see Table 2.5.3.3[link]).

    In addition, the central-beam disc is a source of three-dimensional lattice information from defect-line scattering. Given a sufficiently perfect crystal this fine-line structure overlays the more general intensity modulation, giving this disc a lower and more precisely recorded symmetry.

2.5.3.2.4. Zero-layer absences

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Horizontal glides, a′, n′ (diperiodic, primed notation), generate zero-layer absent rows, or centring, rather than GS bands (see Fig. 2.5.3.3[link]). This is an example of the projection approximation in its most universally held form, i.e. in application to absences. Other examples of this are: (a) appearance of both G and S extinction bands near their intersection irrespective of whether glide or screw axes are involved; and (b) suppression of the influence of vertical, non-primitive translations with respect to observations in the zero layer. It is generally assumed as a working rule that the zero-layer or ZOLZ pattern will have the rotational symmetry of the point-group component of the vertical screw axis (so that [2_{1} \simeq 2]). Elements included in Table 2.5.3.1[link] on this pretext are given in parentheses. However, the presence of [2_{1}] rather than 2 ([3_{1}] rather than 3 etc.) should be detectable as a departure from accurate twofold symmetry in the first-order-Laue-zone (FOLZ) reflection circle (depicted in Fig. 2.5.3.3[link]). This has been observed in the cubic structure of Ba2Fe2O5Cl2, permitting the space groups I23 and [I2_{1} 3] to be distinguished (Schwartzman et al., 1996[link]). A summary of all the symmetry components described in this section is given diagrammatically in Table 2.5.3.2[link].

[Figure 2.5.3.3]

Figure 2.5.3.3 | top | pdf |

Diagrammatic representation of the influence of non-symmorphic elements: (i) Alternate rows of the zero-layer pattern are absent owing to the horizontal glide plane. The pattern is indexed as for an `a' glide; the alternative indices (in parentheses) apply for a `b' glide. (ii) GS bands are shown along the central row of the zero layer, for odd-order reflections.

2.5.3.3. 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 2.5.3.1[link], 2.5.3.3[link] and 2.5.3.4[link], to show better the additional CBED symmetries ([3/m'] and [3 \times \bar{1}'], respectively). [\bar{4}] cannot be decomposed further (Table 2.5.3.1)[link] and generates its own diffraction characteristics in non-projective patterns (see Section 2.5.3.5[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. 2.5.3.3[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. 2.5.3.6[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 2.5.3.2[link] and by actual examples in Section 2.5.3.5[link].

2.5.3.4. Auxiliary tables

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Space groups may very well be identified using CBED patterns from an understanding of the diffraction properties of real-space symmetry elements, displayed for example in Table 2.5.3.2[link]. It is, however, of great assistance to have the symmetries tabulated in reciprocal space, to allow direct comparison with the pattern symmetries.

There are three generally useful ways in which this can be done, and these are set out in Tables 2.5.3.3[link] [link] to 2.5.3.5[link]. The simplest of these is by means of point group, following the procedures of Buxton et al. (1976)[link]. Next, the CBED pattern symmetries can be listed as diperiodic groups which are space groups in two dimensions, allowing identification with a restricted set of three-dimensional space groups (Goodman, 1984b[link]). Finally, the dynamic extinctions (GS bands and zero-layer absences) can be listed for each non-symmorphic space group, together with the diffraction conditions for their observation (Tanaka et al., 1983[link]; Tanaka & Terauchi, 1985[link]). Descriptions for these tables are given below.

  • Table 2.5.3.3[link]. BESR symbols (Buxton et al., 1976[link]) incorporate the subscript R to describe reciprocity-related symmetry elements, R being the operator that rotates the disc pattern by 180° about its centre. The symbols formed in this way are [1_{R}], [2_{R}], [4_{R}], [6_{R}], where [X_{R}] represents [2\pi / X] rotation about the zone axis, followed by R. Of these, [2_{R}] represents the ±H symmetry (two twofold rotations) described earlier [equation (2.5.3.2)[link]] as a transformation of crystalline centrosymmetry; [6_{R}] may be thought of as decomposing into [3 \cdot 2_{R}] for purposes of measurement. The mirror line [m_{R}] (Fig. 2.5.3.2[link]) is similarly generated by [m \cdot 1_{R}].

    Table 2.5.3.3| top | pdf |
    Diffraction point-group tables, giving whole-pattern and central-beam pattern symmetries in terms of BESR diffraction-group symbols and diperiodic group symbols

    IIIIIIIVV
    Whole patternBright field (central beam)BESR groupDiperiodic group (point group)Cubic point groups
    [100][110]
    1 1 1 1    
    1 2 [\underline{1}_{R}] m    
    2 2 2 2    
    1 1 [^{*}2_{R}] [{\bar 1}^{\prime}]    
    2 2 [^{*}\underline{21}_{R}] [2/m']    
    1 m [m_{R}] 2′   23
    m m m m    
    m 2mm [\underline{m1}_{R}] 2′mm   [\bar{4}3m]
    2 2mm [2m_{R}m_{R}] 2′2′2 23 432
    2mm 2mm 2mm mm2    
    m m [^{*}2_{R}mm_{R}] [2'/m]   m [\bar{3}]
    2mm 2mm [^{*}\underline{2mm1}_{R}] mmm m [\bar{3}] m [\bar{3}] m
    4 4 4 4    
    2 4 [4_{R}] [\bar{4}']    
    4 4 [^{*}\underline{41}_{R}] [4/m']    
    4 4mm [4m_{R}m_{R}] [42'2'] [\bar{4}32]  
    4mm 4mm 4mm 4mm    
    2mm 4mm [4_{R}mm_{R}] [\bar{4}'m2'] [\bar{4}3m]  
    4mm 4mm [^{*}\underline{4mm1}_{R}] [4/m'mm] m [\bar{3}] m  
    3 3 3 3    
    3 6 [\underline{31}_{R}] [\bar{6}']    
    3 3m [3m_{R}] 32′    
    3m 3m 3m 3m    
    3m 6mm [\underline{3m1}_{R}] [\bar{6}'m2']    
    6 6 6 6    
    3 3 [^{*}6_{R}] [\bar{3}']    
    6 6 [^{*}\underline{61}_{R}] [6/m']    
    6 6mm [6m_{R}m_{R}] [62'2']    
    6mm 6mm 6mm 6mm    
    3m 3m [^{*}6_{R}mm_{R}] [\bar{3}'m]    
    6mm 6mm [^{*}\underline{6mm1}_{R}] [6/m'mm]    

    Table 2.5.3.3[link] gives the BESR interrelation of pattern symmetries with point group (Buxton et al., 1976[link]; Steeds, 1983[link]). Columns I and II of the table list the point symmetries of the whole pattern and bright-field pattern, respectively; column III gives the BESR diffraction groups. [Note: following the Pond & Vlachavas (1983)[link] usage, `[^{*}]' has been appended to the centrosymmetric groups.]

