Temperature dependence transverse vibration

From the lattice dynamics viewpoint a transition to the ferroelectric state is seen as a limiting case of a transverse optical mode, the frequency of which is temperature dependent. If, as the temperature falls, the force constant controlling a transverse optical mode decreases, a temperature may be reached when the frequency of the mode approaches zero. The transition to the ferroelectric state occurs at the temperature at which the frequency is zero. Such a vibrational mode is referred to as a soft mode . 

The loss frequency dependence e" (v), described by Eqs. (20) and (22), is non-Lorentzian. In view of Fig. 3e and 3f calculated, respectively, for water and ice. For reasonable dimer parameters, this dependence is located in the THz region. Transverse vibrations are very damped, especially at low temperatures, since the parameter mj (23) is very small. As is shown in Table I, in the case of water, m ... 

The adopted molecular constants, along with fitted and estimated parameters, are presented in Tables IV-VI. The absorption frequency dependences are depicted in Figs. lOe, 10c, and 10a, respectively, for the lowest (7 ), room9(74), and highest (77) temperatures. The loss spectra are shown for the same temperatures in Figs. lOf, lOd, and 10b. The dash-dotted lines depict the contribution to loss spectra of transverse vibration (TV), and the dashed lines depict such spectra calculated without account of this vibration. We see that transverse vibrations play an important role in the THz region. 

Figure 14 Estimated temperature dependences mean number of transverse vibration performed during the lifetime z (a) center vj of the partial-loss dependence e i (v) (solid curve) and the second-Debye region frequency vD2 (dashed curve) (b).

It is also important to note that the solid curve in Fig. 14d, which represents the left and right wings of the transverse-frequency dependence v (T), resembles two straight lines with different slopes, depicted in Fig. 17. Hence, behavior of transverse vibrations, aggravated by a molecule association, is perhaps related to the break-point temperature rbreak (near 300 K). 

Figure 23 Absorption (a) and loss (b) frequency dependences in T- and V-bands calculated for ice at the temperature —7°C. Solid lines (1) refer to the total absorption or loss, open circles (2) refer to the experimental data by Warren [49], and lines 3-5 refer to contributions to spectra due to elastic translation along the H-bond (3), elastic reorientation about the H-bond (4), and transverse vibration (5).

Figure 37 Frequency dependences of the loss factor (a) and of the dielectric constant (b) calculated (solid curve) and measured (open circles) in water, (c) Contributions to loss due to libration of a permanent dipole in the hat well (1), vibration of a nonrigid dipole along the H bond (2), reorientation of polar molecules about this bond (3), and transverse vibration of a nonrigid dipole with respect to the H bond (4). Temperature 300 K.

Figure 43 Frequency dependence of the transverse-vibration loss e" (v) calculated for water H20 at the temperature 27°C. p = 0.65 (for curves 1, 3, 4) and p = 1 (for curve 2) yj = 2.8 (for curves 1, 2, 3) and 2 (for curve 4). The vertical lines mark the TV-band center frequency, estimated from Eq. (193).

Single crystal measurements at different temperatures have also been made and analysed to explain in detail the remarkable negative thermal expansivity observed for many microporous solids. Simplistically, enhanced transverse thermal vibration of oxygen atoms in T-O-T bonds can result in the reduction of T-T distances in the frameworks and the shrinkage of cell dimensions that depend on these T T distances. ... 

Inelastic neutron scattering(see Section F.8) performed on two samples has evidenced two types of relaxation, the longitudinal and the transverse. The former corresponds to the usual relaxation between potential wells whereas the latter is related to vibrations inside the potential well. It is shown that several regimes occur for the transverse relaxation. Below a certain temperature about two times smaller than Tg (t , = 10 s), a new regime appears where the interparticle interactions would destroy the local modes. However T depends on the volume and therefore cannot be a transition temperature toward a collective state. This regime could be the approach to such a state. 

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