Fourier transformation dispersion curves

Halle et al. 53) have proposed to describe dispersions such as the one shown in Fig. 13 by a set of unambiguously defined parameters, which are able to fully characterize the whole curve provided that the two plateaus (at low and high frequencies) are clearly visible. Let us denote by C t) the correlation function (see (24)) which, after Fourier transformation, leads to the... 

Whenever the same parameters are available from two different curves (e.g., wq aiid t from Figure 1 or Figure 4a), there is some mathematical relation between the curves. For the "linear" system we have considered (i.e., displacement is proportional to driving amplitude Fq) the time-domain and frequency-domain responses are connected by a Fourier transform. Similarly, absorption and dispersion spectra both yield the same information, and are related by a Hilbert transform (see Chapter 4). In this Chapter, we will next develop some simple Fourier transform properties for continuous curves such as Figures 1-4, and then show the advantages of applying similar relations to discrete data sets consisting of actual physical responses sampled at equally-spaced intervals. 

Fourier Transformation of Dispersion Curves interplanar Force Constants j 181... 

The dispersion curves provide many useful data that can be extracted by the Fourier transformation of these curves. A basic design formula (see derivation of the formula in Appendix D) can be expressed as... 

The use of Fourier Transform instruments eliminates much of the limitations of the EMIRS, since no more potential modulation is needed. The signal-to-noise ratio is far less than the dispersive instrument and can be improved statistically by adding more scans, since the spectral acquisition time is much lower. With the Fourier Transform equipment, also the irreversible processes can be studied, since it is no longer required to return to the same potentials as for modulation (see Sect. 3.4.4). This not only allows the acquisition of derivative or bipolar bands but also the acquisition of integral bands, as in the case of the adsorbed CO on platinum electrodes [20-25], which was impossible with EMIRS. The speed of spectral acquisition of Fourier Transform Infrared (FTIR) instruments allows also the follow-up of a reaction during a dynamic polarization curve [26, 27]. 

In Fig. 40, the P r) vs. r distance distribution functions are presented that were calculated from the scattering curves by inverse Fourier transformation [102,108,109]. It can be clearly seen that the numerical values of the functions indicate the differences in the density of aggregates in the liquid mixtures of different polarities. The interparticle interactions are thus regulated via the selective liquid sorption process on the disperse particles, and it can be established that the interfacial layer composition, the layer thickness, and the heat of wetting are crucial factors for the stability of coUoidal dispersions in nonaqueous liquids. 

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