International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, p. 114

Section 1.6.4.1. Introduction

H. D. Flackb and U. Shmuelia

1.6.4.1. Introduction

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The primary order of presentation of these tables of reflection conditions of space groups is the Bravais lattice. This order has been chosen because cell reduction on unit-cell dimensions leads to the Bravais lattice as described as stage 1 in Section 1.6.2.1[link]. Within the space groups of a given Bravais lattice, the entries are arranged by Laue class, which may be obtained as described as stage 2 in Section 1.6.2.1[link]. As a consequence of these decisions about the way the tables are structured, in the hexagonal family one finds for the Bravais lattice hP that the Laue classes [\overline{3}], [\overline{3}m1], [\overline{3}1m], 6/m and 6/mmm are grouped together.

As an aid in the study of naturally occurring macromolecules and compounds made by enantioselective synthesis, the space groups of enantiomerically pure compounds (Sohncke space groups) are typeset in bold.

The tables show, on the left, sets of reflection conditions and, on the right, those space groups that are compatible with the given set of reflection conditions. The reflection conditions, e.g. h or k + l, are to be understood as [h=2n] or [k+l=2n], respectively. All of the space groups in each table correspond to the same Patterson symmetry, which is indicated in the table header. This makes for easy comparison with the entries for the individual space groups in Chapter 2.3[link] of this volume, in which the Patterson symmetry is also very clearly shown. All space groups with a conventional choice of unit cell are included in Tables 1.6.4.2–1.6.4.30. All alternative settings displayed in Chapter 2.3[link] are thus included. The following further alternative settings, not displayed in Chapter 2.3[link] , are also included: space group [Pb\overline{3}] (205) and all the space groups with an hR Bravais lattice in the reverse setting with hexagonal axes.

Table 1.6.2.1[link] gives some relevant statistics drawn from Tables 1.6.4.2–1.6.4.30. The total number of space-group settings mentioned in these tables is 416. This number is considerably larger than the 230 space-group types described in Part 2 of this volume. The following example shows why the tables include data for several descriptions of the space-group types. At the stage of space-group determination for a crystal in the crystal class mm2, it is not yet known whether the twofold rotation axis lies along a, b or c. Consequently, space groups based on the three point groups 2mm, m2m and mm2 need to be considered.

In some texts dealing with space-group determination, a `diffraction symbol' (sometimes also called an `extinction symbol') in the form of a Hermann–Mauguin space-group symbol is used as a shorthand code for the reflection conditions and Laue class. These symbols were introduced by Buerger (1935[link], 1942[link], 1969[link]) and a concise description is to be found in Looijenga-Vos & Buerger (2002)[link]. Nespolo et al. (2014)[link] use them.

References

Buerger, M. J. (1935). The application of plane groups to the interpretation of Weissenberg photographs. Z. Kristallogr. 91, 255–289.
Buerger, M. J. (1942). X-ray Crystallography, ch. 22. New York: Wiley.
Buerger, M. J. (1969). Diffraction symbols. In Physics of the Solid State, edited by S. Balakrishna, ch. 3, pp. 27–42. London: Academic Press.
Looijenga-Vos, A. & Buerger, M. J. (2002). Space-group determination and diffraction symbols. In International Tables for Crystallography, Volume A, Space-Group Symmetry, 5th ed., edited by Th. Hahn, Section 3.1.3. Dordrecht, Boston, London: Kluwer Academic Publishers.
Nespolo, M., Ferraris, G. & Souvignier, B. (2014). Effects of merohedric twinning on the diffraction pattern. Acta Cryst. A70, 106–125.








































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