Imperfect symmetry

Syntopy and syntopy groups were introduced in an early approach to a fuzzy set representation of approximate symmetry, where imperfect symmetry is regarded as fuzzy symmetry. Whereas any symmetry is a discrete property within a metric space, it is natural to consider a fuzzy set approach for a continuous extension of the discrete symmetry concept to quasisymmetric objects, such as some almost symmetric molecular structures. The syntopy approaches take into account the nonlocalized, quantum-mechanical, fuzzy nature of nuclear arrangements of molecules. 

Imperfect symmetry can be viewed in the broader context of molecular distortions. In this book the emphasis is on descriptions and measures based on topological arguments [54,55,106,108,158,240,243,247,248,345,444,445], or on the related concepts of syntopy [252,394,395] and symmorphy [43,108] discussed briefly in subsequent parts of this chapter. For a variety of alternative approaches and important additional insight, the reader may consult references [58,446-449]. 

Imperfect symmetry can be regarded as fuzzy symmetry. The theory of fuzzy sets has found applications in many fields of engineering and natural sciences (see, e.g., references [386-393]), in particular, for the description of fuzzy molecular arrangements [103,106,251]. It is natural to consider fuzzy sets for a continuous extension of the discrete point symmetry concept to quasi-symmetric molecular structures [252,394,395]. 

Considering the changes of E upon fi for different values of the parameters, Villain has concluded that imperfections in the surface structure may have a very strong influence on the behavior of incommensurate phases and on the C-IC transition. The usual lowering of symmetry during the C-IC transition does not occur here and the nature of the C-IC transition may be quite different from that on the surface free of defects. 

The two extremes of ordering in solids are perfect crystals with complete regularity and amorphous solids that have little symmetry. Most solid materials are crystalline but contain defects. Crystalline defects can profoundly alter the properties of a solid material, often in ways that have usefial applications. Doped semiconductors, described in Section 10-, are solids into which impurity defects are introduced deliberately in order to modify electrical conductivity. Gemstones are crystals containing impurities that give them their color. Sapphires and rubies are imperfect crystals of colorless AI2 O3, red. 

When the incident light is horizontally polarized, the horizontal Ox axis is an axis of symmetry for the fluorescence intensity Iy = Iz. The fluorescence observed in the direction of this axis (i.e. at 90° in a horizontal plane) should thus be unpolarized (Figure 5.3). This configuration is of practical interest in checking the possible residual polarization due to imperfect optical tuning. When a monochromator is used for observation, the polarization observed is due to the dependence of its transmission efficiency on the polarization of light. Then, measurement of the polarization with a horizontally polarized incident beam permits correction to get the true emission anisotropy (see Section 6.1.6). 

Palindromes are often imperfect as is the one shown in Fig. 5-34. Here the two stems in the cruciform structure are related by an exact twofold rotational symmetry but the loops at the ends of the stems are not. 

A number of different multiple pulse sequences (8-, 24- and 52-pulse sequences) have also been introduced in order to obtain better resolution or line narrowing, i.e. to affect the first- and second-order terms in the average Hamiltonian. Since pulse imperfections are the major source of resolution limitations, these composite pulse sequences are designed with corresponding symmetry properties which allows the canceling of specific rf pulse imperfections. 

An example of a symmetry operation is rotation of an object about an axis. To illustrate with a familiar object, if a rectangular table is rotated 180" about an axis perpendicular to and centered on the tabletop (Fig. 4.14), the table looks just the same as it did before rotation (ignoring imperfections such as coffee stains). We say that the table possesses a twofold rotation axis because, in rotating the table one full circle about this axis, we find two positions that are equivalent 0" and 180". The axis itself is an example of a symmetry element. 

Figure 4a. Imperfect bifurcation showing the breaking of the symmetry of the diagram of Figure 3a due to the imposition of an external electric field of strength...

This section has been devoted to the study of the surface excitons of the (001) face of the anthracene crystal, which behave as 2D perturbed excitons. They have been analyzed in reflectivity and transmission spectra, as well as in excitation spectra bf the first surface fluorescence. The theoretical study in Section III.A of a perfect isolated layer of dipoles explains one of the most important characteristics of the 2D surface excitons their abnormally strong radiative width of about 15 cm -1, corresponding to an emission power 10s to 106 times stronger than that of the isolated molecule. Also, the dominant excitonic coherence means that the intrinsic properties of the crystal can be used readily in the analysis of the spectroscopy of high-quality crystals any nonradiative phenomena of the crystal imperfections are residual or can be treated validly as perturbations. The main phenomena are accounted for by the excitons and phonons of the perfect crystal, their mutual interactions, and their coupling to the internal and external radiation induced by the crystal symmetry. No ad hoc parameters are necessary to account for the observed structures. 

Information obtained from any analytical instrument has a certain degree of uncertainty. In structural chemistry, the uncertainty may be in the location of the atoms, as obtained by, e.g., diffraction methods, due to unknown causes (crystal imperfections, thermal motion, etc.) [16]. We address ourselves now, to this problem, again focusing on symmetry issues. 

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