Local fluctuations

We now come back to the important example of two spin 1/2 nuclei with the dipole-dipole interaction discussed above. In simple physical tenns, we can say that one of the spins senses a fluctuating local magnetic field originatmg from the other one. In tenns of the Hamiltonian of equation B 1.13.8. the stochastic fiinction of time F l t) is proportional to Y2 (9,( ))/rjo, where Y, is an / = 2 spherical hannonic and r. is the... 

In conclusion the ionic interactions between the SR and PM are responsible for fine-tuning the smooth muscle Ca2+ signal in terms of fluctuating local cytoplasmic Ca2+ gradients and repetitive Ca2+ waves. 

In spin-lattice relaxation, the excited nuclei transfer their excitation energy to their environment. They do so via interaction of their magnetic vectors with fluctuating local fields of sufficient strengths and a fluctuation frequency of the order of the Larmor frequency of the nuclear spin type. Depending upon the atomic and electronic environment of a nucleus in a molecule and the motion of that molecule, there are five potential mechanisms contributing to spin-lattice relaxation of the nucleus. 

Thus, the C —H nuclei in the para position of monosubstituted benzene derivatives relax faster than those in the ortho or meta positions (Table 3.16 [151]). The reason for this behavior lies in a preferred rotation about the molecular axis passing through the substituent X and the p-carbon. During this motion, the para C —H bond does not change its direction relative to the field B0 fluctuating local fields can only arise at the p-C nucleus by rotations of the molecule perpendicular to the preferred axis. However,... 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

Compared with experiment7 (Fig. 55) it seems that the 1 j g, 3ag and 1 b outer valence holes become shifted by 1-1.5 eV and the 2blu (2 a ) inner valence hole by more than 2 eV due to fluctuation/localization, precisely along the lines previously discussed for N2 and QH2. Concerning the 2ag inner valence region, it looks very similar to the 2ae... 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

In NMR work, spin-lattice relaxation measurements indicated a non-exponential nature of the ionic relaxation.10,11 While this conclusion is in harmony with results from electrical and mechanical relaxation studies, the latter techniques yielded larger activation energies for the ion dynamics than spin-lattice relaxation analysis. Possible origins of these deviations were discussed in detail.10,193 196 The crucial point of spin-lattice relaxation studies is the choice of an appropriate correlation function of the fluctuating local fields, which in turn reflect ion dynamics. Here, we refrain from further reviewing NMR relaxation studies, but focus on recent applications of multidimensional NMR on solid-ion conductors, where well defined correlation functions can be directly measured. 

In this expression, y is the gyromagnetic ratio and is the root-mean-square average of the X component of the fluctuating local field. Note that wo replaces o) in the expression for the relaxation time, which, under the extreme-narrowing condition, becomes 27 [b2l] Tc-Thus, the relaxation time T decreases as Tc increases (i.e., as mobility decreases with, for example, larger molecules or lower temperatures). This regime is depicted for Tj at the left of Figure A5-2 note that Ti is independent of the resonance fi equency coq here. 

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