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IntermetallicsStructures, Properties, and Statistics$
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Walter Steurer and Julia Dshemuchadse

Print publication date: 2016

Print ISBN-13: 9780198714552

Published to Oxford Scholarship Online: September 2016

DOI: 10.1093/acprof:oso/9780198714552.001.0001

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Higher-dimensional approach

Higher-dimensional approach

Chapter:
(p.68) 4 Higher-dimensional approach
Source:
Intermetallics
Author(s):

Walter Steurer

Julia Dshemuchadse

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198714552.003.0004

To understand the structural ordering of aperiodic crystals such as incommensurately modulated phases, composite crystals, quasicrystals and their structurally closely related rational approximants, the higher-dimensional (nD) approach is essential. In a properly selected nD space, aperiodic crystal structures can be described as 3D sections/projections of nD periodic “hypercrystal” structures. Furthermore, “non-crystallographic” symmetries of quasicrystals can become compatible with nD lattice symmetry. For instance, 5-, 8-, 10-, and 12-fold rotational symmetries are proper symmetry operations in 5D and icosahedral point group symmetries in 6D hypercubic lattices. First the so-called “strip-projection” (or “cut-and-project”) method is introduced, because it allows a more intuitive understanding of when and how clusters order quasiperiodically. Then the nD section method is presented, which is the method of choice for nD structure analysis, because it can make use of the reciprocal space information experimentally accessible by diffraction methods.

Keywords:   nD approach, higher-dimensional approach, strip-projection, method, cut-and-project method, nD section method, nD structure analysis, Fibonacci sequence, 2D Penrose tiling, 3D Ammann tiling

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