    Inspection of columns I and II shows that 11 of the 31 diffraction groups can be determined from a knowledge of the whole pattern and bright-field (central-beam disc) point symmetries alone. The remaining 10 pairs of groups need additional observation of the dark-field pattern for their resolution. Disc symmetries [1_{R}], [m_{R}] (Fig. 2.5.3.2[link]; Table 2.5.3.2[link]) are sought (a) in general zero-layer discs and (b) in discs having an [m_{R}] line perpendicular to a proposed twofold axis, respectively; the ±H test is applied for centrosymmetry, to complete the classification.

    Column IV gives the equivalent diperiodic point-group symbol, which, unprimed, gives the corresponding three-dimensional symbol. This will always refer to a non-cubic point group. Column V gives the additional cubic point-group information indicating, where appropriate, how to translate the diffraction symmetry into [100] or [110] cubic settings, respectively.

    Of the groups listed in column III, those representing the projection group of their class are underlined. These groups all contain [1_{R}], the BESR symbol for m′. When the projection approximation is applicable, only those groups underlined will apply. The effect of this approximation is to add a horizontal mirror plane to the symmetry group.

  • Table 2.5.3.4[link]. This lists possible space groups for each of the classified zero-layer CBED symmetries. Since the latter constitute the 80 diperiodic groups, it is first necessary to index the pattern in diperiodic nomenclature; the set of possible space groups is then given by the table.

    Table 2.5.3.4| top | pdf |
    Tabulation of principal-axis CBED pattern symmetries against relevant space groups given as IT A numbers

    Three columns of diperiodic groups (central section) correspond to (i) symmorphic groups, (ii) non-symmorphic groups (GS bands) and (iii) non-symmorphic groups (zero-layer absences arising from horizontal glide planes). Cubic space groups are given underlined in the right-hand section with the code: underlining = [001](cyclic) setting; italics + underlining = [110](cyclic) setting. Separators `;' and `:' indicate change of Bravais lattice type and change of crystal system, respectively.

