Debye mode

In the simplest model investigated, including a single Debye mode (X(f) -exp(-t/ r, ), xL being the longitudinal dielectric relaxation time), the spectral effect was found to be small and negative -0.2 <[Pg.332]

Fig. 2. Wavepacket trajectories on the excited free energy surface for a two Debye modes model. The term crossing line A and B are for weakly and moderately exergonic CR, respectively.

In Figure 4, A gives a graph of the contribution due to three torsional Debye modes with 6 = 200° K., while B represents the heat capacity contribution of three hindered rotational degrees of freedom with ... 

From the low-temperature fluorescence work of Vanderkooi, we would expect that if the red edge of the absorption spectrum of the complex is excited, then only a few of the collective (Debye) modes will be excited. Work of Vanderkooi was steady state, and thus only the time-averaged distribution of collective modes is seen in a time-resolved experiment with a subpicosecond time scale resolution the actual occupation of the collective mode can be observed directly, since the fact that the mode is coupled to the chromophore makes the chromophore absorbance a function of the collective mode occupation. If the coherence time of the collective mode is sufficiently long, one might actually expect to find a periodic modulation of the chromophore absorbance as collective mode energy oscillates between the few modes excited. If we assume that the Debye spectrum has a maximum at about 100 cm , then these modes should show up with a frequency of about 3 x 10 Hz, or a period of about... 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

In the energy region examined, there exist at least two modes. One is a localized mode (Boson peak mode) and the other is an extended mode (Debye mode). The decoupling of both modes is needed to understand which mode contributes to the decrease in G co) with thickness in the energy region below 10 meV. The contribution of the Debye mode in G(m) was evaluated from the Debye frequency cod and the contribution from the Boson peak mode was obtained by subtracting the Debye contribution from the total G(co) [45]. For the thin films, the Debye contributions were estimated assuming that the amplitude of Debye mode and Boson peak mode are independent of film thickness. The Debye contribution from bulk and thin films... 

The density of phonon states of Debye mode Gd(co) is related to the average sound velocity V through the relation Gd((w) = where Vis the average atomic volume... 

The root. i simply indicates that infinite distances are correlated with infinite time, S2 is the reciprocal of the Debye relaxation time, and 3 is the kinetic relaxation frequency of the system. Depending on the kinetic parameters of the chemical process, the kinetic relaxation frequency can be faster or slower than the Debye frequency of the system. If the kinetic relaxation frequency is much smaller than the Debye mode, it can be determined experimentally by conductance fluctuation analysis. 

In the coupled diffusion, regime III, the interpretation of the diffusion processes in molecular terms is more complicated and certain linear combinations of the D) such as interdiffusion modes or cooperative diffusion coefficients are used to describe the underlying physical processes. As can be seen from Table 2, two relaxation times are proportional to q, indicating diffusive behaviour. The third relaxation (t3) is q-independent and is often referred to as the Debye mode plasmon mode or interdiffusion mode (although not diffusive at all). Its relaxation time is, however, related to the polyion and coion diffusion according to... 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

It is noteworthy that it is the lower cross-over temperature T 2 that is usually measured. The above simple analysis shows that this temperature is determined by the intermolecular vibration frequencies rather than by the properties of the gas-phase reaction complex or by the static barrier. It is not surprising then, that in most solid state reactions the observed value of T 2 is of order of the Debye temperature of the crystal. Although the result (2.77a) has been obtained in the approximation < ojo, the leading exponential term turns out to be exact for arbitrary cu [Benderskii et al. 1990, 1991a]. It is instructive to compare (2.77a) with (2.27) and see that friction slows tunneling down, while the q mode promotes it. 

Debye) is the permanent dipole of H2O and the factor 2 (in our calculation only 1.4) is the enhancement due to the image. It has been observed that with the formation H2O layers this mode stiffens and becomes even stronger (15). 

It is important from a practical viewpoint to predict the shear viscosity of mixtures from those of pure melts. For alkali nitrate melts, a linear dependence has been found between the reorientational line width obtained by Raman measurements and the ratio of temperature divided by shear viscosity.For NO3 ions, the depolarized Raman scattering from 1050cm" total stretching vibrational mode (Al) has a contribution to the line width L, which is caused by the reorientational relaxation time of the Csv axis of this ion. The Stokes-Einstein-Debye(SED) relation establishes a relation between the shear viscosity r of a melt and the relaxation time for the reorientation of a particle immersed in it ... 

The spectrum recorded at 230 K was discarded in the fit procedure because above 200 K the effective thickness decreases drastically because of a significant softening of protein-specific modes [16]. From the simultaneous fit of the spectra in the temperature range 3.2-200 K, the Debye temperature was determined as do = 215 K. AEq proved to be a temperature-dependent quantity, which is discussed later (see Sect. 9.4.2). 

The so-called Boson peak is visible as a hump in the reduced DOS, g(E)IE (Fig. 9.39b), and is a measure of structural disorder, i.e., any deviation from the symmetry of the perfectly ordered crystal will lead to an excess vibrational contribution with respect to Debye behavior. The reduced DOS appears to be temperature-independent at low temperatures, becomes less pronounced with increasing temperature, and disappears at the glass-liquid transition. Thus, the significant part of modes constituting the Boson peak is clearly nonlocalized on FC. Instead, they represent the delocalized collective motions of the glasses with a correlation length of more than 20 A. 

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

The Debye temperature characterizes the rigidity of the lattice it is high for a rigid lattice but low for a lattice with soft vibrational modes. The mean squared displacement of the atom, , can be calculated in the Debye model and depends on the mass of the vibrating atom, the temperature and the Debye temperature. 

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