Symmetry breakdown

Fig. 14 Temperature dependence of modes observed in ST018. The soft u mode closed circles above Tc) in the tetragonal Dih phase is divided into two modes presented by open circles (below T ) and closed circles (at Td. Closed squares and open squares denote the modes split from the doubly degenerated Eg mode. Closed triangles above Tc indicate the Raman-inactive Aiu mode observed by local symmetry breakdown [11]...

The first term in (7) describes the coupling between the polarization splay and tilt of the molecules with respect to the smectic layer normal. This coupling is responsible for the chiral symmetry breakdown in phases where bent-core molecules are tilted with respect to the smectic layer normal [32, 36]. The second term in (7) stabilizes a finite polarization splay. The third term with positive parameter Knp describes the preferred orientation of the molecular tips in the direction perpendicular to the tilt plane (the plane defined by the nematic director and the smectic layer normal). However, if Knp is negative, this term prefers the molecular tips to lie in the tilt plane. The last term in (7) stabilizes some general orientation (a) of the polar director (see Fig. 7) which leads to a general tilt (SmCo) structure. 

Higgs, P.W. (1966). Spontaneous symmetry breakdown without massless bosons, Phys. Rev. 145, 1156-1163. 

The formation of an interface in a two-component homogeneous fluid happens when interaction between like entities becomes dominant. The resulting rearrangement is an example of symmetry breakdown that leads to the segregation of components into separate layers. On the cosmic scale a phase separation between matter and anti-matter is assumed to create two three-dimensional worlds in 4D space, such as >4 of Thierrin space. The interface can therefore not be crossed in three dimensions. 

Raman spectra as a function of temperature are shown in Fig. 21.6b for the C2B4S2 SL. Other superlattices exhibit similar temperature evolution of Raman spectra. These data were used to determine Tc using the same approach as described in the previous section, based on the fact that cubic centrosymmetric perovskite-type crystals have no first-order Raman active modes in the paraelectric phase. The temperature evolution of Raman spectra has indicated that all SLs remain in the tetragonal ferroelectric phase with out-of-plane polarization in the entire temperature range below T. The Tc determination is illustrated in Fig. 21.7 for three of the SLs studied SIBICI, S2B4C2, and S1B3C1. Again, the normalized intensities of the TO2 and TO4 phonon peaks (marked by arrows in Fig. 21.6b) were used. In the three-component SLs studied, a structural asymmetry is introduced by the presence of the three different layers, BaTiOs, SrTiOs, and CaTiOs, in each period. Therefore, the phonon peaks should not disappear from the spectra completely upon transition to the paraelectric phase at T. Raman intensity should rather drop to some small but non-zero value. However, this inversion symmetry breakdown appears to have a small effect in terms of atomic displacement patterns associated with phonons, and this residual above-Tc Raman intensity appears too small to be detected. Therefore, the observed temperature evolution of Raman intensities shows a behavior similar to that of symmetric two-component superlattices. 

Analysis of the vibrational bands revealed that at earliest growth stages the film is amorphous. In particular, a broad band at 1373 cm-1 proves the amorphous nature of the film. On the other hand, the mode at 1606 cm-1, usually an infrared active band, proves a symmetry breakdown of the molecule at this growth stage. Additionally, the amorphous phase lacks of vibrational activity at the phenyl groups, and tetracene backbone. Therefore, it is likely that the geometry of the rubrene molecule is dramatically distorted. 

The emergence of (27t) (0) simply means that the quantity P ) is proportional to the system s volume. The quantity U ) is called a potential, or an effective potential. The condition of symmetry breakdown of a system (continuous phase transition) results from f/(4>) s minimum for 0. 

The symmetry breakdown of AFne, given in Table 8, shows that AVmio) is greatly reduced from that in DME and AFne( r) becomes <0, i.e., antibarrier. Consequently a bonding and cr reorganization effects appear to represent the key barrier energy determinant in methanol. 

Second, due to the crystal field the five-fold degeneracy of the d atomic orbital is partially lifted, and fiiis symmetry breakdown results in a rather large effect in the splitting of the levels. 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electi onic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164-168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

Hisatsune and co-workers [290—299] have made extensive kinetic studies of the decomposition of various ions in alkali halide discs. Widths and frequencies of IR absorption bands are an indication of the extent to which a reactant ion forms a solid solution with the matrix halide. Sodium acetate was much less soluble in KBr than in KI but the activation energy for acetate breakdown in the latter matrix was the larger [297]. Shifts in frequency, indicating changes in symmetry, have been reported for oxalate [294] and formate [300] ions dispersed in KBr. 

Guslev et al. [1155] confirm the increase in stability of (Ni,Mg) oxalates with increase in magnesium content. These workers suggest that the impurity cation in the solid solution (i.e. that which is present at the lower concentration) disturbs the symmetry of the oxalate ion, so promoting its breakdown. 

The next qualitative change that occurs is the breakdown of the bilateral symmetry of the flow — the counterclockwise (left) eddy begins to grow at the expense of the clockwise (right) eddy, the latter effectively disappearing at t 300 psec. The symmetry is restored by / 370 psec, but at t 470 psec only the... 

It was demonstrated by Higgs [50] that the appearance of massless bosons can be avoided by combining the spontaneous breakdown of symmetry under a compact Lie group with local gauge symmetry. The potential V() which is invariant under the local transformation of the charged field... 

Barkemeyer et al. [8 a] showed previously that high enhancement can also be achieved at high magnetic fields when hydrogenating symmetric systems, where the breakdown of symmetry is caused by the naturally abundant 13C nuclei occurring individually in the two other equivalent carbon atoms of the double bond of the substrate (see Scheme 12.8) [8a]. 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

Hence, according to the symmetry selection rule, n —> n transitions are allowed but n —> ti transitions are forbidden. However, in practice the n —> it transition is weakly allowed due to coupling of vibrational and electronic motions in the molecule (vibronic coupling). Vibronic coupling is a result of the breakdown of the Born-Oppenheimer approximation. 

Bi-Pb-Cu-Ca-Sr-0 composition space 155 BN nanotube 265 Bom-Oppenheimer approximation 211,220 Bosch, C. 55 breakdown field 163 broken symmetry 178 buckyballs 263 bulk modulus 259 burning of air 210 by N-methyltriazolinedione 38... 

The lack of a center of symmetry in HD+, due to the difference in nuclear masses, creates a particularly interesting situation that requires a theoretical approach that may differ from those used to describe the parent cation, Hj, and its symmetric isotopomer, Dj. The asymmetry of the HD+ system has been investigated both experimentally [109,110] and theoretically [111-114]. In recent work, Ben-Itzhak et al. [109] studied the dissociation of the electronic ground state of HD+ following ionization of HD by fast proton impact and found the H++D(li) dissociation channel is more likely than the H(li) + D" dissociation channel by about 7%. They attributed this asymmetry breakdown to... 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

The scientific interest in solid surfaces whether in the context of heterogeneous catalysis, solid state electronics, or corrosion is a direct consequence of the fact that their properties are unique with relation to the corresponding properties of the bulk solid. This is not difficult to appreciate since at the surface there is a breakdown of translational symmetry, extreme gradients of chemical composition are feasible, and perturbation of both bulk structure and charge are possible. There are, therefore, formidable problems to overcome if we are to arrive at a situation where surface structure, electronic... 

EXPONENTIAL BREAKDOWN SYMBOLIC COMPUTING Symmetry-conserving allosteric model, MONOD-WYMAN-CHANGEUX MODEL SYMPORT Symproportionation,... 

---