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Towards a more comprehensive crystallography

by A. Janner
Acta Crystallographica Section B ()
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Despite the intrinsic difference from the point of view of\nstructure of various types of crystals, such as\ncommensurate, incommensurately modulated, intergrowth\ncrystals and quasicrystals, a common approach to their\nsymmetry seems to be possible which eventually will lead to\na more comprehensive crystallography. The unifying elements\nbecome apparent through treatments which, at first, seem to\nbe contradictory with the geometry of the crystal\nstructures involved. Examples are the description of\naperiodic crystals in terms of lattice-periodic structures\n(going beyond three-dimensionality), the investigation of\nscaling symmetry in quasicrystals by means of a Z-module of\ntranslations generating a dense set of translationally\nequivalent atomic positions (going beyond discreteness) and\nfinally the characterization of Euclidean properties of\nnormal crystals through non-Euclidean symmetries (going\nbeyond Euclidean metric). These changing approaches do not\nmodify, however, the fundamental nature of crystals to be\nthree-dimensional, discrete and Euclidean. They only allow\nimplicit symmetry groups like the superspace groups\n(unifying the crystallography of incommensurate and\ncommensurate crystals) and the multimetrical space groups\n(unifying the possible symmetries of quasicrystals and\nnormal crystals) to be made explicit. Aspects of crystal\ndiffraction, morphology and crystal structure are presented\nfrom this unifying point of view, without intending to\ncover the whole crystallography.

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