Wyckoff Positions  Practical Electron Microscopy and Database   An Online Book  

Microanalysis  EM Book http://www.globalsino.com/EM/  


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Except few space groups, there are always several different equivalent sets of atomic coordinates for exactly the same crystal structure with an unchanged setting of the space group. A procedure of obtaining another equivalent coordinate set from one is described in Section 15.3 of International Tables, Volume A. [1] An infinite number of crystallographic orbits for a given space group E can be subdivided into sets of socalled Wyckoff positions of E. Wyckoff positions tell where the atoms in a crystal we can find. All the crystallographic orbits that have the same sitesymmetry group belong to the same Wyckoff position. In other words, if the corresponding coordinates are completely fixed by the symmetry, the orbit is then identical with the Wyckoff position. However, if one or more coordinates are variable, the Wyckoff position comprises many orbits. A Wyckoff position is designated by the Wyckoff label, for instance, 8a. The '8' is the multiplicity, indicating how many points belonging to an orbit of the Wyckoff position are included in an unit cell. The 'a' is an alphabetical label, i.e. small Roman letter, (a, b, c, ...), representing different Wyckoff positions. [1] The maximum number of different Wyckoff positions in the 230 space groups is 27 (in Group (47) Pmmm). The various possible sets of Wyckoff positions for all the space groups are given in the International Tables for Crystallography [1] and an Internet site (you can find it at page1474), for instance, Wyckoff positions for space group P4_{2}/mnm (136).
[1] International Tables for Crystallography. Volume A: Spacegroup symmetry (Th. Hahn, ed.). Kluwer, Dordrecht, (editions of 1983, 1987, 1992, 1995, 2002, 2005)


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