Space Group
- Practical Electron Microscopy and Database -
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The full information on space groups is given in the International Tables for Crystallography. The basic knowledge of the space groups on the EM online book (see link) is your good start since it is necessary to know the general principles of the space-group designations in order to use the crystal-structure databases correctly. A table (see link) gives the list of the known 230 three-periodic space groups and the space groups that have not been completely determined. The 230 space groups were proposed in the late 19th century independently by Fedorov[1], Barlow[2] and Schoenflies[3].

Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be self-consistently arranged in space. There are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Figure 4547 schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. By combining the 32 point groups with the translation operations of 14 Bravais lattices, 73 symmorphic space groups are obtained. The other 157 (=230-73) space groups have point-symmetry operations with improper (partial) translations, i.e. rotations through screw axes and reflections (mirrors) in glide planes.

The relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups

Figure 4547. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.

Table 4547 lists the relation between three-dimensional crystal families, crystal systems, and lattice systems.

Table 4547. The relation between three-dimensional crystal families, crystal systems, and lattice systems.

Crystal
family
Crystal
system
Required
symmetries
of point group
Point
group
Space
group
Bravais
lattices
Lattice
system
None 2 2 1 Triclinic
1 two-fold axis of rotation or 1 mirror plane 3
13
2
Monoclinic

Orthorhombic

3 two-fold axes of rotation or 1 two-fold axis of rotation and two mirror planes 3 59 4
Orthorhombic

Tetragonal

1 four-fold axis of rotation

7

68

2

Tetragonal
Hexagonal
Trigonal 1 three-fold axis of rotation 5 7 1 Rhombohedral
18 1 Hexagonal
Hexagonal 1 six-fold axis of rotation 7 27
4 three-fold axes of rotation 5 36 3 Cubic
Total: 6 7   32 230 14 7

XRD obeys Friedel's Law because it can be approximately described by kinematical scattering. This limits the number of the space groups of crystals that can be determined by XRD to fifty. For electron diffraction, multiple (dynamical) scattering of the incident electrons in the TEM specimen occurs so that the Friedel's law is broken down, and thus the crystals with all the 230 space groups can be identified.

Note that the d-spacing and the intensities of the diffracted spots can be different for different crystals even though their space group are the same, for instance, as discussed in page3013 for space group Pnma (62).

 

 

 

 

 

[1] E.S. Fedorov (1891) ”Симмтрія правильныхъ системъ фигуръ” (“The symmetry of regular systems of figures”), Zapiski Imperatorskogo S. Petersburgskogo Mineralogichesgo Obshchestva (Proceedings of the Imperial St. Petersburg Mineralogical Society), 2(28), pp 1-146. English translation: David and Katherine Harker, (1971) “Symmetry of Crystals”, American Crystallographic Association Monograph No. 7, Buffalo, N.Y. Am. Cryst. Ass. pp 50-131.
[2] W. Barlow (1894) “Über die Geometrischen Eigenschaften homogener starrer Strukturen und ihre Anwendung auf Krystalle” (“On the geometrical properties of homogeneous rigid structures and their application to crystals”), Zeitschrift für Krystallographie und Minerologie, 23, pp 1-63.
[3] A. Schoenflies (1892) Krystallsysteme und Krystallstruktur.

 

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