Quasi-periodic crystals

According to Yamamoto (1996) quasi-periodic structures belong to the following 

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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. 

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. 

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... 

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. 

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. 

We have developed in previous work a practical scheme for calculating the self-energy corrections and quasi-particle states in periodic crystals. One main iijgredient of this scheme is the short-ran e property of E in r-r [, which has been proven by Sham and Kohn on the basis of general considerations of many-body perturbation theory. As a consequence, S(r,r E) will have matrix elements in a set of local orbitals (Wannier, LCAO s, muffin-tin etc.) appreciably different from zero only about the diagonal, r = r. ... 

H. Da, C. Xu, Z. Li, Omnidirectional reflection fiom one-dimensional quasi-periodic photonic crystal containing left-handed material. Phys. Lett. A 345(4), 459-468 (2005)... 

Figure 10 Time evolution of heat capacity during quasi-isothermal crystallization of PHBat 296K, Ai=OAK, fp = 100s (curve (a)). Curves (b)and (c) correspond to solid and liquid heat capacities available from the ATHAS-DB, respectively. Curve (d) was estimated from a two-phase model and curve (e) from a three-phase model by using /soiid(Tg) from ACp (see Reference 85 for details). The subscript solid denotes the solid fraction of the polymer consisting of crystalline and glassy fractions. The subscript CRF denotes the crystalline fraction alone. The squares represent measurements at modulation periods ranging from 240 to 1200 s. Curve (f) shows the exothermal effect in the total heat flow, and curves (g) and (h) show the expected values from model calculations (see text and Reference 85). Data from PetIdnElmer Pyris I DSC. Reproduced with permission from Schick, C. Wurm, A. Mohammed, A. Thermochim. ActaZOOS, 396,119-132. ...

Where b is Planck s constant and m and are the effective masses of the electron and hole which may be larger or smaller than the rest mass of the electron. The effective mass reflects the strength of the interaction between the electron or hole and the periodic lattice and potentials within the crystal stmcture. In an ideal covalent semiconductor, electrons in the conduction band and holes in the valence band may be considered as quasi-free particles. The carriers have high drift mobilities in the range of 10 to 10 cm /(V-s) at room temperature. As shown in Table 4, this is the case for both metallic oxides and covalent semiconductors at room temperature. 

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