Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 286-288   | 1 | 2 |

Section Background theory and analytical approach

P. Goodmanb Background theory and analytical approach

| top | pdf | Direct and reciprocity symmetries: types I and II

| top | pdf |

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 ([link]] and Chapter 5.2 [equation (][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[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| 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
    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.[link]). [{\bf K}_{g} = {\bf 0}] also defines the symmetry centre within the two-dimensional disc diagram (Fig.[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 | 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 | 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]. Reciprocity and Friedel's law

| top | pdf |

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(] 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 ([link] to be complex-conjugated giving the useful CBED pattern equation [{\bf u}_{g, \, {\bf K}} = {\bf u}_{\bar{g}, \, {\bf K}}. \eqno(] 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 ([link] holds only for centrosymmetric crystals. This condition ([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). In-disc symmetries

| top | pdf |

  • (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.[link], and given in operational detail in Table[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.[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| 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[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. Zero-layer absences

| top | pdf |

Horizontal glides, a′, n′ (diperiodic, primed notation), generate zero-layer absent rows, or centring, rather than GS bands (see Fig.[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[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.[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[link].


Figure | 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.


First citation Bilhorn, D. E., Foldy, L. L., Thaler, R. M. & Tobacman, W. (1964). Remarks concerning reciprocity in quantum mechanics. J. Math. Phys. 5, 435–441.Google Scholar
First citation Buxton, B., Eades, J. A., Steeds, J. W. & Rackham, G. M. (1976). The symmetry of electron diffraction zone axis patterns. Philos. Trans. R. Soc. London Ser. A, 181, 171–193.Google Scholar
First citation Cowley, J. M. (1969). Image contrast in transmission scanning electron microscopy. Appl. Phys. Lett. 15, 58–59.Google Scholar
First citation Goodman, P. (1984a). A matrix basis for CBED pattern analysis. Acta Cryst. A40, 522–526.Google Scholar
First citation Gunning, J. & Goodman, P. (1992). Reciprocity in electron diffraction. Acta Cryst. A48, 591–595.Google Scholar
First citation Laue, M. von (1935). Die Fluoreszenzrontgenstrahlung von Einkristallen. Ann. Phys. (Leipzig), 23, 703–726.Google Scholar
First citation Moodie, A. F. (1972). Reciprocity and shape function in multiple scattering diagrams. Z. Naturforsch. Teil A, 27, 437–440.Google Scholar
First citation Pogany, A. P. & Turner, P. S. (1968). Reciprocity in electron diffraction and microscopy. Acta Cryst. A24, 103–109.Google Scholar
First citation Schwartzman, A., Goodman, P. & Johnson, A. W. S. (1996). IUCr XVII Congress and General Assembly, Seattle, Washington, USA, August 8–16, Collected Abstracts, p. C-54, Abstract PS02.03.18.Google Scholar

to end of page
to top of page