DESCRIPTION AND SYMMETRY OF APERIODIC CRYSTALS
DESCRIPTION AND SYMMETRY OF APERIODIC CRYSTALS
This chapter discusses aperiodic crystals. The structure and symmetry of quasiperiodic crystals can be described by embedding them into a higher-dimensional space as lattice periodic structures. Their intersection with the physical space gives the real structure, shifting the physical space parallel to the original one gives another possible realization of the crystal with the same energy. The Fourier transform of the quasiperiodic structure, and its diffraction pattern, are projections of the corresponding quantities in higher dimensions. The symmetry groups of quasiperiodic structures are superspace groups, higher-dimensional space groups for which the point group can be decomposed into a component in physical space and one in the additional, internal space. The structure determination reduces to the determination of number and positions of atomic surfaces in the higher-dimensional unit cell, and that of the shape of the atomic surfaces.
Keywords: aperiodic crystals, quasiperiodic crystals, quasicrystals, symmetry, Fourier transformation
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