    DGIIISGIII
    Point groupsDiperiodic groupsSpace groups
    H–MBESR(i)(ii)(iii)Subgroups IIb (Subgroups 1)
    Oblique Triclinic
    1 1 1 p1     1  
    2* [\bar{1}] [2_{R}] [p\bar{1}']     2  
      Monoclinic (Oblique)
    3 12 2 p2     3 4, 5
    4 1m [1_{R}] pm     6 8
    5 1m       pb 7 9
    6* [2/m] [21_{R}] [p2/m']     10 11, 12
    7* [2/m] [21_{R}]     [p2/b'] 13 14, 15
    Rectangular (Rectangular)
    8 21 [m_{R}] p2′     3 52: 195; 197, 199
    9 21 [m_{R}]   [p2'_{1}]   4 198
    10 21 [m_{R}] c2′     5 196
    11 m1 m pm     62 7, 82
    12 m1 m   pa   72 92
    13 m1 m cm     8 9
    14* [12/m] [2_{R}mm_{R}] [p2'/m]     10 13, 122: 200, 201; 204
    15* [12/m] [2_{R}mm_{R}]   [p2'_{1}/m]   11 14
    16* [12/m] [2_{R}mm_{R}]   [p2'/a]   132 152: 206
    17* [12/m] [2_{R}mm_{R}]   [p2'_{1}/m]   142 205
    18* [12/m] [2_{R}mm_{R}] [c2'/m]     12 15: 202, 203
      Orthorhombic
    19 222 [2m_{R}m_{R}] p2′2′2     16 17; 212; 22: 195; 196, 207, 206; 211, 214
    20 222 [2m_{R}m_{R}]   [p2'_{1}2'2]   172 182; 202: 212, 213
    21 222 [2m_{R}m_{R}]   [p2'_{1}2'_{1}2]   18 19: 198
    22 222 [2m_{R}m_{R}] c2′2′2     21 20; 23, 24: 197, 199, 209, 210
    23 mm2 2mm pmm2     25 26, 27; 38, 39; 42
    24 mm2 2mm   pbm2   28 29, 30, 312; 40, 41
    25 mm2 2mm   pba2   32 33, 34; 43
    26 mm2 2mm cmm2     35 36, 37; 44, 45, 46
    27 mm2 [m1_{R}] p2′mm     252 281; 352, 422; 382, 392: 215; 217
    28 mm2 [m1_{R}]   [p2'_{1}m'a]   261 311; 361
    29 mm2 [m1_{R}]   [p2'_{1}ab'] [(p2'_{1}ab')] 292 332
    30 mm2 [m1_{R}]     [p2'_{1}ma'] 262 291; 362
    31 mm2 [m1_{R}]     [p2'_{1}mn'] 312 332
    32 mm2 [m1_{R}]     p2′mb 282 322, 402, 412
    33 mm2 [m1_{R}]     p2′aa 272 302; 372
    34 mm2 [m1_{R}]     [pb2'n'] 301 342; 432: 218; 219
    35 mm2 [m1_{R}] c2′mm     381 401; 442, 461: 216; 220
    36 mm2 [m1_{R}]     c2′mb 391 411; 452, 462
    37* mmm [2mm1_{R}] pmmm     47 49, 511; 652, 672; 69:
                  200; 202, 221, 224, 226, 228, 229
    38* mmm [2mm1_{R}]   [pbmm'\; (2'_{1})]   512 531, 57, 592; 631, 641
    39* mmm [2mm1_{R}]   [pbam'\; (2'_{1}2'_{1})]   55 58, 622
    40* mmm [2mm1_{R}]   [pmab'\; (2'_{1}2'_{1})] [(pmab')] 571 602, 61, 62: 205
    41* mmm [2mm1_{R}]   [pbaa'\; (2'_{1})] [(pbaa')] 542 52, 562, 601
    42* mmm [2mm1_{R}]     [pmma'\; (2'_{1})] 51 54, 552, 572; 632, 642
    43* mmm [2mm1_{R}]     pmmn[(2'_{1}2'_{1})] 59 56, 621
    44* mmm [2mm1_{R}]     [pbmn'\; (2'_{1})] 532 521, 581, 60
    45* mmm [2mm1_{R}]     pmaa 492 502, 53, 541; 662, 681: 222, 223
    46* mmm [2mm1_{R}]     pban 50 522, 48; 70: 201; 203, 230
    47* mmm [2mm1_{R}] cmmm     65 63, 66; 72, 742, 71: 204, 225, 227
    48* mmm [2mm1_{R}]     cmma 67 64, 68; 721, 74, 73: 206
    Square Tetragonal
    49 4 4 p4     75 77, 76, 78; 79, 80
    50 [4/m] [41_{R}] [p4/m']     83 84; 87
    51 [4/m] [41_{R}]     [p4/n'] 85 86, 88
    52 422 [4m_{R}m_{R}] [p42'2']     89 93, 91, 95; 97, 98:
                  207, 208; 209, 210; 211, 214
    53 422 [4m_{R}m_{R}]   [p42'_{1}2']   90 94, 92, 96: 212, 213
    54 4mm 4mm [p4mm]     99 101, 103, 105; 107, 108
    55 4mm 4mm   [p4bm]   100 102, 104, 106; 109, 110
    56* [4/mmm] [4mm1_{R}] [p4/m'mm]     123 124, 131, 132; 139, 140;
                  221, 223; 225, 226; 229
    57* [4/mmm] [4mm1_{R}]   [p4/m'bm\; (2'_{1})]   127 128, 135, 136
    58* [4/mmm] [4mm1_{R}]     [p4/n'bm] 125 126, 133, 134; 141, 142:
                  222, 224; 227, 228; 230
    59* [4/mmm] [4mm1_{R}]     [p4/n'mm\; (2'_{1})] 129 130, 137, 138
    60 [\bar{4}] [4_{R}] [p\bar{4}']     81 82
    61 [\bar{4}2m] [4_{R}mm_{R}] [p\bar{4}'m\bar{2}']     115 116; 119, 120
    62 [\bar{4}2m] [4_{R}mm_{R}]   [p\bar{4}b2']   117 118; 122: 220
    63 [\bar{4}2m] [4_{R}mm_{R}] [p\bar{4}'2'm]     111 112; 121: 215; 216; 217; 218; 219
    64 [\bar{4}2m] [4_{R}mm_{R}]   [p4'2'_{1}m]   113 114
    Hexagonal Trigonal
    65 3 3 p3     143 144, 145; 146
    66 [\bar{3}] [6_{R}] [p\bar{3}']     147 148
    67 32 [3m_{R}] p312′     149 151, 153
    68 32 [3m_{R}] [p32'1]     150 152, 154; 155
    69 3m 3m p31m     157 159
    70 3m 3m p3m1     156 158; 160, 161
    71* [\bar{3}m] [6_{R}mm_{R}] [p\bar{3}'1m]     162 163
    72* [\bar{3}m] [6_{R}mm_{R}] [p\bar{3}'m1]     164 165; 166, 167
      Hexagonal
    73 6 6 p6     168 171, 172, 173, 169, 170
    74 [\bar{6}] [31_{R}] [p3/m'\; (p\bar{6}')]     174  
    75 622 [6m_{R}m_{R}] [p62'2']     177 180, 181, 182, 178, 179
    76 6mm 6mm [p6mm]     183 184, 185, 186
    77* [6/m] [61_{R}] [p6/m']     175 176
    78* [6/mmm] [6mm1_{R}] [p6/m'mm]     191 192, 193, 194
    79 [\bar{6}m2] [3m1_{R}] [p3/m'2'm] [(p\bar{6}'m2')]     189 190
    80 [\bar{6}m2] [3m1_{R}] [p3/m'm2'] [(p\bar{6}'2'm)]     187 188

    A basic requirement for diperiodic group nomenclature has been that of compatibility with IT A and I. This has been met by the recent Pond & Vlachavas (1983) tabulation. For example, DG: [(^{*})pban'], where [^{*}] indicates centrosymmetry, becomes space group Pban when, in Seitz matrix description, the former group matrix is multiplied by the third primitive translation, [a_{3}]. Furthermore, in textual reference the prime can be optionally omitted, since the lower-case lattice symbol is sufficient indication of a two-dimensional periodicity (as pban).

    The three sections of Table 2.5.3.4[link] are:

    • I. Point-group entries, given in H–M and BESR symbols.

    • II. Pattern symmetries, in diperiodic nomenclature, have three subdivisions: (i) symmorphic groups: patterns without zero-layer absences or extinctions. Non-symmorphic groups are then given in two categories: (ii) patterns with zero-layer GS bands, and (iii) patterns with zero-layer absences resulting from a horizontal glide plane; where the pattern also contains dynamic extinctions (GS bands) and so is listed in column (ii), the column (iii) listing is given in parentheses.

      The `short' (Pond & Vlachavas) symbol has proved an adequate description for all but nine groups for which the screw-axis content was needed: here [(2'_{1})], or [(2'_{1}2'_{1})], have been added to the symbol.

    • III. Space-group entries are given in terms of IT A numbers. The first column of each row gives the same-name space group as illustrated by the example pban′ → Pban above. The groups following in the same row (which have the same zero-layer symmetry) complete an exhaustive listing of the IIb subgroups, given in IT A . Cubic space groups are underlined for the sake of clarity; hence, those giving rise to the zero-layer symmetry of the diffraction group in the [100] (cyclic) setting have a single underline: these are type I minimal supergroups in IT A nomenclature. The cubic groups are also given in the [110] setting, in underlined italics, since this is a commonly encountered high-symmetry setting. (Note: these then are no longer minimal supergroups and the relationship has to be found through a series of IT A listings.)

    The table relates to maximal-symmetry settings. For monoclinic and orthorhombic systems there are three equally valid settings. For monoclinic groups, the oblique and rectangular settings appear separately; where rectangular C-centred groups appear in a second setting this is indicated by superscript `2'. For orthorhombic groups, superscripts correspond to the `incident-beam' system adopted in Table 2.5.3.5[link], as follows: no superscript: [001] beam direction; superscript 1: [100] beam direction; superscript 2: [010] beam direction. The cubic system is treated specially as described above.

