Relaxation terms

Here rj is the viscosity of the dewetting liquid. Note that a relaxational term proportional to a has been added, with fi(j)) being the chemical potential of the vapor. This term alone guarantees that a homogeneous liquid film will relax to its equilibrium value hooip) by evaporation or condensation. For h = hooip) this term vanishes. 

Nevertheless, the overall structural problem can be solved from combined n.O.e. and single-selective relaxation-measurements through the evaluation of individual cross-relaxation terms, (Ty. According to Noggle and Shirmer, the n.O.e. value is a function of the cross-relaxation between spins i and j and the relaxation contributions of the neighboring protons to spin i, that is. 

According to these equations, the effect of selectively perturbing the spin states of spins i and j is to isolate the cross-relaxation paths common to these two spins. Combining Eqs. 15 and 19, the individual cross-relaxation terms are readily determined from single-selective and double-selective relaxation-rate measurements, that is. 

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

Whereas in CF4 we can ignore the surface relaxation term, this term is significant for c-C4F8 at 291 K, with the relative weighting becoming increasingly important as we add additional molecular layers, as shown below. This is equivalent to Eq. (3.5.6), with the bulk fluid term set to the spin-rotation relaxation of the bulk gas. It is clear that in such a system, in comparison with CF4 at 294 K, the effect of liquid phase surface relaxation cannot be ignored. 

For highly conductive liquids and solids the loss term not only results from a single relaxation term, as given by Eq. (8), but also from term resulting from ionic conductivity, cr, as described by Eq. (14) ... 

The value objective function is extended by the relaxation terms penalizing the model infeasibility. The term (3) is withdrawn from the profit z in the objective function. 

In this section, we shall only discuss the relaxation term however, before performing the detailed calculations corresponding to the precise Brownian model we have in mind, we shall first consider simpler cases, which already give a qualitatively correct description of the relaxation effect. 

In the four preceding sections, we have developed various approximations for the relaxation term in the limiting law for the conductance of electrolytes, starting from the generalized transport equation (111). 

Our discussion of Section V has indicated that the electrophoretic effect has to be found in the Ta term defined in Eq. (301) (see also Eq. (312)) moreover we have already found a diagram (Fig. 14a) in which the solvent is transmitting the wave number —k from ion /S to ion a, as we expect to find from the classical theory. This term was not calculated in Section V because it gives a contribution of order ei to while the relaxation term is of order e6 it will be considered presently. 

Having now found the classical electrophoretic term from a microscopic analysis of the system ions plus solvent, we should analyze more complicated diagrams in order to see whether there is any other contribution than (468) in the limiting form of the electrophoretic term. Indeed, it is very tempting to develop an analysis similar to the one followed in the study of the relaxation term and to consider the possibility of more complicated collective effects. These would have essentially three origins ... 

These equations, obtained by a simple superposition of the external force term and the relaxation term, give a quantitative agreement with experiment in the case of liquids. The stationary solution of Eq. (6) allows the calculation of %. 

The FPA experiment can also be performed by exciting into the S2band of ethidium using 315-nm radiation and detecting at 630 nm (J. H. Shibata, J. C. Thomas, and J. M. Schurr, unpublished results). Substantial cancellation of the two main relaxing terms (n = 1, 2) in r(t) considerably diminishes the statistical accuracy of the best-fit a., but nevertheless it remains the same as found for excitation into S, which provides a reassuring check. Using intercalated quinacrine as the probe dye, Fan et al. recently obtained a... 

Rapid unresolved excitation transfers would decrease r0. Their predominantly azimuthal motion would preferentially decrease the amplitude of the n = 2 term relative to the = 1 term and the n = 1 term relative to the = 0 term, thereby shifting relative amplitude into the more slowly relaxing terms. That in turn would increase the apparent torsion constant. 

Now, the most direct interpretation of Eq. (11.5) follows from the observation, suggested by Eqs. (11.1) and (11.2), that/is essentially a relaxation term. In fact, Vk — represents the difference between the electrostatic potential at the h nucleus in the given molecule and the potential that the same nucleus would feel if the atomic orbitals and the equilibrium distances remained the same as in the reference molecule in spite of the change in electron populations. 

The relative magnitude of both second-order terms determines whether the molecule is stable with respect to distortion along the vibrational coordinate Qt. If the relaxation term is of larger magnitude than the distortion term then the geometry is unstable and the potential energy surface has negative curvature for mode Qi. 

Here p is the density matrix for all molecular states in the three-level system depicted in Fig. 4, and all incoherent relaxation terms caused, for example, by collisions, spontaneous emission, or decay in a (quasi)continuum are incorporated in the relaxation matrix rreiax. 

Figure H. The calculated dme-dependent total friction (f(r)) and its relative contributions from the binary term ( CB(0) and the density relaxation term RPI> t) for the iodine atom in CH3I. The reduced temperature T (= kaT/f) is 0.363 and the reduced density p for CH3I is 0.918. The time-dependent friction is scaled by t 2, where z,c = [miaj/kBT 2 3.3 ps. The friction is found to be much higher for Iodine than CH3. This figure has been taken from Ref. 133.

Figure 5.3 shows the measured temperature dependence of the loss tangent of LaA103 single crystals and a fit employing the skm model plus a Debye relaxation term. The best fit was achieved for an activation energy of 31 meV, the estimated defect concentration is only 1016/cm3. It was suggested that aluminium atoms on interstitial lattice positions are responsible for the observed relaxation phenomena. This indicates, that the dielectric losses are extremely sensitive to very small concentrations of point defects [21],... 

By rewriting the solute electronic density (in terms of the one-particle density matrix on a given basis set) corresponding to the excited state A as a sum of the GS and a relaxation term PA, and by assuming a complete equilibration between the solute in the excited state K and the solvent, we obtain... 

This equation shows that vertical excitations in solvated systems are obtained as a sum of two terms, the difference in the excited and ground state energies in the presence of a frozen ground state solvent and a relaxation term determined by the mutual polarization... 

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