Band decomposition

Figure 8 Inelastic neutron scattering spectrum of the KHCO3 crystal at 4 K in the "tunnelling" region and band decomposition into Gaussian profiles.

However, for signals/spectra rich in high-frequency components, this decomposition scheme may not be satisfactory. A full decomposition over the high frequency components may be preferable. The one-dimensional 2-band decomposition scheme can be viewed as a full two-way frequency-time tree. 

In the molten form, the three bands at 1463, 1441, and 1418 cm , observed in the highly crystallized forms of polyethylene, change to a broad asymmetrical band with its maximum at 1440 cm In a similar manner, the narrow band at 1296 cm of crystalline polyethylene is broadened and shifted to the band at 1303 cm in the melt. Finally, the bands at 1130 and 1064 cm disappear and coalesce to form a single weak, broad band centered at 1080 cm [34], The band decomposition of the Raman spectrum from a partially crystallized polyethylene produces a fractional sum of the crystalline and amorphous forms of polyethylene, as illustrated in Fig. 14. 

Figure 8.9 INS spectrum of KHCOj at 4 K in the tunnelling region and band decomposition into Gaussian profiles. Reproduced with permission from Chem. Phys. 124, F. Fillaux, J. Tomkinson and J. Penfold, Chem. Phys., Proton Dynamics in the Hydrogen Bond. The Inelastic Neutron Scattering Spectrum of Potassium Hydrogen Carbonate at 5K, 425, Copyright (1988) Elsevier.

Figure 8.21 Left the INS tunnelling spectrum of lithium acetate dihydrate at 1.5 K and band decomposition. Reproduced with permission from Chem. Phys. 290, B. Nicolai, A. Cousson and F FUlaux, The effect of methyl deuteration on the crystalline structure in Lithium acetate dihydrate, 101, Copyright (2003) Elsevier.

In measured infrared spectra of many samples, especially organic materials, a large number of bands are usually present. Many overlap each other to differing extents, and weak bands are sometimes buried beneath intense bands. By looking at such spectra, it is often difficult to determine precisely the true number of existing bands and their intensities. To solve these difficulties, at least to a certain extent, the methods of difference spectroscopy, derivative spectroscopy, Fourier self-deconvolution (FSD), and band decomposition (curve fitting) have been developed. 

Figure 6.8 Band decomposition for the infrared spectrum of n-butyl stearate in the CH stretching region, (a) Measured spectrum, (b) spectrum synthesized from the six decomposed bands in (c), and (c) decomposed six bands.

Mixtures of components, that caimot be physically separated but whose molar fractions can be changed under a number of factors are considered as undefined. Such mixtures cannot be analyzed by means of classical spectrophotometric analysis (lack of calibration as shown above) and tautomeric mixtures are a typical example. Therefore, there are two approaches to treat tautomeric mixtures presented as a set of spectra with different tautomeric ratios direct quantitative analysis based on overlapping band decomposition or nonlinear optimization based on existing physical relations between the tautomeric constant and the external factor causing the shift in the equilibrium. The first one is the only option to analyze changes caused by the solvent or by salt addition. Both could be used to estimate the effects of temperature, acidity, or concentration and a critical comparison is available in Section 2.2.3 in this respect... 

As a result, the parameters of the individual bands constituting the pure spectra of the tautomers as well as the molar fractions of the tautomers in each solution are obtained. The problem with the assignment of the individual bands to the tautomers is solved by a tree-like optimization procedure [30]. The simultaneous band decomposition of the whole set of spectra increases the convergence of the optimization procedure and makes possible the analysis of tautomeric systems with a slight shift in the position of the equilibrium. 

Band decomposition of each absorption spectram from Figure 2.1a into individual Gaussian bands and determination of their areas, according to... 

At first sight, this approach is very easy to perform, but there are two possible pitfalls in the last two steps, namely the effects of the integration limits (Xj -X2) and the band decomposition interval (V-X"). The band areas (called also integral intensities [17]) can be easily calculated according to Eq. (2.35) and then used for band assignment, but this equation cannot be applied in the transformation of a spectrum into area. The problem is that Eq. (2.35) assumes integration from —00 to +00, while the spectra are recorded and decomposed in a limited interval X k . 

Figure 2.3 Molar fraction of 2b in ethanol as a function of the band decomposition interval (X -X" (solid line-) 240-600 nm (dashed line) 290-600 nm (open circles) 340-600 nm) and limits of integration (varying X, with Xj = 600 nm).

On the other hand, the molar fraction must he constant in the particular solution and this is the fact when Xj is in the interval 320-360 nm, that is, statistically meaningful result can be obtained when there is a plateau. The narrowing of the decomposition interval, as shown in Figure 2.3, leads to narrowing of this plateau or even to the impossibihty to obtain statistically valid results for molar fractions. The latter happens when the band decomposition procedure is apphed only in the spectral interval 340-600 nm. Therefore, when this algorithm for quantitative analysis of tautomeric mixtures is apphed, the band decomposition interval must be as wide as the available spectral instrument and the used solvent would allow. 

From the point of view of quantitative analysis, these spectra are not suitable for simultaneous band decomposition by Fqs. (2.25) and (2.26), because the substantial reduction of the temperature causes narrowing of the vibronic bands, that is, the individual spectra of the tautomers are not temperature-independent. As a result, the spectra from Figure 2.6 were processed using the approach described... 

The process of biprotonation can be treated in a similar way [47], leading to the observation that the biammonium band (8b) at 320 nm decreases slightly, while the ammonium-azonium band (8a) at 520 nm increases with increase in water content The calculated tautomeric constants for the equilibrium 8b tr 8a (0.53 and 0.88 for 10% and 30% water) show that the ammonium-azonium form, which is strongly polar and more acidic, is favored by the increase of water in the solvent composition. As a final result, omitting chemical details, within several steps this compHcated four-component tautomeric equilibrium has been analyzed using band decomposition techniques. 

Comparison of K and I band decompositions yield good agreement. The E band amplitudes of the even Fourier components are somewhat higher, because dust attenuates the spiral arm crest in the I band images. This comparison also shows that the measured non-axisymmetries are not a result of the data analysis process. 

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