Aperiodic structures

The determination of the atomic structure of surfaces is the cornerstone of surface science. Before the invention of STM, various diffraction methods are applied, such as low-energy electron diffraction (LEED) and atom beam scattering see Chapter 4. However, those methods can only provide the Fourier-transformed information of the atomic structure averaged over a relatively large area. Often, after a surface structure is observed by diffraction methods, conflicting models were proposed by different authors. Sometimes, a consensus can be reached. In many cases, controversy remains. Besides, the diffraction method can only provide information about structures of relatively simple and perfectly periodic surfaces. Large and complex structures are out of the reach of diffraction methods. On real surfaces, aperiodic structures such as defects and local variations always exist. Before the invention of the STM, there was no way to determine those aperiodic structures. 

In the STM study of the reconstructed Au(lll) surface, in addition to the confirmation of the model suggested from scattering experiments with unambiguous determination of the positions of atoms, aperiodic structures are also observed. 

Progress in nanotechnology also depends critically on new developments in microscopy [42-45]. Compared to other investigation methods that help to explore the relation between the molecular structure and macroscopic properties, microscopy gives the most direct information. Particularly, in the case of disordered or aperiodic structures, visualisation of the structure is often more useful than indirect measurement and interpretation of its scattering properties. In practice, the utilisation and value of microscopes depends on their spatial resolution, the contrast and the imaging conditions. 

Another approach to liquid glass transition is the self-consistent phonon theory or density functional theory applied to aperiodic structures [112-114]. These theories predict the Lindemann stability criterion for the emergence of a density wave of a given symmetry. Although the finite Lindemann ratio implies a first-order phase transition, the absence of latent heat in glassy systems suggests the presence of an exponentially large number of aperiodic structures that are frozen in at Tg [94,95,110,111],... 

The perturbation (modulation) function used in the description of aperiodic structures is obtained by associating interatomic distances (or larger fragments in the crystal structure) with length ratio x to 1 to letters L... 

For the case of dumbbells formed from tangent spheres, L = ct, it is also possible to pack the dumbbells in orientationally disordered structures in which the spheres of the dumbbell lie on an fee lattice but the centers of mass form an aperiodic structure. This idea has been explored more extensively for two-dimensional dumbbells by Wojciechowski et al. [241,242]. The configurational degeneracy of the aperiodic structure renders it more stable than the orientationally ordered structures at all densities, even though the effect upon the equation of state and the free energy without the contribution from the degeneracy is quite small [60]. The freezing of hard dumbbells into such structures has also been studied by Bowles and Speedy [243],... 

The nucleotide frequencies in the second codon positions of genes are remarkably different for the coding regions that correspond to different secondary structures in the encoded proteins, namely, helix, /3-strand ami aperiodic structures (Fig. 10.19). Indeed, hydropho-... 

Based on this hypothesis, further details can be discussed, such as the clear separation of aperiodic structure and //-strand structure, shown by the second codon position analysis. 

The summation is over all the p atoms in the unit cell vvdth atomic scattering factor The modulation describes the individual atomic displacement relative to its basic position r. The integration variable t is related to the fourth dimension by the expression X4 = q. r -1- r (see Fig. 2). The first line of this expression corresponds to the structure factor of a conventional crystal (/i4 is equal to zero, and thus H = h). The second line of the expression is specific to aperiodic structures and contains the information related to the atomic modulation function in superspace. In addition to a displacement parameter, the modulation can also be affected by a variable occupation parameter p. ... 

Shown in Fig. 5 is a selection of electron-density maps related to N13 and CIO, with the vertical axis representing the V4 coordinate. We observe that the modulation function is specific for each atom. For N13, the atomic modulation function is continuous along the X4 axis with extrema lying approximately 1.8 A apart in the b direction. This deviation indicates that the departure from the average structure is particularly important throughout the crystal structure. The modulation of CIO describes another aspect of the diversity of aperiodic structures. In the crystal, this atom exhibits two stable positions a and b which are best represented in superspace by slightly modulated crenel functions. Thus, the domain of existence of ClOa and Cl Ob is colnplementary and mutually exclusive. 

The discussion so far has focussed on the calculation of valences (and bond lengths) in periodic crystals, but the bond valence method is equally plicable to aperiodic structures and is potentially very useful for predicting relaxations around defects in crystals (Brown, 1988) and at surfaces and interfaces (O Keeffe, 1991b). 

Theoretical assessments of DNA-based computation by self-assembly indicate computation power far beyond that which has yet been experimentally implemented. Still, it remains to be seen if DNA computers will ever find a computational niche in which they can efficiently compete with electronic computers. One realm in which algorithmic DNA self-assembly appears to provide very great promise is in the nanofabrication of specific aperiodic structures for templating of nanoelectronics devices. Such nanopatterned materials could be used not only for communications and computational devices but also for sensors, biosensors, medical diagnostics, and molecular robotics applications. 

Along with symmetrical structures or the periodic structures, there is as a rule the existence of asymmetric or aperiodic structures. It has also been introduced that materials present in their asymmetric structural existence show drastically different characteristics, which are not found to exist in their periodic structural states of existence. This asymmetries, which are present in the aperiodic structures of materials and which are incommensurate with the classical structures, leads to the revelation of many more fantastic properties, which are hitherto not found in perfect structural existence of the same materials. 

An Outline of the Diffraction Theory of Periodic and Aperiodic Structures... 

The detailed theory of X-ray diffraction from periodic and aperiodic structures is, however, out of the scope of this book. However, a brief introduction is given here. The readers are requested to find the detailed analysis from [1-4] and also they may consult the references given in Further Reading. ... 

Though the outline of this book is designed to serve those having only the basic introduction to mathematics, a brief introduction to the diffraction theory of perfect periodic to aperiodic structures is given in Appendix A. Some solved problems are also given in Appendix B to help students. 

Whether the bubbles are spherical, polyhedral, or in between, they typically have a distribution of sizes and pack together into a disordered, aperiodic structure. In Fig. 18.1 the average bubble diameter is 2 mm, but similar structures are also found in foams where the average bubble diameter is varied from 10 p,m to 1 cm. In practice, the average bubble size and shape in a foam can be altered for a given liquid according to the production method, the surface-active ingredients, and other chemical additives such as viscosity modifiers or polymeric stabilizers. 

It is clear that real atomic structures are always manifestations of matter in 3-D real, physical space. The cutting of the 2-D hyperspace to obtain real 1-D atoms illustrated in Fig. 1.3-8 may serve as an instructive basic example of the concept of higher-dimensional (n-D, > 3) crystallography. The concept is also called a superspace description it applies to all aperiodic structures and provides a convenient finite set of variables that can be used to compute the positions of aU atoms in the real 3-D structure. 

Aperiodic structures of this type are also seen in the florets of daisies, sunflowers etc., wit grain boundaries forming one-dimensional quasicrystals. ... 

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