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

International Tables for Crystallography (2006). Vol. A, ch. 14.2, p. 848

Section 14.2.2.1. The degrees of freedom

W. Fischera and E. Kocha*

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:  kochelke@mailer.uni-marburg.de

14.2.2.1. The degrees of freedom

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The number of coordinate parameters that can be varied independently within a Wyckoff position is called its number of degrees of freedom. For most lattice complexes, the number of degrees of freedom is the same as for any of its Wyckoff positions. The lattice complex with characteristic Wyckoff position [Pm\bar{3}\ 12j\ m..\ 0yz], for instance, has two degrees of freedom. If, however, the variation of a coordinate corresponds to a shift of the point configuration as a whole, one degree of freedom is lost. Therefore, [I4_{1}\ 8b\ xyz] is the characteristic Wyckoff position of a lattice complex with only two degrees of freedom, although position 8b itself has three degrees of freedom. Another example is given by [P4/m\ 4j\ m..\ xy0] and P4 4d 1 xyz. Both Wyckoff positions belong to lattice complex [P4/m\ j] with two degrees of freedom.

According to its number of degrees of freedom, a lattice complex is called invariant, univariant, bivariant or trivariant. In total, there exist 402 lattice complexes, 36 of which are invariant, 106 univariant, 105 bivariant and 155 trivariant. The 30 plane lattice complexes are made up of 7 invariant, 10 univariant and 13 bivariant ones.

Most of the invariant and univariant lattice complexes correspond to several types of Wyckoff set. In contrast to that, only one type of Wyckoff set belongs to each trivariant lattice complex. A bivariant lattice complex may either correspond to one type of Wyckoff set (e.g. [Pm\bar{3}\ j]) or to two types (P4 d, for example, belongs to the lattice complex with the characteristic position [P4/m\ j]).








































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