    Table 2.5.3.5| top | pdf |
    Conditions for observation of GS bands for the 137 space groups exhibiting these extinctions

    Point groups 2, m, [{\bf 2/{\bi m}}] (2nd setting unique axis [\|b])

    Space groupIncident-beam direction
    [u0w]
    4 [P2_{1}] 0k0 S
    [2_{1}]
    7 Pc h0l G
    c
    9 Cc h0l G
    c
    11 [P2_{1}/m] 0k0 S
    [2_{1}]
    13 [P2/c] h0l G
    c
    14 [P2_{1}/c] 0k0 S
    [2_{1}]
    h0l G
    c
    15 [C2/c] h0l G
    c

    Point groups 222, mm2

    Space groupIncident-beam direction
    [100][010][001][uv0][0vw][u0w]
    17 [P222_{1}] 00l S 00l S     00l S        
    [2_{1}] [2_{1}]     [2_{1}]        
    18 [P2_{1}2_{1}2] 0k0 S h00 S h00, 0k0 S     h00 S 0k0 S
    [2_{1}] [2_{1}] [2_{1}]     [2_{1}] [2_{1}]
    19 [P2_{1}2_{1}2_{1}] 0k0, 00l S h00, 00l S h00, 0k0 S 00l S h00 S 0k0 S
    [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}]
    20 [C222_{1}] 00l S 00l S     00l S        
    [2_{1}] [2_{1}]     [2_{1}]        
    26 [Pmc2_{1}] 00l GS 00l c′−     00l S     h0l G
    [c + 2_{1}] [2_{1}]     [2_{1}]     c
    27 [Pcc2] 00l c′− 00l c′−         0kl G h0l G
    c c         c c
    28 [Pma2]         h00 G         h0l G
            a         a
    29 [Pca2_{1}] 00l c′− 00l GS h00 G 00l S 0kl G h0l G
    [2_{1}] [c + 2_{1}] a [2_{1}] c a
    30 [Pnc2] 00l n′− 00l c′− 0k0 G     0kl G h0l G
    c n n     n c
    31 [Pmn2_{1}] 00l GS 00l n′− h00 G 00l S     h0l G
    [n + 2_{1}] [2_{1}] n [2_{1}]     n
    32 [Pba2]         h00, 0k0 G     0kl G h0l G
            a b     b a
    33 [Pna2_{1}] 00l n′− 00l GS h00, 0k0 G 00l S 0kl G h0l G
    [2_{1}] [n + 2_{1}] a b [2_{1}] n a
    34 [Pnn2] 00l n′− 00l n′− h00, 0k0 G     0kl G h0l G
    n n n     n n
    36 [Cmc2_{1}] 00l GS 00l c′−     00l S     h0l G
    [c + 2_{1}] [2_{1}]     [2_{1}]     c
    37 [Ccc2] 00l c′− 00l c′−         0kl G h0l G
    c c         c c
    39 [Abm2]                 0kl G    
                    b    
    40 [Ama2]         h00 G         h0l G
            a         a
    41 [Aba2]         h00 G     0kl G h0l G
            a     b a
    43 [Fdd2] 00l d′− 00l d′− h00, 0k0 G     0kl G h0l G
      d d     d d
    45 [Iba2]   b′−   a′−         0kl G h0l G
                b a
    46 [Ima2]       a′−             h0l G
                      a

    Point group mmm

    Space groupIncident-beam direction
    [100][010][001][uv0][0vw][u0w]
    48 P 2/n 2/n 2/n 00l, 0k0 n′− 00l, h00 n′− 0k0, h00 n′− hk0 G 0kl G h0l G
    n n n n n n
    49 P 2/c 2/c 2/m 00l c′− 00l c′−         0kl G h0l G
    c c         c c
    50 P 2/b 2/a 2/n 0k0 b′− h00 a′− 0k0, h00 n′− hk0 G 0kl G h0l G
    n n b a n b a
    51 [P\;2_{1}/m\;2/m\;2/a]     h00 GS h00 a′− hk0 G h00 S    
        [a + 2_{1}] [2_{1}] a [2_{1}]    
    52 [P\;2/n\;2_{1}/n\;2/a] 00l, 0k0 a′− 00l, h00 n′− 0k0 GS hk0 G 0kl G h0l G
    [n + 2_{1}] n
    [2_{1}] n a h00 n′− a n 0k0 S
    n [2_{1}]
    53 [P\;2/m\;2/n\;2_{1}/a] 00l GS h00, 00l a′− h00 a′− hk0 G     h0l G
    a
    [n + 2_{1}] [2_{1}] n 00l S n
    [2_{1}]
    54 [P\;2_{1}/c\;2/c\;2/a] 00l c′− h00 GS h00 a′− hk0 G 0kl G h0l G
    [a + 2_{1}] c
    c 00l c′− [2_{1}] a h00 S c
    c [2_{1}]
    55 [P\;2_{1}/b\;2_{1}/a\;2/m] 0k0 b′− h00 a′− 0k0, h00 GS     0kl G h0l G
    b a
    [2_{1}] [2_{1}] [b + 2_{1}, a + 2_{1}] h00 S 0k0 S
    [2_{1}] [2_{1}]
    56 [P\;2_{1}/c\;2_{1}/c\;2/n] 0k0 GS h00 GS 0k0, h00 n′− hk0 G 0kl G h0l G
    [n + 2_{1}] [n + 2_{1}] c c
    00l c′− 00l c′− [2_{1}] n h00 S 0k0 S
    c c [2_{1}] [2_{1}]
    57 [P\;2/b\;2_{1}/c\;2_{1}/m] 00l GS 00l c′− 0k0 GS 00l S 0kl G h0l G
    [c + 2_{1}] c
    0k0 b′− [2_{1}] [b + 2_{1}] [2_{1}] b 0k0 S
    [2_{1}] [2_{1}]
    58 [P\;2_{1}/n\;2_{1}/n\;2/m] 00l, 0k0 n′− 00l, h00 n′− 0k0, h00 GS     0kl G h0l G
    n n
    n [2_{1}] n [2_{1}] [n + 2_{1}] h00 S 0k0 S
    [2_{1}] [2_{1}]
    59 [P\;2_{1}/m\;2_{1}/m\;2/n] 0k0 GS h00 GS 0k0, h00 n′− hk0 G h00 S 0k0 S
    [n + 2_{1}] [n + 2_{1}] [2_{1}] n [2_{1}] [2_{1}]
    60 [P\;2_{1}/b\;2/c\;2_{1}/n] 00l GS h00 GS 0k0, h00 n′− hk0 G 0kl G h0l G
    [c + 2_{1}] [n + 2_{1}] n b
    0k0 b′− 00l c′− b [2_{1}] 00l S h00 S c
    n [2_{1}] [2_{1}] [2_{1}]
    61 [P\;2_{1}/b\;2_{1}/c\;2_{1}/a] 00l GS h00 GS 0k0 GS hk0 G 0kl G h0l G
    [c + 2_{1}] [a + 2_{1}] [b + 2_{1}] a b c
    0k0 b′− 00l c′− h00 a′− 00l S h00 S 0k0 S
    [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}]
    62 [P\;2_{1}/n\;2_{1}/m\;2_{1}/a] 0k0, 00l n′− 00l GS 0k0 GS hk0 G 0kl G 0k0 S
    [n + 2_{1}] [n + 2_{1}] a n
    [2_{1}] h00 h00 a′− 00l S h00 S [2_{1}]
    [a + 2_{1}] [2_{1}] [2_{1}] [2_{1}]
    63 [C\;2/m\;2/c\;2_{1}/m] 00l GS 00l c′−     00l S     h0l G
    [c + 2_{1}] [2_{1}] [2_{1}] c
    64 [C\;2/m\;2/c\;2_{1}/a] 00l GS 00l c′−     hk0 G     h0l G
    a
    [c + 2_{1}] [2_{1}] 00l S c
    [2_{1}]
    66 [C\;2/c\;2/c\;2/m] 00l c′− 00l c′−         0kl G h0l G
    c c c c
    67 [C\;2/m\;2/m\;2/a]             hk0 G        
    a
    68 [C\;2/c\;2/c\;2/a] 00l c′− 00l c′−     hk0 G 0kl G h0l G
    c c a c c
    70 [F\;2/d\;2/d\;2/d] 00l, 0k0 d′− h00, 00l d′− 0k0, h00 d′− hk0 G 0kl G h0l G
    d d d d d d
    72 [I\;2/b\;2/a\;2/m]   b′−   a′−         0kl G h0l G
    b a
    73 [I\;2/b\;2/c\;2/a]   b′−   c′−   a′− hk0 G 0kl G h0l G
    a b c
    74 [I\;2/m\;2/m\;2/a]             hk0 G        
    a

