Brillouin-zone databases

The k-vector layer-group data of the Brillouin-zone database of the Bilbao Crystallographic Server are accessed by the retrieval tool LKVEC which uses as input the sequential layer-group number as given in International Tables for Crystallography, Vol. E (abbreviated as ITE)[1]. The output consists essentially of wave-vector tables and figures. There are several sets of figures and tables for the same layer group when its Brillouin-zone shape depends on the lattice parameters of the reciprocal lattice. The k-vector data are the same for layer groups of the same arithmetic crystal class.

In the k-vector tables, the wave-vector data of Litvin & Wike, 1991 (referred to as L&W) [2] are compared with the Wyckoff-position data of International Tables for Crystallography, Vol. A (referred to as ITA) [3]. Each k-vector type is specified by its label and coefficients as listed in L&W. The corresponding Wyckoff positions are described by Wyckoff letters, multiplicities and site-symmetry groups. The parameter descriptions specify the independent parameter ranges chosen in such a way that each orbit of the Wyckoff positions of ITE, i. e. each k-vector orbit is also listed exactly once. In the figures, the Brillouin zones of L&W and the conventional unit cells of ITE are displayed. The asymmetric units play the role of the representation domains of the Brillouin zones and they are chosen often in analogy to those of ITE.

[1] International Tables for Crystallography, Volume E: Subperiodic groups (2010). Edited by V. Kopsky & D. Litvin, 2nd edition. Chichester: Wiley. [Abbreviated as ITE].

[2] Litvin, D. B. & Wike, T. R. (1991). Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press. [Abbreviated as L&W].

[3] International Tables for Crystallography, Volume A: Space-Group Symmetry (2016). Edited by M. I. Aroyo, 6th edition. Chichester: Wiley. [Abbreviated as ITA].

Bilbao Crystallographic Server
www.cryst.ehu.es

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