Mode softening

Therefore the resulting TAi mode softening in ferrous martensite can be understood... 

In our approach to membrane breakdown we have only taken preliminary steps. Among the phenomena still to be understood is the combined effect of electrical and mechanical stress. From the undulational point of view it is not clear how mechanical tension, which suppresses the undulations, can enhance the approach to membrane instability. Notice that pore formation models, where the release of mechanical and electrical energy is considered a driving force for the transition, provide a natural explanation for these effects [70]. The linear approach requires some modification to describe such phenomena. One suggestion is that membrane moduli should depend on both electrical and mechanical stress, which would cause an additional mode softening [111]. We hope that combining this effect with nonlocality will be illuminating. 

A number of other models were considered and tested (for example, direct B—H bonding). The most significant test was the IR vibrational spectrum, where a sharp absorption band at 1875 cm-1 was found, corresponding to the Si—H stretch mode softened by the proximity of the B-atom. Had the hydrogen been bonded to boron, a sharp absorption band at 2560 cm-1 would have been expected. Also, Johnson (1985) showed that deuteration produced the expected isotopic shift. The most definitive and elegant proof of the correctness of the Si-H-B bonding model was provided by Watkins and coworkers (1990), on the basis of a parametric vibrational interaction between the isotopes D and 10B. 

In the parent phase (F > Tc), co is proportional to 2a T - Tc), i.e. it decreases linearly to zero at Tc. For this property the oscillation is called soft mode, i.e. on lowering the temperature the mode softens above Tc and hardens again below Tc, where co is proportional to 4a(Tc - T). 

A AH < kT has important consequences. As the temperature is lowered to where AHg, kT, strong electron-phonon interactions must manifest themselves. Direct evidence for mode softening and strong electron-phonon coupling in the internal Ty < T < 250 K has been provided by measurements of the Mdssbauer recoiless fraction and the X-ray Debye-Waller factor as well as of muon-spin rotation Therefore, it would be... 

There are several arguments that support a causality between the phonon anomalies and the pseudo gap formation in the spin fluctuation spectrum. The connecting element is a coupling of the lattice displacement of the a phonon to the occupancy and admixture of interlayer dyZ and dxz orbitals. The energy of this mode softens from 155 to 130 cm-1 in the temperature range from 200 to... 

Next, we discuss the concept of phonon-assisted reactions. In relation to thermal reactions, they can be assisted by phonon-mode softening leading to large-amplitude overdamped oscillations. In the case of a photochemical reaction, a strong electron-phonon coupling can assist in polymerization. Then some non-linear spectroscopic studies are discussed which illuminate on the dynamics of photopolymerization process. Then follows a discussion of results on reaction in a different kind of molecular assembly, the Langmuir-Blodgett films. Finally, some gas-solid interface reactions which produce polymers in a doped state are discussed. 

For further confirmation of the mode-softening and a possible identification of the molecular nature of the over-damped mode, we used the rigid-body motion analysis of the thermal- parameters of the room temperature x-ray diffraction study. A thermal-motion analysis (TMA) program was used to calculate the components of the librational (L) and the translational (T) tensors with a least-square fit of the published thermal parameters ( ) of all nonhydrogen atoms of the molecule. The librational frequencies were calculated by the method of Cruickshank (7), using the appropriate eigenvalues of the L-tensor and the corresponding moments of inertia. 

Second-order phase transitions also show up via the critical slowing down of the critical fluctuations (Hohenberg and Halperin, 1977). In structural phase transitions, one speaks about soft phonon modes (Blinc and Zeks, 1974 Bruce and Cowley, 1981) in isotropic magnets, magnon modes soften as T approaches Tc from below near the critical point of mixtures the interdiffusion is slowed down etc. This critical behavior of the dynamics of fluctuations is characterized by a dynamic critical exponent z one expects that some characteristic time r exists which diverges as T - TCl... 

The H-atom disordering in OH—O hydrogen bonds is a common feature of many crystals. Whereas a dynamical disorder of the H atom is commonly observed, the origin of this disorder and its relation to the crystal structure and lattice vibrations are poorly understood. As is well known, the H hopping becomes coupled with lattice vibrations in particular modes. When the temperature is decreased to 7), these modes soften and their frequencies become zero at and below T. In structural terms, the vibrations propagating in the crystal lattice involve motion of types that... 

However, contrary to CeBCu this mode softening in Cei La Bei3 for 0.8 is also observed for all other symmetry modes with respect to the average behavior of the reference materials. The phonon softening in Cej La, Be,3 for 0.1 X 0.8, independent of the mode symmetry is also reflected by the behavior of the Debye temperature 0 (Besnus et a. 1983), which is displayed at the bottom of fig. 35. No temperature dependent phonon anomaly has been observed for the optical phonons of CeBcij, contrary to the anomalous softening of the bulk modulus upon cooling down below 350 K (Lenz et al. 1984). 

As indicated in connection with Eq. (11), the energy supply Sj may occur at any, or all, frequencies of the polar modes. The subsequent excitation of a particular mode, e.g., coi, then requires some time even when the total rate of supply exceeds o- The system thus possesses storing ability. One would expect, however, that this time is particularly short if the energy is supplied at the frequency of the mode that will be excited coherently. Furthermore, arising from the considerations of mode softening, this mode, when excited, may be detached from the band and hence provide a very frequency-sensitive target for further energy supply. Detailed calculations on this question have not been carried out yet. 

It has also been suggested that the assembly of biological oscillators may be modeled as parameter tunnel diode memory system with couplings. The frequencies of such a system can include values very much lower than the design frequency, which thus supports our prediction of mode softening. 

B. S. Thornton, Solid State Memory Problems Support Mode-Softening for Larger Assemblies of Biological Dipole Oscillations, Phys. Lett. 102A, 77-79 (1984). 

Dipole oscillations in an assembly of molecules in the membrane of cells can be modeled as phase-locked solid state oscillators by a basic circuit as in Figure 1. Loose coupling between such circuits imposes an eigenvalue problem from which significant mode softening can be shown to result and this has been suggested to be an important requirement in the energetics associated with the reproduction and mutation of cells. As each individual unit oscillator can operate at subharmonics as well as harmonics, the above model is consistent with the idea that in vivo a number of discrete frequencies exist in the cell. 

Completely delocalized ionization is obtained in the coqjngated case when only one single bond separates the donble bonds. More than one separating bond leads to mode softening and partial localization, whereas a completely localized, ionized double bond is obtained if many single bonds separate the donble bonds. 

Fig.4.5-TI (CH3NHCH2C00H)3 CaCl2. vq versus T. Vo is the phonon mode frequency. Triangles measured by millimeter spectroscopy. Brown circles measured from far-infrared spectra. Gray circles measured from electric-field-induced Raman spectra. In the paraelectric phase T > 0f), Vo decreases as the temperature decreases toward 0f, that is, the phonon mode softens...

Fig. A.5-22 BaTiOs. Phonon dispersion relation determined by neutron scattering along the [100] direction in the cubic phase, v is the phonon frequency. LA, longitudinal acoustic branch TA, transverse acoustic branch TO, transverse optical branch. The frequency of the TO branch is lower (softer) at 230 °C than at 430 " C, indicating mode softening...

Fig. tt.5-23 BaTiOs. Avq and F versus T, obtained from hyper-Raman scattering in the cubic phase. Avq and F are the optical mode frequency and damping constant, respectively. The different symbols brown and gray) show results from different authors. Avq decreases as the temperature decreases to the Curie point, showing the presence of mode softening, c ligth velocity... 

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