The relaxation theory

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

The relaxation theory used in the Appendix to describe the principle of TROSY clearly tells us what to expect, but it is always a little more satisfying if one can obtain a simple physical picture of what is happening. We consider a system of two isolated scalar coupled spins of magnitude %, 1H (I) and 15N (S), with a scalar coupling constant JHN. Transverse relaxation of this spin system is dominated by the DD coupling between spins XH and 15N and by the CSA of each individual spin. The relaxation rates of the individual multiplet components of spin 15N are now discussed assuming an axially symmetric 15N CSA tensor with the axial principal component parallel to the 15N-XH vector as shown in Fig. 10.2. 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

The combination of the modified Solomon-Bloembergen Eqs. (7-11) with the equations for electron spin relaxation (14-16) constitutes a complete theory to relate the observed paramagnetic relaxation rate enhancement to the microscopic properties, and it is generally referred as to Solomon-Bloembergen-Mor-gan (SBM) theory. Detailed discussions of the relaxation theory have been published [13,14]. 

The relaxation theory of vitrification. I. Solution and investigation of the basic equation. Sov. Phys. Techn. Phys. 26, 2204 —2222 (1956). Traduction anglaise American Institute of Phys. Inc. New York, J. Techn. Phys. Acad. Sci. URSS. 1, 2138-2156 (1957). 

Effects of addition of n-tetradecyltrimethylammonium bromide (C14TAB) on the micelle-monomer exchange processes of /f-decyltrimethylammonium bromide (CioTAB) were investigated by the ultrasonic relaxation method. The relaxation frequency increased and the relaxation strength decreased with increasing amount of CuTAB added. The dependence of the relaxation frequency on the amount of CuTAB added was in fair agreement with the relaxation theory of Annianson for mixed micelle... 

The primary key to this successful cowork (Figure 1) is the matching of time scales, accessed using MD simulations and NMR relaxation experiments. Correlation times (characteristic motional time constants) for translational, angular rotational and reorientational motions are a few of the basic components in the relaxation theories. These quantities are standard dynamical properties, obtained in MD simulations. The real gain in using MD is that it can be used to calculate not only the various correlation times, but even the entire correlation functions, whose shapes and other characteristic features are a very rich source of informa-... 

Interestingly, the amplitude of spectral modulation increased toward higher wind speed and wave number (Fig. 3). This trend is opposite to the prediction of the relaxation theory that assumes a monotonic increase in... 

In the free-volume model,/ becomes the dominant structural mode of the relaxation theories. The rearrangement of the cage structure requires diffusion, which is slowly being frozen out. As T approaches 7, p can no longer follow its equilibrium value, but becomes frozen at a value p>p. Since the variation in p no longer contributes to the heat capacity, decreases. The relevant contribution to arises from the communal entropy, which depends primarily on p and ceases to change. However, at temperatures above 7, the structure can equilibrate rapidly compared to the measurement time and both p and approach their equilibrium value. 

Both the phase shift and the amplitude of the dynamic component are used for the calculation of a complex (frequency-dependent) heat capacity. These quantities can be interpreted in the context of the relaxation theory or irreversible thermodynamics. 

In the present paper the equation derived on the basis of the relaxation theory and the surface thermodynamics will be checked by the experimental results. 

In light of the similarity between the relaxation theory for friction and that for adhesion we may use the analogy of an activated complex to describe the bond-breaking and the bond-forming processes as suggested by Hatfield and Rathmann. In the free-energy... 

This treatment is closely related to the relaxation theory of Wangsness and Bloch. The U functions are further simplified by examining, for example,... 

Before discussing some raultiexponential relaxation functions, we will consider some specific aspects of theoretical methods describing the relaxation phenomena in nuclear magnetic resonance. Various methods have been proposed in the literature since the work of Bloembergen, Purcell, and Pound (1). Many of them have some common features like the use of density matrix theory, correlation functions, and their associated spectral density functions. People interested in the foundations of these theories will find some excellent book chapters and review articles in the literature (7-9,21). Slichter s book (Chapter 5) contains a very good introduction to the density matrix which is of crucial importance in all these theories (3). On the other hand, Abragam s book (2) remains a "Bible" in this field. The more recent book of Lenk (5) contains very concise definitions of many terms and concepts widely used in all the relaxation theories. Moreover, this author presents an excellent overview of many recent theories, especially those based on irreversible thermodynamics ... 

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