    Point groups 4, [\bar{\bf{4}}], [\bf{4/{\bi m}}]

    Space groupIncident-beam direction
    [uv0]
    76 [P4_{1}] 00l S
    [4_{1}]
    78 [P4_{3}] 00l S
    [4_{3}]
    85 [P4/n] hk0 G
    n
    86 [P4_{2}/n] hk0 G
    n
    88 [I4_{1}/a] hk0 G
    a

    Point group 422

    Space groupIncident-beam direction
    [uv0][0vw]
    90 [P42_{1}2]     h00 S
    [2_{1}]
    91 [P4_{1}22] 00l S    
    [4_{1}]
    92 [P4_{1}2_{1}2] 00l S h00 S
    [4_{1}] [2_{1}]
    94 [P4_{2}2_{1}2]     h00 S
    [2_{1}]
    95 [P4_{3}22] 00l S    
    [4_{3}]
    96 [P4_{3}2_{1}2] 00l S h00 S
    [4_{3}] [2_{1}]

    Point group 4mm

    Space groupIncident-beam direction
    [100][001][110][u0w] and [0vw] [[u\bar{u}w]]
    100 [P4bm]     h00, 0k0 G     h0l, 0kl G    
    a b a b
    101 [P4_{2}cm] 00l c′−         h0l, 0kl G    
    c c
    102 [P4_{2}nm] 00l n′− h00, 0k0 G     h0l, 0kl G    
    n n n
    103 [P4cc] 00l c′−     00l c′− h0l, 0kl G hhl G
    c c c c
    104 [P4nc] 00l n′− h00, 0k0 G 00l c′− h0l, 0kl G hhl G
    n n c n c
    105 [P4_{2}mc]         00l c′−     hhl G
    c c
    106 [P4_{2}bc]     h00, 0k0 G 00l c′− h0l, 0kl G hhl G
    a b c a b c
    108 [I4cm]             h0l, 0kl G    
    c
    109 [I3_{1}md]     hh0 G 00l d′−     hhl G
    [{\bar h}h0]    
    d d d
    110 [I4_{1}cd]     hh0 G 00l d′− h0l, 0kl G hhl G
    [{\bar h}h0]      
    d d c d

