Quasi-periodic structures quasicrystals

Notes on the crystallography of quasi-periodic structures. A general way to face the problems related to the interpretation of quasi-periodic structures (modulated structures, quasicrystals) is based on the introduction and application of higher-dimensional crystallography (de Wolff 1974, 1977, Janner and Janssen 1980, Yamamoto 1982, 1996, Steurer 1995). 

In this chapter, general aspects and structural properties of crystalline solid phases are described, and a short introduction is given to modulated and quasicrystal structures (quasi-periodic crystals). Elements of structure systematics with the description of a number of structure types are presented in the subsequent Chapter 7. Finally, both in this chapter and in Chapter 6, dedicated to preparation techniques, characteristic features of typical metastable phases are considered with attention to amorphous and glassy alloys. 

It was reported recently, that polymeric can also form quasicrystals. Hayashida et al. [50] demonstrated that certain blends of polyisoprene, polystyrene, and poly(2-vinylpyridine) form starshaped copolymers that assemble into quaskrystals. By probing the samples with transmission electron microscopy and X-ray diffraction methods, they conclude that the films are composed of periodic patterns of triangles and squares that exhibit 12-fold symmetry. These are signs of quasicrystalline ordering. Such ordering differ from conventional crystals lack of periodic structures yet are well-ordered, as indicated by the sharp diffraction patterns they generate. Quasi-crystals also differ from ordinary crystals in another fundamental way. They exhibit rotational symmetries (often five or tenfold). There are still some basic questions about their stracture. 

Another characteristic point is the special attention that in intermetallic science, as in several fields of chemistry, needs to be dedicated to the structural aspects and to the description of the phases. The structure of intermetallic alloys in their different states, liquid, amorphous (glassy), quasi-crystalline and fully, three-dimensionally (3D) periodic crystalline are closely related to the different properties shown by these substances. Two chapters are therefore dedicated to selected aspects of intermetallic structural chemistry. Particular attention is dedicated to the solid state, in which a very large variety of properties and structures can be found. Solid intermetallic phases, generally non-molecular by nature, are characterized by their 3D crystal (or quasicrystal) structure. A great many crystal structures (often complex or very complex) have been elucidated, and intermetallic crystallochemistry is a fundamental topic of reference. A great number of papers have been published containing results obtained by powder and single crystal X-ray diffractometry and by neutron and electron diffraction methods. A characteristic nomenclature and several symbols and representations have been developed for the description, classification and identification of these phases. 

Quasi-crystals have macroscopic symmetries which are incompatible with a crystal lattice (Section 2.4.1). The first example was discovered in 1984 when the alloy AlMn is rapidly quenched, it forms quasi-crystals of icosahedral symmetry (Section 2.5.6). It is generally accepted that the structure of quasicrystals is derived from aperiodic space filling by several types of unit cell rather than one unique cell. In two-dimensional space, the best-known example is that of Penrose tiling. It is made up of two types of rhombus and has fivefold symmetry. We assume that the icosahedral structure of AlMn is derived from a three-dimensional stacking analogous to Penrose tiling. As is the case for incommensurate crystals, quasi-crystals can be described by perfectly periodic lattices in spaces of dimension higher than three in the case of AlMn, we require six-dimensional space. 

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