Debye time

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

Experiments in the picosecond time range show that C(t) is non-exponential in most solvents with an average spectral relaxation time greater than the longitudinal relaxation time tl and smaller than the Debye time td-... 

We assume that if many of the liquids of interest, such as propylene carbonate, were studied by higher frequency (measurement techniques, new, high frequency components would be discovered which would account at least partially for the short time scale dynamics we see in the solvation C(f) data. Indeed, the apparent observation of a single Debye time is inconsistent with theories of liquids that take into account dipole-dipole interactions (see Kivelson [109]). Furthermore, some of the liquids studied have extraordinarily large apparent infinite frequency dielectric constants (e.g., = 10... 

Bagchi et al. have derived analogous equations for a solvent with two Debye times associated with two overlapping dispersion regimes [53]. 

Note the close resemblance between the reference time of the internal rotary diffusion of the magnetic moment and the Debye time... 

For simplicity we assume that the particles are magnetically hard.1 Then, the already developed formalism [Eqs. (4.90) and (4.293)] applies in full under two conditions e is identified with v and the internal relaxation time Tq is replaced by the external rotational diffusion (Debye) time x. In the modified equations, the orientation order parameter is given by (Pi)- Similarly to (4.294), we set... 

A comparison between experimental and simulated main Debye relaxation time is presented in Figure 16-7. Simulation and experimental results show excellent agreement for not so dilute systems (p > 0.4g/cm3). However, below this density the experimental Debye time increases with decreasing density, whereas simulation results for this quantity keep decreasing and approaching the limiting behavior of a collection of free rotors. The extent of the loss of dynamic correlation between... 

The first case, Eq. (8.2) corresponds to the magnetic moment being frozen or blocked as considered in [16]. Since M will maintain its direction relative to axes fixed in the particle for a long time compared with the Debye time Tp. The second case corresponds to the calculation in [17] where the effect of the rotational Brownian motion of the fluid on the magnetic susceptibility is ignored since the directional fluctuations of... 

In the new calculation scheme the hat model may be involved twice. First, this model may be applied for describing the relaxation frequency band, characterized by the Debye time td- Second, this model may be used for describing the libration band, characterized by the lifetime Tor. Therefore, the parameters of the two used hat models (or similar to them) should be quite different. We emphasize that in the present calculation scheme, applied to water, we employ the hat model only once—for describing both Debye and libration bands, for which the same set of the model parameters is used. 

Finally, we should enquire as to whether or not it is reasonable to expect a linear response in describing the rotational diffusion of dipoles in the presence of a very strong local field, such as presented by the excess electron. The time window of the optical Kerr gate driven by a picosecond laser pulse depends on the relaxation of the molecules of the Kerr medium from an aligned orientation to an isotropic spatial distribution, once the applied optical field is switched off. For many liquids this relaxation time r is the low field limit, namely, the Debye time. We might anticipate an asynunetry in the temporal response 5 (0 of... 

In Debye solvents, x is tire longitudinal relaxation time. The prediction tliat solvent polarization dynamics would limit intramolecular electron transfer rates was stated tlieoretically [40] and observed experimentally [41]. 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

This is nothing other than a long-time asymptotics of orientational relaxation valid at t > xj. Sometimes deviations from Debye s relaxation were also found at t > xg/. This is an effect of long-range Coulomb and... 

Its experimental confirmation provides information about the free rotation time tj. However, this is very difficult to do in the Debye case. From one side the density must be high enough to reach the perturbation theory (rotational diffusion) region where i rotational relaxation which is valid at k < 1. The two conditions are mutually contradictory. The validity condition of perturbation theory... 

The discovery of a transition which we identify with this has been reported by Simon, Mendelssohn, and Ruhemann,16 who measured the heat capacity of hydrogen with nA = 1/2 down to 3°K. They found that the heat capacity, after following the Debye curve down to about 11°K, rose at lower temperatures, having the value 0.4 cal/deg., 25 times that of the Debye function, at 3°K. The observed entropy of transition down to 3°K, at which the transition is not completed, was found to be about 0.5 E.U. That predicted by Eq. (15) for the transition is 2.47 E.U. 

Hoffman, D. (2006) Peter Debye (1884-1966) - ein typischer Wissenschaftler in untypischer Zeit [Peter Debye (1884-1966) - a Typical Scientist in Untypical Times]. Preprint des Max Plank Institutfiir Wissenschafisgeschichte Nr, 314, Berlin MPIWG,... 

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