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By Ted Janssen, Gervais Chapuis, Marc de Boissieu

Till the Seventies all fabrics studied consisted of periodic arrays of unit cells, or have been amorphous. within the final a long time a brand new category of strong kingdom topic, known as aperiodic crystals, has been stumbled on. it's a lengthy diversity ordered constitution, yet with out lattice periodicity. it truly is present in a variety of fabrics: natural and anorganic compounds, minerals (including a considerable element of the earths crust), and steel alloys, below quite a few pressures and temperatures. due to the loss of periodicity the standard recommendations for the examine of constitution and actual houses now not paintings, and new suggestions need to be constructed. This booklet bargains with the characterisation of the constitution, the constitution selection and the learn of the actual homes, specifically dynamical and digital homes of aperiodic crystals. The remedy is predicated on an outline in an area with extra dimensions than 3, the so-called superspace. this permits us to generalise the normal crystallography and to appear in a different way on the dynamics. the 3 major periods of aperiodic crystals, modulated stages, incommensurate composites and quasicrystals are handled from a unified standpoint, which stresses similarities of some of the platforms. The booklet assumes as a prerequisite a data of the basic recommendations of crystallography and the speculation of condensed subject, and covers the literature on the leading edge of the sphere.

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Note that the triacontahedron can be obtained as the union of an icosahedron and a dodecahedron. These clusters with icosahedral symmetry are quasiperiodically packed. INTRODUCTION 28 Fig. 15. Top: The Mackay cluster: an icosahedron in the centre of a larger cell with 42 atoms on the vertices and mid-edges of an icosahedron. Bottom: The Bergman cluster: an icosahedron inside a rhombic triacontahedron. 6 Morphology Just as lattice periodic crystals show flat facets, with an orientation connected to the reciprocal lattice, quasiperiodic crystals have facets connected to the Fourier module.

If the tiles are decorated by putting 'atoms' in each tile, this is an example of a perfectly ordered quasi-periodic structure (Fig. 14). Soon after the discovery of quasicrystals their relation with the Penrose tiling was seen. A first type of model was a three-dimensional tiling of the space by means of rhombohedra. Using two types of rhombohedra, a three-dimensional analogue of the Penrose tiling can be constructed. The first attempts were made using a decoration of such a tiling. These structure models were made simpler by the fact that some of the quasicrystals had related compounds with slightly different composition and with locally similar structure.

With respect to this basis, a five-fold and a three-fold rotation are given by the matrices QUASIPERIODIC CRYSTALS 25 These two matrices generate, together with the total inversion (the opposite of the identity), a point group with 120 elements, the symmetry group of an icosahedron (or a dodecahedron). This group contains 60 rotations: 6 five- fold axes (giving 24 rotations), 10 three-fold axes (for 20 rotations), 15 two-fold axes (for 15 rotations), and the identity. Fig. 13. ,6) and their inverses form the vertices of an icosahedron.

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