    Point groups [\bar{\bf {4}}]2m, 4/mmm

    Space groupIncident-beam direction
    [100][001][110][u0w] and [0vw][uuw][uv0]
    112 [P{\bar 4}2c]         00l c′−     hhl G    
    c c
    113 [P{\bar 4}2_{1}m] 0k0 S h00, 0k0 S     0k0, h00 S        
    [2_{1}] [2_{1}] [2_{1}]
    114 [P{\bar 4}2_{1}c] 0k0 S h00, 0k0 S 00l c′− 0k0, h00 S hhl G    
    [2_{1}] [2_{1}] c [2_{1}] c
    116 [P{\bar 4}c2] 00l c′−         h0l, 0kl G        
    c c
    117 [P{\bar 4}b2]     h00, 0k0 G     h0l, 0kl G        
    a b a b
    118 [P{\bar 4}n2] 00l n′− h00, 0k0 G     h0l, 0kl G        
    n n n
    120 [I{\bar 4}c2]             h0l, 0kl G        
    c
    122 [I{\bar 4}2d]     hh0 G 00l d′−     hhl G    
    [{\bar h}h0]    
    d d d
    124 [P\;4/m\;2/c\;2/c] 00l c′−     00l c′− h0l, 0kl G hhl G    
    c c c c
    125 [P\;4/n\;2/b\;2/m] 0k0 b′− h00, 0k0 n′−     h0l, 0kl G     hk0 G
    n a b a b n
    126 [P\;4/n\;2/n\;2/c] 0k0 n′− h00, 0k0 n′− 00l c′− h0l, 0kl G hhl G hk0 G
    n
    00l n c n c n
    n
    127 [P\;4/m\;2_{1}/b\;2/m] 0k0 b′− h00 GS     h0l, 0kl G        
    [a + 2_{1}] a b
    [2_{1}] 0k0 0k0, h00 S
    [b + 2_{1}] [2_{1}]
    128 [P\;4/m\;2_{1}/n\;2/c] 00l, 0k0 n′− h00, 0k0 GS 00l c′− h0l, 0kl G hhl G    
    n
    [n\quad 2_{1}] [n + 2_{1}] c 0k0, h00 S c
    [2_{1}]
    129 [P\;4/n\;2_{1}/m\;2/m] 0k0 GS h00, 0k0 n′−     0k0, h00 S     hk0 G
    [n + 2_{1}] [2_{1}] [2_{1}] n
    130 [P\;4/n\;2_{1}/c\;2/c] 0k0 GS h00, 0k0 n′− 00l c′− h0l, 0kl G hhl G hk0 G
    [n + 2_{1}] c
    00l c′− [2_{1}] c 0k0, h00 S c n
    c [2_{1}]
    131 [P\;4_{2}/m\;2/m\;2/c]         00l c     hhl G    
    c c
    132 [P\;4_{2}/m\;2/c\;2/m] 00l c′−         h0l, 0kl G        
    c c
    133 [P\;4_{2}/n\;2/b\;2/c] 0k0 b′− h00, 0k0 n′− 00l c′− h0l, 0kl G hhl G hk0 G
    n a b c a b c n
    134 [P\;4_{2}/n\;2/n\;2/m] 0k0, 00l n′− h00, 0k0 n′−     h0l, 0kl G     hk0 G
    n n n n
    135 [P\;4_{2}/m\;2_{1}/b\;2/c] 0k0 b′− h00, 0k0 GS 00l c′− h0l, 0kl G hhl G    
    a b
    [2_{1}] [a + 2_{1}\;b + 2_{1}] c 0k0, h00 S c
    [2_{1}]
    136 [P\;4_{2}/m\;2_{1}/n\;2/m] 00l, 0k0 n′− h00, 0k0 GS     h0l, 0kl G        
    n
    n 2 [n + 2_{1}] 0k0, h00 S
    [2_{1}]
    137 [P\;4_{2}/n\;2_{1}/m\;2/c] 0k0 GS h00, 0k0 n′− 00l c′− 0k0, h00 S hhl G hk0 G
    [n + 2_{1}] [2_{1}] c [2_{1}] c n
    138 [P\;4_{2}/n\;2_{1}/c\;2/m] 0k0 GS h00, 0k0 n′−     h0l, 0kl G     hk0 G
    [n + 2_{1}] c
    00l c′− [2_{1}] 0k0, h00 S n
    c [2_{1}]
    140 [I\;4/m\;2/c\;2/m]             h0l, 0kl G        
    c
    141 [I\;4_{1}/a\;2/m\;2/d]     hh0 a′− 00l, [{\bar h}h0] d′−     hhl G hk0 G
    [{\bar h}h0]      
    d d a d a
    142 [I\;4_{1}/a\;2/c\;2/d]     hh0 a′− 00l, [{\bar h}h0] d′− h0l, 0kl G hhl G hk0 G
    [{\bar h}h0]        
    d d a c d a

    Point groups 3m, [{\bar{\bf 3}}] m, 6, 6/m, 622, 6mm, [{\bar{\bf 6}}] m2, 6/mmm

    Space groupIncident-beam direction
    [100][210][2u u w][v0w]
    158 P3c1     000l G [hh2{\bar h}l] G    
    c c
    159 P31c 000l G         [h{\bar h}0l] G
    c c
    161 R3c     000l G [hh2{\bar h}l] G    
    [l = 6n + 3]  
    c c
    163 [P{\bar 3}1c] 000l G         [h{\bar h}0l] G
    c c
    165 [P{\bar 3}c1]     000l G [hh2{\bar h}l] G    
    c c
    167 [R{\bar 3}c]     000l G [hh2{\bar h}l] G    
    [l = 6n + 3]  
    c c
    169 [P6_{1}] 000l S 000l S        
    [6_{1}] [6_{1}]
    170 [P6_{5}] 00l S 00l S        
    [6_{5}] [6_{5}]
    173 [P6_{3}] 000l S 000l S        
    [6_{3}] [6_{3}]
    176 [P6_{3}/m] 000l S 000l S        
    [6_{3}] [6_{3}]
    178 [P6_{1}22] 000l S 000l S        
    [6_{1}] [6_{1}]
    179 [P6_{5}22] 000l S 000l S        
    [6_{5}] [6_{5}]
    182 [P6_{3}22] 000l S 000l S        
    [6_{3}] [6_{3}]
    184 [P6cc] 000l c′− 000l c′− [hh2{\bar h}l] G [h{\bar h}0l] G
    c c c c
    185 [P6_{3}cm] 000l c′− 000l GS [hh2{\bar h}l] G    
    [6_{3}] [c + 6_{3}] c
    186 [P6_{3}mc] 000l GS 000l c′−     [h{\bar h}0l] G
    [c + 6_{3}] [6_{3}] c
    188 [P{\bar 6}c2]     000l G [hh2{\bar h}l] G    
    c c
    190 [P{\bar 6}c2] 000l G         [h{\bar h}0l] G
    c c
    192 [P6/mcc] 000l c′− 000l c′− [hh2{\bar h}l] G [h{\bar h}0l] G
    c c c c
    193 [P6_{3}/mcm] 00l c′− 000l GS [hh2{\bar h}l] G    
    [6_{3}] [c + 6_{3}] c
    194 [P6_{3}/mmc] 000l GS 000l c′−     [h{\bar h}0l] G
    [c + 6_{3}] [6_{3}] c

