Quasi-periodic structures crystals

Apart from the simulation of ideal surfaces, increasing interest in real 2-D crystals now exists, which are quasi-periodic structures in two dimensions but only a few atomic layers thick, and which may present new and useful properties precisely because of their limited thickness. This branch of nanoscience is then an ideal ground for application of the slab model. 

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 may be mentioned that in 2D and 3D the possible rotations (the symmetry axes) that superimpose an infinitely periodic structure on itself are limited to angles 360°/n with n = 1, 2, 3, 4 or 6. Notice that for non-periodic, noncrystalline, quasi-crystalline structures, other symmetry axes are possible. See 3.11.3 and Fig. 3.45 on quasi-periodic crystals. 

In this section we show that photo-induced light scattering is a powerful tool, especially to obtain information about the polar structure in the crystal bulk. The polar structure in SBN Ce can be considered as the composition of different periodical and/or quasi-periodical assemblies of ferroelectric 180°-domains distributed in the bulk aligned along the c-axis. The existence of... 

The intensity variation along the rod (i.e. as a function of or /) is solely contained in the structure factor it is thus related to the z-co-ordinates of the atoms within the unit-cell of this quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements. This is the case of a reconstructed surface, for which rods are found for fractional order values of h and k, i.e. outside scattering from the bulk. 

The action of compression plasma flows (CPF), generated by quasi-stationary plasma accelerators, upon solid surfaces leads to a substantial modification of surface properties of exposed materials [1-3]. It was found that exposure of silicon crystals to CPF causes formation of sub-micron bulk periodic structures on its surface. These structures are of great interest for development of nanoelectronic devices. 

These are crystal structures which have perfect long-range order but which are only approximately periodic, incommensurate crystals on the one hand, and quasi-crystals on the other. 

Quasi-crystals are currently of great scientific interest. Nonetheless, the vast majority of all solid bodies are made up of crystals which possess periodic structures. For this reason, the following chapters will deal exclusively with three-dimensional crystallography. 

Exact, perfect order of a periodic or quasi-periodic crystal is never obtained by a real atomic arrangement. All real crystals are more or less disordered. We describe the disorder by the term structural defect with respect to the idealized periodic structure. Many crystal properties (e.g. electrical conductivity and mechanical properties) are strongly dependent on the defect structure. 

The host system is treated as a perfect crystalline structure, and the exploitation of periodicity or quasi-periodicity is an essential ingredient when treating the defect as an impurity. From a quantum-mechanical point of view, the defect is treated as a perturbation to the electronic structure of the perfect crystal environment. 

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. 

It was therefore suggested that there is a possible aggregation of the domains into superdomains , similarly with the growth of spherulites in crystalline polymers, inducing formation of lamellar and fibrillar crystals (a model proposed by Kabanov [694]). Such a quasi-crystalline structure may have a characteristic period of ordering of 1 pm. 

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. 

While the period of the interference curve does not depend on the properties of a metal surface, the magnitude of effect, that is the amplitude of the interference curve, can reveal a strong dependence of such a kind. Indeed, Kadomtsev s theory is based on the assumption, that the atom interacts with quasi-free electrons in the thin surface layer. Therefore, the state of such electrons must be tightly connected with the properties of such a layer-for instance its temperature and crystal structure. 

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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