Exponential relaxation analysis

In the case of hi- or multi-exponential relaxation curves the treatment involved can be rather complex (119-123). It becomes even more problematic. Needles to say, the same is true for systems with suspected continuous distributions of relaxation rates, whose evaluation by numerical analysis of the decay curves (124-128) represents one of the most arduous mathematical problems (124-128). In general, evaluation tasks of this kind need to be treated off-line, using specific programs and algorithms. 

T2-relaxation analysis may be used to study the effect of ingredient composition on the properties of water in whippable emulsions16. In food systems non-exponential relaxation curves are often found. This can be accounted for by the presence of 2, 3 or more recognizable components representing species of hydrogen atoms with different mobility51. 

Equation 4.35 shows that the concentration deviations based on a linearization analysis of the rate laws in Eqs. 1.54a and 1.54c will decay to zero exponentially ( relax ) as governed by the two time constants, r, and r2. These two parameters, in turn, are related to the rate coefficients for the coupled reactions whose kinetics the rate laws describe (Eqs. 4.36c-4.36e and 4.38). If the rate coefficients are known to fall into widely different time scales for each of the coupled reactions, their relation to the time constants can be simplified mathematically (Eq. 4.39 and Table 4.3). Thus an experimental determination of the time constants leads to a calculation of the rate coefficients.20 In the example of the metal complexation reaction in Eq. 1.50, with the assumptions that the outer-sphere complexation step is much faster than the inner-sphere complexation step and that dissociation of the inner-sphere complex is negligible (k b = 0 in Eq. 1.54c), the results for tx and r2 in the first row of Table 4.3 can be applied. The expression for tx indicates that measurements of this parameter as a function of differing equilibrium concentrations of the complexing metal and ligand will produce a straight line whose slope is kf and whose y-intercept is kb. The measured values of l/r2 at these same two equilibrium concentrations then lead to a calculation of kf. 

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. 

Kohlrausch (1847) used this analysis to explain his original observation of stretched exponential relaxation. 

The theoretical developments [95,129,148] revealed that the exponential relaxation is influenced by the kinetics of micellization, and from the data analysis we could determine the rate constant of the fast process, k. The observation of different kinetic regimes for different surfactants and/or experimental methods makes the physical picture rather complicated. 

Bertero, M., Boccacci, P., Pike, E. R., On the Recovery and Resolution of Exponential Relaxation Rates from Experimental Data A Singular-value Analysis of the Laplace Transform Inversion in the Presence of Noise, Proc. R Soc. London, Ser. A., 1982, 383, 15-29. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Exponential decay often occurs in measurements of diffusion and spin-relaxation and both properties are sensitive probes of the electronic and molecular structure and of the dynamics. Such experiments and analysis of the decay as a spectrum of 7i or D, etc., are an analog of the one-dimensional Fourier spectroscopy in that the signal is measured as a function of one variable. The recent development of an efficient algorithm for two-dimensional Laplace inversion enables the two-dimensional spectroscopy using decaying functions to be made. These experiments are analogous to two-dimensional Fourier spectroscopy. 

In order to be able to evaluate data with a reasonable number of parameters, the mode analysis assumes, as a first approximation, that the exponential correlation of the correlations [Eq. (18)] is maintained, and only the relaxation rates 1/tp are allowed to depend on a general form on the mode index... 

Cross-correlation effects between 15N CSA and 1H-15N dipolar interactions [10] will result in different relaxation rates for the two components of the 15N spin doublet, which could significantly complicate the analysis of the resulting bi-exponential decay of the decoupled signal in T, or T2 experiments. To avoid this problem, 180° 1H pulses are applied during the 15N relaxation period [3, 4], which effectively averages the relaxation rates for the two components of the 15N spin doublet... 

In the following, we shall be concerned with the determination of the cross-relaxation rate a, essentially when 7 and S are nuclear spins. The bi-exponential character of the evolution of 7 and Sz (see (13)) and the difficulties associated with the analysis of such evolutions have prompted the design of more direct methods based on the initial behavior after an... 

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