Breaking of symmetry

At the organism level, there are three issues that must be tackled, besides that of visual representation the first division, breaking of symmetry, and intercellular communication. The simulation forces the first cell to divide by initializing the node (gene) signifying division to 1. This ensures that the organism divides at least once. 

Ferroelectric mesophase that appears through the breaking of symmetry in a tilted smectic mesophase by the introduction of molecular chirality and, hence, mesophase chirality. 

As shown in Fig. 13a, An for the (llO)c face is composed of two contributions from the antiferrodistortive phase transition and the ferroelectric transition (see data for 7 x 2 x 0.3 mm ). On the other hand, only the ferroelectric transition is seen for the (OOl)c face. The inequality Px 7 Py means the breaking of symmetry in the (OOl)c plane. Therefore, the symmetry of the ferroelectric phase is below orthorhombic. 

One other reason why many chemists and biologists are skeptical about parity violation and other subtle physical effects, is that the breaking of symmetry can be realized rather simply in the chemistry laboratory. According to Meir Lahav, one of the best known researchers in the field, breaking of symmetry is not the problem. He means by that, that the problem is rather the propagation and amplification of chirality. In sidebox 3.3 he summarizes some of the main concepts in particular, he considers crystals as agents of symmetry breaking (Weissbruch et al, 2003). 

I have mentioned that there are two main aspects to consider on the origin of homochirality, one being the breaking of symmetry, the other the preservation and amplification of this initial inbalance. The latter will be considered further in the next chapter, which deals with macromolecules. 

In this chapter the question of homochirality has also been considered according to Meir Lahav breaking of symmetry is not the problem. I do not know how many scientists would agree with him, but it is certainly true that in the laboratory chiral compounds can be obtained starting from racemic mixtures -and this by simple means, without invoking subtle effects of parity violation. Of course we do not know how homochirality really evolved in nature however, it is comforting to know that there is in principle an experimental solution to the problem. 

The natural tendency of polypeptide chains to grow homochirally may suggest an alternative mechanism for the breaking of symmetry, based on macromolecules instead of monomers. The argument is that it should be easier to separate enantiomeric homochiral chains, rather than racemic low-molecular-weight monomers, from each other. It has been shown for example that when the NCA-polycondensation is performed on mineral support, the oligomeric product remained absorbed on the surface. The lower oligomers are, however, easily... 

Rule 3.2 (Corollary). The breaking of symmetry is always the consequence of an identifiable chemical or spatial constraint. 

The examples discussed in this chapter show that there are many different ways in which lattice-induced strain can be relaxed or aeeommodated, the partieular mode depending on the properties of the elements and the struetures involved. Many of these compounds have unusual properties resulting from non-integral stoichiometry, the presence of non-integral oxidation states, or the spontaneous breaking of symmetry, all of which are the direct consequence of lattice-induced strain. 

In the transition from Pair B to Pair C, the overall symmetry of the site is restored by an analogous 180° rotational translation of the second radical. In Pair C there is rapid internal rotation about the Ca—Cf bond, and the anisotropy of hyperfine splitting by the alpha hydrogens establishes the direction of the first C—C bonds in the radicals. This direction confirms the supposed screw motion. The breaking of symmetry in the first step, its subsequent restoration in the third, and the overall similarity in the motions of the two radicals give support to the hypothesis that the radicals move one by one. Infrared evidence discussed in Sections IX and VII-B.4.a confirms this inference. 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

A hint that such a possibility exists was provided by the example of molecular spectra (see section 2.32) where the breaking of symmetry was shown to have a dramatic effect on the apparent complexity of spectra (see in particular figs. 2.16 and 2.17). 

It is easy to see that the breaking of symmetry leads to a pair of fixed points, 0 and —[Pg.421]

The phase transitions in solids still attract substantial attention of scientists and engineers due to many anomalies in their vicinity. Usually, at the phase transition point, the breaking of symmetries including translational, orientational and time inversion ones [1] takes place. 

The statistical mechanics of the Curie Weiss mean field or the van der Waals mean field can likewise be discussed by the method of random fields (Siegert (1963) Jalickee (1969)). In these cases the mean field analogous to 0(r) is a position-independent vector. The existence of this mean field, however, implies the destruction of the isotropy of space, i.e., the breaking of a symmetry. As Edwards (1970a, b) notes, therefore, there must also be a breaking of symmetry in order to obtain electron localization in the transla-tionally invariant averaged system. 

Breaking of symmetry has been reported in stirred l crystallization. NaClOs crystallization in an unstirred solution produces a statistically equal number of I- or d-crystals, but crystallization in a stirred achiral solution can produce 99% crystal enantiomeric excess. This is due to a secondary nucleation phenomenon. Dendritic or needle-like structures on the surface of a crystal break off in a stirred solution. The result is an amplification of the corresponding enantiomeric phase. 

Let the director of the nematic phase be perpendicular to a flat interface. Then we can anticipate two effects. First, the polar surface layer due to the break of symmetry n 7 —n appears. Second, due to a correlation in position of the centers of mass (parallel to the interface) in a surface adjacent to the layer the translational invariance is broken and the surface induces the smectic A order. 

Unlike the homogeneous bulk, the interface is unique in the break of symmetry, which may induce order or disorder. Numerous techniques have been used to acquire structure and composition information at the interface of ILs, many of which require UHV. The negligible vapor pressure of ILs makes those techniques possible to interrogate their air-liquid interface. Other surface-sensitive methods with ambient conditions including NR, SFG, AFM, and tensiometry are also applicable to measure air-liquid, solid-liquid, and liquid-liquid interfaces. 

We have shown [8] that this more formal property is responsible for the asymmetry of the optical absorption peaks of transitions involving intragap states. Both symmetries are absent in the real systems imder consideration, but for most questions this breaking of symmetries is merely a question of quantity rather than of importance for the existence of these localized states. 

In more general terms The creation of an interface means a break of symmetry of the homogeneous starting situation. Given a sufficient mobility of charge caarriers, this necessarily leads to a charging. ... 

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