Relaxation formalism

Figure 8 Temperature-dependent Mossbauer spectra of metallic iron nanoparticles in zeolite NaX (a). The superparamagnetic blocking temperature Ti is aroimd 40 K. The solid lines have been calculated by a relaxation formalism assuming a hyperfine field distribution as shown in (b) with a bimodal size distribution of metallic iron particles. (Reprinted from Schiinemann, Winkler, Butzlaff and Trautwein. With kind permission from Springer Science Business Media)...

Rast et al. studied the mechanisms of the intermolecular NMR relaxation dispersion of the tetramethylammonium protons in Gd " heavy-water solutions. The standard dipolar nuclear relaxation formalism of Solomon-Bloembergen, valid for the used frequency range between 10 and 800 MHz, leads to overall good agreement with the measured data without any adjustable parameters. Madhu et al. reported experimental observation and numerical simulation of a two-dimensional multiplet effect in the heteronuclear correlation spectrum of a paramagnetic protein that depends on molecular geometry. Bertini et performed solution structure calculations through self-... 

The suggested scheme for closing the fluctuation energy conservation equation is reminiscent of methods specific to the relaxational formalism of the thermodynamics of irreversible processes [27,28]. This scheme could be brought into play empirically on the basis of this formalism. In essence, this scheme enables us to... 

While the model can be formulated for three-dimensional stresses and strain states, here only the imiaxial deformation is dealt with. It is important to add that the model has both a creep formalism and a relaxation formalism. The relevant equations are presented in the next paragraphs. In addition, although this is a nonlinear theory, it is a small strain theory... 

The relaxation formalisms of dipolar-coupled two-spin 1/2 systems (protons) on the one hand, and of quadrupolar-coupled spin 1 nuclei (deuterons) on the other, have very much in common and lead to largely equivalent analytical expressions [2, 17]. This very much facilitates comparisons of experimental results obtained with either technique. 

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

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In phenomenological terms, the presence of a distribution of traps is equivalent to an increase of the number of tail states of the DOS and, concomitantly, opens additional pathways to the relaxation of charge earners towards sleeper states. A zero order analytic description of the effect can be based upon the Hocsterey and Letson [75] formalism. The latter is premised on the argument that the carrier mo-... 

There exists one more formally strict method for determining yield stress. It is based on the fact that during relaxation, the stress x(t) cannot become lower that yield... 

Let us compare in detail the differential theory results with those obtained for rotational relaxation kinetics from the memory function formalism (integral theory). Using R(t) from Eq. (1.107) as a kernel of Eq. (1.71) we can see that in the low-density limit... 

When other relaxation mechanisms are involved, such as chemical-shift anisotropy or spin-rotation interactions, they cannot be separated by application of the foregoing relaxation theory. Then, the full density-matrix formalism should be employed. 

Ideas concerning the ionic atmosphere can be used for a theoretical interpretation of these phenomena. There are at least two effects associated with the ionic atmosphere, the electrophoretic effect and the relaxation effect, both lowering the ionic mobilities. Formally, this can be written as... 

The Mossbauer spectra of the complex [Fe(acpa)2]PF6 shown in Fig. 26 have also been interpreted on the basis of a relaxation mechanism [168]. For the calculations, the formalism using the modified Bloch equations again was employed. The resulting correlation times x = XlXh/(tl + Xh) are temperature dependent and span the range between 1.9 x 10 s at 110 K and 0.34 x 10 s at 285 K. Again the correlation times are reasonable only at low temperatures, whereas around 200 K increase of the population of the state contributes to... 

Upon illumination, photons having energy higher than the band gap (eg = ec — v) are absorbed in the semiconductor phase and the electron-hole-pairs (e //i+) are generated. This effect can be considered equivalent to the photoexcitation of a molecule (Fig. 5.57) if we formally identify the HOMO with the ec level and LUMO with the v level. The lifetime of excited e //i+ pairs (in the bulk semiconductor) is defined analogously as the lifetime of the excited molecule in terms of a pseudo-first-order relaxation (Eq. 5.10.2). 

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

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