Fast relaxation

In this case all relaxation times tj are small compaured to to and the thermal equilibrium takes place, that is 

The kinetic equation (8.1.2) acquires then the following singularly pertui bed form 

If Knudsen number eg for the gas phase is small, one comes to the following singularly perturbed problem for Qc 

Different forms of the kernel Rcd b, b fc,9c), that depend functionally on the distribution functions /c, gc, are discussed in Part IV. 

The condition (8.2.1) allows one to expand the solution of (8.2.2), (8.2.3) in power series over small parameters e, g 

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Other properties of association colloids that have been studied include calorimetric measurements of the heat of micelle formation (about 6 kcal/mol for a nonionic species, see Ref. 188) and the effect of high pressure (which decreases the aggregation number [189], but may raise the CMC [190]). Fast relaxation methods (rapid flow mixing, pressure-jump, temperature-jump) tend to reveal two relaxation times t and f2, the interpretation of which has been subject to much disagreement—see Ref. 191. A fast process of fi - 1 msec may represent the rate of addition to or dissociation from a micelle of individual monomer units, and a slow process of ti < 100 msec may represent the rate of total dissociation of a micelle (192 see also Refs. 193-195). 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

Figure B2.5.4. Periodic displacement from equilibrium through a sound wave. The frill curve represents the temporal behaviour of pressure, temperature, and concentrations in die case of a very fast relaxation. The other lines illustrate various situations, with 03Xj according to table B2.5.1. 03 is the angular frequency of the sound wave and x is the chemical relaxation time. Adapted from [110].

X-ray scattering studies at a renewed pc-Ag/electrolyte interface366,823 provide evidence for assuming that fast relaxation and diffu-sional processes are probable at a renewed Sn + Pb alloy surface. Investigations by secondary-ion mass spectroscopy (SIMS) of the Pb concentration profile in a thin Sn + Pb alloy surface layer show that the concentration penetration depth in the solid phase is on the order of 0.2 pm, which leads to an estimate of a surface diffusion coefficient for Pb atoms in the Sn + Pb alloy surface layer on the order of 10"13 to lCT12 cm2 s i 820 ( p,emicai analysis by electron spectroscopy for chemical analysis (ESCA) and Auger ofjust-renewed Sn + Pb alloy surfaces in a vacuum confirms that enrichment with Pb of the surface layer is probable.810... 

The intrinsic relaxation rate of a paramagnetic center can he enhanced hy spin—spin coupling with a nearhy fast-relaxing species. 

The expression in Eq. (29) can be evaluated numerically for all values of t, and the results for three different waiting times are shown in Fig. 11 for c = 0.1. The value of Tmin = 2.0 ps at E/To = 5.7 x lO", derived from the present theory (also consistent with Goubau and Tait [101]) was used. The results for t = 10 ps demonstrate that, due to a lack of fast relaxing systems at low energies, short-time specific heat measurements can exhibit an apparent gap in the TLS spectrum. Otherwise, it is evident that the power-law asymptotics from Eq. (30) describes well Eq. (29) at the temperatures of a typical experiment. 

This procedure is applicable if the relaxation between the spin states is fast (t<1 X 10 s) and thus the quadrupole doublets of the two spin states collapse into one. It is found that, in the intermediate temperature range, the widths of the two lines are significantly enlarged. This shows that the assumption of fast relaxation strictly does not apply. In spite of this, the areas of the lines ean be well reproduced within the Debye model employing the same Debye temperature for both spin states, p 123 K. 

In the low-field condition, the quantization axis is defined by the EFG main component In this situation, and rj can both be determined from powder spectra when recorded in an externally applied field. Figure 4.14 shows simulated spectra as is often encountered in practice such as in applied-field measurements of diamagnetic compounds or fast-relaxing paramagnetic compounds at high temperatures. The simulated traces differ in detail from a single-crystal spectrum as shown in Fig. 4.13, but their features still correlate in a unique manner with rj and the sign of... 

Powder spectra of paramagnetic compounds measured with applied fields are generally more complicated than those shown in Fig. 4.14. Large internal fields at the Mossbauer nucleus that are temperature- and field-dependent give rise to this complication. If, however, the measurement is performed at sufficiently high temperature, which is above ca. 150 K, the internal magnetic fields usually collapse due to fast relaxation of the electronic spin system (vide infra, Chap. 6). Under... 

Computation of Mossbauer Spectra in Slow and Fast Relaxation Limit... 

Fig. 6.5 Mossbauer spectra of NH4Fe(S04)2-12H20 at 4.2 K and with the indicated magnetic fields applied parallel to the y-ray direction. The lines indicate fits in accordance with a theoretical relaxation model [19, 29]. The bar diagrams indicate the theoretical line positions in the case of infinitely fast relaxation. (Adapted from [29] copyright 1973 by Springer-Verlag)...

Thus, in the case of fast relaxation between the two canted states the total average magnetic field, which is experienced by the nucleus, is given by [84, 85]... 

Anti-Stokes picosecond TR spectra were also obtained with pump-probe time delays over the 0 to 10 ps range and selected spectra are shown in Figure 3.33. The anti-Stokes Raman spectrum at Ops indicates that hot, unrelaxed, species are produced. The approximately 1521 cm ethylenic stretch Raman band vibrational frequency also suggests that most of the Ops anti-Stokes TR spectrum is mostly due to the J intermediate. The 1521 cm Raman band s intensity and its bandwidth decrease with a decay time of about 2.5 ps, and this can be attributed the vibrational cooling and conformational relaxation of the chromophore as the J intermediate relaxes to produce the K intermediate.This very fast relaxation of the initially hot J intermediate is believed to be due to strong coupling between the chromophore the protein bath that can enable better energy transfer compared to typical solute-solvent interactions. ... 

D is the dipole-dipole interaction between the slow relaxing carotenoid radical and the fast relaxing Ti3+ ion r is the interspin distance... 

Fig. 32. NSE spectra of labelled cross-links in a four-functional PDMS network at four different Q-values. Included is a common fit with Eq. (63). The shaded area displays the time independent EISF part in the spectra. Note that the spectra do not approach 1 for t 0. This is related to a fast relaxation process of the deuterated network strands which has not been substracted. (Reprinted with permission from [84]. Copyright 1988 The American Physical Society, Maryland)...

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