Periodic systems, topological structure

Drawing of Toroidal and Planar Graphs 6.4.1 Topological Structure of Periodic Systems... 

Fig. 6.3 The topological structure for periodic systems and drawing with four bi-lobal eigenvectors of the Laplacian. The unit cell with perpendicular unit vectors ai and a2 which contains 60 vertices (a). Drawing the nanotorus of parameters (m,n,p,q) = (1,0,0,5) (b). Drawing the nanombe of parameters (m,n,p,q) = (1,—1,5,5) (c). The unit cell which was generated by four bi-lobal eigenvectors of the Laplacian (d)...

Figure 8.17. The topological information on the structure of a multiphase system that is related to one-dimensional projections / 1 (sj) in different directions. The demonstration shows two directions indicated by arrows and the related chords. From /jj ( ) the distributions of the chord segments between domain edges are retrieved. Long periods are indicated by broken lines...

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

In this chapter we will show that the tight binding ( ) description of the covalent bond is able to provide a simple and unifying explanation for the above structural trends and behaviour. We will see that the ideas already introduced in chapter 4 on the structures of small molecules may be taken over to these infinite bulk systems. In particular, we will find that the trends in structural stability across the periodic table or within the structure maps can be linked directly to the topology of the local atomic environment through the moments theorem of Ducastelle and Cyrot-Lackmann (1971). 

In the case of nonlinear lattice topology, electron hops mix different spin configurations and the corresponding eigenvalue problem becomes much more complicated. A complete analytical solution of this problem is known only for some special cases (e.g. for the linear chain with periodic boundary conditions [22]). In the absence of exact results a reliable way to describe the properties of nonlinear systems is to perform a numerical study of the electron structure for finite lattice clusters. 

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