In-disc symmetries

Goodman, P.

doi:10.1107/97809553602060000558

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 287-288 | 1 | 2 |

Section 2.5.3.2.3. In-disc symmetries

P. Goodman^b ^‡

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, and given in operational detail in Table 2.5.3.2. The notation subscript R, for reciprocity-induced symmetries, introduced by Buxton et al. (1976) is now adopted (and referred to as BESR notation). Fig. 2.5.3.2 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 `+'

Type	Symmetry element	Observation and action	In combination	Interpretation
Vertical	4
Vertical	m; a
Horizontal	2′; $[2'_{1}]$
	$[i(\bar{1}')]$
	m′; a′

(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 and II) discussed above apply (see Table 2.5.3.3).

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.

References

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

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 287-288