    Point groups 23, m3, 432, m3m

    Space groupIncident-beam direction
    [100][110][uv0][uuw]
    (cyclic)(cyclic)(cyclic)(cyclic)
    198 P [2_{1}]3 00l, 0k0 S 00l S 00l S    
    [2_{1}] [2_{1}] [2_{1}]
    201 [Pn\bar{3}] 00l, 0k0 n′−     [{\bar k}h0] G    
    [P2/n{\bar 3}] n n
    203 Pd [{\bar 3}] 00l, 0k0 d′−     [{\bar k}h0] G    
    [F2/d{\bar 3}] d d
    205 Pa [{\bar 3}] 00l GS 00l S 00l S    
    [c + 2_{1}] [2_{1}] [2_{1}]
    [P2_{1}/a{\bar 3}] 0k0 b′− [{\bar h}h0] G [{\bar k}h0] G
    [2_{1}] a a
    206 Ia [{\bar 3}]     [{\bar h}h0] G [{\bar k}h0] G    
    [I2_{1}/a{\bar 3}] a a
    212 [P4_{3}32]         00l S    
    [4_{3}]
    213 [P4_{1}32]         00l S    
    [4_{1}]
    218 [P{\bar 4}3n]     00l n hhl G    
    c c
    219 [F{\bar 4}3c]         hhl G    
    c
    220 [I{\bar 4}3d] 0kk G 00l d hhl G    
    [0{\bar k}k]    
    d d d
    222 [Pn{\bar 3}n] 00l, 0k0 n′− 00l n hk0 G hhl G
    n c n c
    223 [Pm{\bar 3}n]     00l n′−     hhl G
    c c
    224 [Pn{\bar 3}m] 00l, 0k0 n′−     hk0 G    
    n n
    226 [Fm{\bar 3}c]             hhl G
    c
    227 [Fd{\bar 3}m] 00l, 0k0 d′−     hk0 G    
    d d
    228 [Fd{\bar 3}c] 00l, 0k0 d′−     hk0 G hhl G
    d d c
    230 [Ia{\bar 3}d] 0kk b′− 00l, [{\bar h}h0] d′− hk0 G hhl G
    [0{\bar k}k]      
    d d a a d
    Conditions in this column are cyclic on h and k.
  • Table 2.5.3.5[link]. This lists conditions for observation of GS bands for the 137 space groups exhibiting these extinctions. These are entered as `G', `S', or `GS', indicating whether a glide plane, screw axis, or both is responsible for the GS band. All three possibilities will lead to a glide line (and hence to both extinction bands) in projection, and one of the procedures (a), (b), or (c) of Section 2.5.3.3[link](iv)[link] above is needed to complete the three-dimensional interpretation. In addition, the presence of horizontal glide planes, which result in systematic absences in these particular cases in the zero-layer pattern, is indicated by the symbol `−'. Where these occur at the site of prospective `G' or `S' bands from other glide or screw elements the symbol of that element is given and the `G' or `S' symbol is omitted.

    The following paragraphs give information on the real-space interpretation of GS band formation, and their specific extinction rules, considered useful in structural interpretation.

    Real-space interpretation of extinction conditions . Dynamic extinctions (GS bands) are essentially a property of symmetry in reciprocal space. However, since diagrams from IT I and A are used there is a need to give an equivalent real-space description. These bands are associated with the half-unit-cell-translational glide planes and screw axes represented in these diagrams. Inconsistencies between `conventional' and `physical' real-space descriptions, however, become more apparent in dynamical electron diffraction, which is dependent upon three-dimensional scattering physics, than in X-ray diffraction. Also, the distinction between general (symmorphic) and specific (non-symmorphic) extinctions is more basic (in the former case). This is clarified by the following points:

    • (i) Bravais lattice centring restricts the conditions for observation of GS bands. For example, in space group Abm2 (No. 39), `A' centring prevents observation of the GS bands associated with the `b' glide at the [001] zone-axis orientation; this observation, and hence verification of the b glide, must be made at the lower-symmetry zone axes [0vw] (see Table 2.5.3.5)[link]. In the exceptional cases of space groups [I2_{1}2_{1}2_{1}] and [I2_{1}3] (Nos. 24 and 199), conditions for the observation of the relevant GS bands are completely prevented by body centring; here the screw axes of the symmorphic groups I222 and I23 are parallel to the screw axes of their non-symmorphic derivatives. However, electron crystallographic methods also include direct structure imaging by HREM, and it is important to note here that while the indistinguishability encountered in data sets acquired in Fourier space applies to both X-ray diffraction and CBED (notwithstanding possible differences in HOLZ symmetries), this limitation does not apply to the HREM images (produced by dynamic scattering) yielding an approximate structure image for the (zone-axis) projection. This technique then becomes a powerful tool in space-group research by supplying phase information in a different form.

    • (ii) A different complication, relating to nomenclature, occurs in the space groups [P\bar{4}3n], Pn3n and Pm3n (Nos. 218, 222 and 223) where `c' glides parallel to a diagonal plane of the unit cell occur as primary non-symmorphic elements (responsible for reciprocal-space extinctions) but are not used in the Hermann–Maugin symbol; instead the derivative `n' glide planes are used as characters, resulting in an apparent lack of correspondence between the conventionally given real-space symbols and the reciprocal-space extinctions.

      (Note: In IT I non-symmorphic reflection rules which duplicate rules given by lattice centring, or those which are a consequence of more general rules, are given in parentheses; in IT A this clarification by parenthesizing, helpful for electron-diffraction analysis, has been removed.)

    • (iii) Finally, diamond glides (symbol `d') require special consideration since they are associated with translations [{1 \over 4}], [{1 \over 4}], [{1 \over 4}], and so would appear not to qualify for GS bands; however, this translation is a result of the conventional cell being defined in real rather than reciprocal space where the extinction symmetry is formed. Hence `d' glides occur only in F-centred lattices (most obviously Nos. 43, 70, 203, 277 and 228). These have correspondingly an I-centred reciprocal lattice for which the zero-layer two-dimensional unit cell has an edge of [a^{*\prime\prime} = 2a^{*}]. Consequently, the first-order row reflection along the diamond glide retains the reciprocal-space anti-symmetry on the basis of this physical unit cell (halved in real space), and leads to the labelling of odd-order reflections as [4n + 2] (instead of [2n + 1] when the cell is not halved). Additionally, although seven space groups are I-centred in real space with the conventional unit cell (Nos. 109, 110, 122, 141, 142, 220 and 230), these space groups are F-centred with the transformation [a'' = [110]], [b'' = [1\bar{1}0]], and correspondingly I-centred in the reciprocal-space cell as before, but the directions [100], [010] and reflection rows h00, 0k0 become replaced by directions [110] (or [[1\bar{1}0]]) and rows hh0, [h\bar{h}0], in the entries of Table 2.5.3.5[link].

    Extinction rules for symmetry elements appearing in Table 2.5.3.5[link]. Reflection indices permitting observation of G and S bands follow [here `zero-layer' and `out-of-zone' (i.e. HOLZ or alternative zone) serve to emphasize that these are zone-axis observations].

