Reflection conditions and diffraction symbol

Looijenga-Vos, A.; Buerger, M. J.

doi:10.1107/97809553602060000506

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 3.1, pp. 44-45

Section 3.1.3. Reflection conditions and diffraction symbol

A. Looijenga-Vos^a ^‡ and M. J. Buerger^b ^§

^a Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands, and ^bDepartment of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

3.1.3. Reflection conditions and diffraction symbol

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In Section 2.2.13 , it has been shown that `extinctions' (sets of reflections that are systematically absent) point to the presence of a centred cell or the presence of symmetry elements with glide or screw components. Reflection conditions and Laue class together are expressed by the Diffraction symbol, introduced by Buerger (1935, 1942, 1969); it consists of the Laue-class symbol, followed by the extinction symbol representing the observed reflection conditions. Donnay & Harker (1940) have used the concept of extinctions under the name of `morphological aspect' (or aspect for short) in their studies of crystal habit (cf. Crystal Data, 1972). Although the concept of aspect applies to diffraction as well as to morphology (Donnay & Kennard, 1964), for the present tables the expression `extinction symbol' has been chosen because of the morphological connotation of the word aspect.

The Extinction symbols are arranged as follows. First, a capital letter is given representing the centring type of the cell (Section 1.2.1 ). Thereafter, the reflection conditions for the successive symmetry directions are symbolized. Symmetry directions not having reflection conditions are represented by a dash. A symmetry direction with reflection conditions is represented by the symbol for the corresponding glide plane and/or screw axis. The symbols applied are the same as those used in the Hermann–Mauguin space-group symbols (Section 1.3.1 ). If a symmetry direction has more than one kind of glide plane, for the diffraction symbol the same letter is used as in the corresponding space-group symbol. An exception is made for some centred orthorhombic space groups where two glide-plane symbols are given (between parentheses) for one of the symmetry directions, in order to stress the relation between the diffraction symbol and the symbols of the `possible space groups'. For the various orthorhombic settings , treated in Table 3.1.4.1, the top lines of the two-line space-group symbols in Table 4.3.2.1 are used. In the monoclinic system, dummy numbers `1' are inserted for two directions even though they are not symmetry directions, to bring out the differences between the diffraction symbols for the b, c and a settings.

Example

Laue class: [12/m1]

Reflection conditions: $[\matrix{hkl: h + k = 2n{\hbox{;}} & &\cr h0l: h, l = 2n{\hbox{;}} \hfill &0kl: k = 2n{\hbox{;}} &hk0: h + k = 2n{\hbox{;}}\cr h00: h = 2n{\hbox{;}} \hfill &0k0: k = 2n{\hbox{;}} &00l: l = 2n. \hfill \cr}]$ As there are both c and n glide planes perpendicular to b, the diffraction symbol may be given as $[1{\hbox to 2pt{}}2/m{\hbox to 2pt{}}1C1c1]$ or as $[1{\hbox to 2pt{}}2/m{\hbox to 2pt{}}1C1n1]$ . In analogy to the symbols of the possible space groups, C1c1 (9) and $[C1{\hbox to 2pt{}}2/c{\hbox to 2pt{}}1\;(15)]$ , the diffraction symbol is called $[1{\hbox to 2pt{}}2/m{\hbox to 2pt{}}1C1c1]$ .

For another cell choice, the reflection conditions are: $[\matrix{hkl: k + l = 2n{\hbox{;}} & &\cr h0l: h, l = 2n{\hbox{;}} \hfill &0kl: k + l = 2n{\hbox{;}} &hk0: k = 2n{\hbox{;}}\cr h00: h = 2n{\hbox{;}} \hfill &0k0: k = 2n{\hbox{;}} \hfill &00l: l = 2n.\cr}]$ For this second cell choice, the glide planes perpendicular to b are n and a. The diffraction symbol is given as $[1\;2/m\;1A1n1]$ , in analogy to the symbols A1n1 (9) and $[A1\;2/n\;1\;(15)]$ adopted for the possible space groups.

References

Crystal Data (1972). Vol. I, General Editors J. D. H. Donnay & H. M. Ondik, Supplement II, pp. S41–52. Washington: National Bureau of Standards.Google Scholar

Buerger, M. J. (1935). The application of plane groups to the interpretation of Weissenberg photographs. Z. Kristallogr. 91, 255–289.Google Scholar

Buerger, M. J. (1942). X-ray crystallography, Chap. 22. New York: Wiley.Google Scholar

Buerger, M. J. (1969). Diffraction symbols. Chap. 3 of Physics of the solid state, edited by S. Balakrishna, pp. 27–42. London: Academic Press.Google Scholar

Donnay, J. D. H. & Harker, D. (1940). Nouvelles tables d'extinctions pour les 230 groupes de recouvrements cristallographiques. Nat. Can. 67, 33–69, 160.Google Scholar

Donnay, J. D. H. & Kennard, O. (1964). Diffraction symbols. Acta Cryst. 17, 1337–1340.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 3.1, pp. 44-45