Symmetry domains

Molecular nuclear configurations within the nuclear configuration space and potential energy hypersurface model of conformational changes and chemical reactions, in particular, the distribution and representation of symmetry domains in the nuclear configuration space, as well as reaction mechanisms " "... 

On subbranches B and B we find surfaces consisting of spherelike regions lying inside the symmetry domains Schoen 1970, with nearly unduloidal necks piercing through each of the four faces of each symmetry domain and... 

Catchment Regions and Symmetry Domains of Nuclear Configuration Spaces... 

One interesting consequence of the above theorems on the point symmetry domain partitioning of M into subsets Gy is the following "non-crossing rule" ... 

No steepest descent (relaxation) pathof the potential energy hypersurface E(K) of any electronic state can cross the boundary of any point symmetry domain Gy of a nuclear corfiguration space M. 

The vertex set V(g(M,sym)) and edge set E(g(M,sym)) define the graph g(M,sym). This graph provides a concise representation of the mutual arrangements of point symmetry domains Gy within the nuclear configuration space M. 

By replacing the point symmetry domains Gy, and Gj.j. of equations (8)-(10) with the D y and D. j. domains of the eigenvalue sign distribution of local Hessian matrices within the nuclear configuration space M, and by replacing the symbol sym with the symbol hess, one obtains the analogous neighbor relation... 

Several groups have studied the structure of chiral phases illustrated in Fig. IV-15 [167,168]. These shapes can be understood in terms of an anisotropic line tension arising from the molecular symmetry. The addition of small amounts of cholesterol reduces X and produces thinner domains. Several studies have sought an understanding of the influence of cholesterol on lipid domain shapes [168,196]. 

The normal component of velocity and tangential component of surface force are set to zero along a line of symmetry. For the domain shown in Figure 3.3 these are expressed as... 

The most important materials among nonlinear dielectrics are ferroelectrics which can exhibit a spontaneous polarization PI in the absence of an external electric field and which can spHt into spontaneously polarized regions known as domains (5). It is evident that in the ferroelectric the domain states differ in orientation of spontaneous electric polarization, which are in equiUbrium thermodynamically, and that the ferroelectric character is estabUshed when one domain state can be transformed to another by a suitably directed external electric field (6). It is the reorientabiUty of the domain state polarizations that distinguishes ferroelectrics as a subgroup of materials from the 10-polar-point symmetry group of pyroelectric crystals (7—9). 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

The S domains form the viral shell by tight interactions in a manner predicted by the Caspar and Klug theory and shown in Figure 16.8. The P domains interact pairwise across the twofold axes and form protrusions on the surface. There are 30 twofold axes with icosahedral symmetry that relate the P domains of C subunits (green) and in addition 60 pseudotwofold axes relating the A (red) and B (blue) subunits (Figure 16.9). By this arrangement the 180 P domains form 90 dimeric protrusions. 

An ordering phase transition is characterized by a loss of symmetry the ordered phase has less symmetry than the disordered one. Hence, an ordering process leads to the coexistence of different domains of the same ordered phase. An interface forms whenever two such domains contact. The thermodynamic behavior of this interface is governed by different forces. The presence of the underlying lattice and the stability of the ordered domains tend to localize the interface and to reduce its width. On the other hand, thermal fluctuations favor an interfacial wandering and an increase of the interface width. The result of this competition depends strongly on the order of the bulk phase transition. 

Some of the above discussed precursor phenomena are also observed prior to diffusion driven phase transformations. A typical example are the conventional EM tweed images obtained in the tetragonal parent phase in high Tc superconductors and other ceramics. In a recent survey by Putnis St e of such observations it was concluded that in these cases the tweed contrast resulted from underlying microstructures fomied by symmetry changes driven by cation ordering. These symmetry changes yield a fine patchwork of twin related domains which coarsen when the transfomiation proceeds. However, in view of the diffusion driven character of the latter examples, these cases should be clearly separated from those in the field of the martensites. 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

The nAChR is cylindrical with a mean diameter of about 6.5 nm (Fig. 1). All five rod-shaped subunits span the membrane. The receptor protrades by <6 nm on the synaptic side of the membrane and by <2 nm on the cytosolic side [2]. The pore of the channel is along its symmetry axis and includes an extracellular entrance domain, a transmembrane domain and a cytosolic entrance domain. The diameter of the extracellular entrance domain is <2.5 nm and it becomes narrower at the transmembrane domain. The... 

The dependence of P (PeL) and g (PeL) is shown in Fig. 11.4. The parameter P (PeL) is a parabola with an axis of symmetry left of the line Pcl = 0. Since the Peclet number is positive, for any value of the operating parameters, the physical meaning is that only for the right branch of this parabola, which intersects the axis of the abscissa at some critical value of Peclet number, Pcl = Peer- The vertical line PeL = Peer subdivides the parametrical plane P - Pcl into two domains, corresponding to positive (PeL < Peer) or negative (PeL > Peer) values of the parameter P . The critical Peclet number is... 

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