    • (i) Vertical glide planes lead to `G' bands in reflections as listed (`a', `b', `c' and `n' glides):

      h0l, hk0, 0kl out-of-zone reflections (for glide planes having normals [010], [001] and [100]) having [h + l], [h + k], [k + l = 2n + 1], respectively, in the case of `n' glides, or h, k, l odd in the case of `a', `b' or `c' glides, respectively;

      h00, 0k0, 00l zero-layer reflections with h, k or l odd.

      Correspondingly for `d' glides:

      • (a) In F-centred cells:

        h0l, hk0, 0kl out-of-zone reflections (for glide planes having normals [010], [100] and [001], having [h + l], [k + l], or [h + k = 4n + 2], respectively, with h, k and l even; and (space group No. 43 only) zero-layer reflections h00, 0k0 with h, k even and [= 4n + 2].

      • (b) In I-centred cells:

        hhl (cyclic on h, k, l for cubic groups) out-of-zone reflections having [2h + l = 4n + 2], with l even; and zero-layer reflections hh0, [h\bar{h}0] (cyclic on h, k, l for cubic groups) having h odd.

    • (ii) Horizontal screw axes, namely [2_{1}] or the [2_{1}] component of screw axes [4_{1}], [4_{3}], [6_{1}], [6_{3}], [6_{5}], lead to `S' bands in reflection rows parallel to the screw axis, i.e. either h00, 0k0 or 00l, with conventional indexing, for h, k or l odd.

    • (iii) Horizontal glide planes lead to zero-layer absences rather than GS bands. When these prevent observation of a specific GS band (by removing the two-dimensional conditions), the symbol `−' indicates a situation where, in general, there will simply be absences for the odd-order reflections. However, Ishizuka & Taftø (1982)[link] were the first to observe finite-intensity narrow bands under these conditions, and it is now appreciated that with a sufficient crystal thickness and a certain minimum for the z-axis repeat distance, GS bands can be recorded by violating the condition for horizontal-mirror-plane (m′) extinction while satisfying the condition for G or S, achieved by appropriate tilts away from the exact zone-axis orientation [see Section 2.5.3.3[link](iv)[link]].

2.5.3.5. Space-group analyses of single crystals; experimental procedure and published examples

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2.5.3.5.1. Stages of procedure

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  • (i) Zone-axis patterns . The first need is to record a principal zone-axis pattern. From this, the rotational order X of the vertical axis and associated mirror (including glide-line) components are readily observed (see all examples).

    This pattern may include part of the higher-order Laue zone; in particular the closest or first-order Laue zone (FOLZ) should be included in order to establish the presence or absence of horizontal glide planes, as illustrated in Fig. 2.5.3.3[link]. The projection approximation frequently applies to the zone-axis pattern, particularly when this is obtained from thin crystals (although this cannot apply by definition to the FOLZ). This is indicated by the absence of fine-line detail in the central beam particularly; identification of the projected symmetry is then straightforward.

  • (ii) Laue circle patterns . Next, it is usual to seek patterns in which discs around the Laue circle include the line [m_{R}] (Fig. 2.5.3.2[link]). The internal disc symmetries observed together with those from the zone-axis pattern will determine a diffraction group, classifying the zero-layer symmetry. [Fig. 6(c) of Goodman & Whitfield (1980)[link] gives an example of Laue-circle symmetries.]

  • (iii) Alternative zone axes or higher-order Laue zones . Finally, alternative zone or higher-order Laue-zone patterns may be sought for additional three-dimensional data: (a) to determine the three-dimensional extinction rules, (b) to test for centrosymmetry, or (c) to test for the existence of mirror planes perpendicular to the principal rotation axis. These procedures are illustrated in the following examples.

2.5.3.5.2. Examples

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  • (1) Determination of centrosymmetry; examples from the hexagonal system. Fig. 2.5.3.4[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.

    [Figure 2.5.3.4]

    Figure 2.5.3.4 | top | pdf |

    (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. 2.5.3.4[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. 2.5.3.4[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 2.5.3.3.[link] Fig. 2.5.3.5[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 2.5.3.3)[link]; this same observation establishes the crystallographic point group as m3m (column V of Table 2.5.3.3[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 2.5.3.3)[link], and identifies the cubic point group as [\bar{4}3m].

    [Figure 2.5.3.5]

    Figure 2.5.3.5 | top | pdf |

    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 2.5.3.4[link] and 2.5.3.5[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. 2.5.3.6[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]).

    [Figure 2.5.3.6]

    Figure 2.5.3.6 | top | pdf |

    (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 2.5.3.4[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. 2.5.3.3[link]). The only possible cubic group from Table 2.5.3.4[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. 2.5.3.6b[link]), confirmed by mirror symmetries across the G and S lines. From Table 2.5.3.5[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 2.5.3.4[link]. This zone-axis pattern has the characteristics illustrated in Figs. 2.5.3.3[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 2.5.3.5[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 2.5.3.5[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 2.5.3.5[link].

    From the above discussions it will be clear that Tables 2.5.3.4[link] and 2.5.3.5[link] present information in a complementary way: in Table 2.5.3.4[link] the specific pattern group is indexed first with the possible space groups following, while in Table 2.5.3.5[link] the space group is indexed first, and the possible pattern symmetries are then given, in terms of the standard International Tables setting.

2.5.3.6. 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].

2.5.3.7. 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 2.5.3.8[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].

2.5.3.8. Computer programs available

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  • (1) A FORTRAN source listing of program TCBED for simulating three-dimensional convergent-beam patterns with absorption by the Bloch-wave method: Zuo et al. (1989)[link] [see also Electron microdiffraction (Spence & Zuo, 1992[link]) for other useful programs and worked examples for the analysis of these diffraction patterns]. Contact J. M. Zuo or J. C. H. Spence, Physics Department, Arizona State University, Tempe, Arizona, USA.

  • (2) A package for CBED pattern simulation by both Bloch-wave and multi-slice methods is available from P. Stadelmann (pierre.stadelmann@cime.uhd.edfl.ch ), Lausanne, Switzerland, in UNIX for workstations [Silicon Graphics, Dec Alpha (OSF), IBM RISC 6000, SUN and HP-9000].

  • (3) HOLZ line simulations: Listing for PC 8801 (NEC): Tanaka & Terauchi (1985[link], pp. 174–175).

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