Analysis multi-frequency dynamic

These results clearly indicate that the multi-frequency dynamic analysis method allows us to estimate the contribution of different relaxation mechanisms during curing of elastomers, and the changes in chemical and physical networks densities can be studied separately. 

The main advantage of a multi-frequency study is that it provides information on the frequency dispersion of magnetic resonance parameters. This approach (dispersion), for example, is the power in NMRD studies. Several laboratories pioneered in the application of multi-frequency EPR as a route to a more accurate evaluation of key spectroscopic parameters (g, A, Q, D, E), as well as a more sensitive methodology for studying dynamical processes, where an interplay between the frequency dependence of the spin process and the frequency dependence of the EPR observation often can provide exceptionally detailed information [64,65]. In order to take advantage of the method, the frequency dependence of spin systems must be understood. This has led to the development of several theoretical approaches for better analysis of multi-frequency data, and especially in BPCA research, for the analysis of the frequency dependence of geffective, Tle, T2e, and the overall EPR line shape in frozen glasses and in room-temperature aqueous solutions. 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Experimental NMR data are typically measured in response to one or more excitation pulses as a function of the time following the last pulse. From a general point of view, spectroscopy can be treated as a particular application of nonlinear system analysis [Bogl, Deul, Marl, Schl]. One-, two-, and multi-dimensional impulse-response functions are defined within this framework. They characterize the linear and nonlinear properties of the sample (and the measurement apparatus), which is simply referred to as the system. The impulse-response functions determine how the excitation signal is transformed into the response signal. A nonlinear system executes a nonlinear transformation of the input function to produce the output function. Here the parameter of the function, for instance the time, is preserved. In comparison to this, the Fourier transformation is a linear transformation of a function, where the parameter itself is changed. For instance, time is converted to frequency. The Fourier transforms of the impulse-response functions are known to the spectroscopist as spectra, to the system analyst as transfer functions, and to the physicist as dynamic susceptibilities. 

Laskar, J. (1993). Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Physica D, 67 257-281. 

UV, F) (Hi) multi-probe measurements and (iv) dynamic performance monitoring based on multivariate analysis. However, at plant sites there is a tendency to minimise the frequency of on/at-line analyses. It would appear that rather few new applications of on-line spectroscopic techniques (UVWIS, mid-IR, NIR) or LR-NMR are currently under development. 

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