Applications of Group Cohomology to the Classification of Fourier-space Quasicrystals
نویسندگان
چکیده
In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. Recently, the method has been applied to quasicrystals and modulated crystals. This paper phrases Bienenstock and Ewald’s definitions in terms of group cohomology. A Fourier quasicrystal is defined, along with its space group, without requiring that it come from a quasicrystal in real (direct) space. A certain cohomology group classifies the space groups associated to a given point group and quasilattice, and the dual homology group gives all gauge invariants. This duality is exploited to prove several results that were previously known only in special cases, including the classification of space groups for quasilattices of arbitrary rank in two dimensions. Extinctions in X-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of non-zero homology classes. PACS 2003: 02.10.Hh, 61.50.Ah, 61.44.Br, 61.44.Fw
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Applications of Group Cohomology to the Classification of Crystals and Quasicrystals
In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. Recently, the method has been applied to quasicrystals and modulated crystals. We interpret this method in terms of group cohomology. A certain cohomology group classifies the space groups, and the dual homology group gives all gauge invariants. We exploit this d...
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