Band-fitting procedure

Fig. 6. FTIR spectrum of a solution-crystallized polyethylene (NIST standard SMR1482, Mw = 1.36 X 10 ) in the CH2 rocking and bending (scissoring) regions. Pictured are the crystalline and amorphous components in each of these regions, as resolved using a band fitting procedure. From Ref. 38.

Spectra of s.o. samples differed markedly from those of a.p. samples and were unaffected by a subsequent evacuation up to 673 K (Fig. 4, a). Spectra consisted of a composite envelope of heavily overlapping bands at 980-1070 cm-, with two weak bands at 874 and 894 cm-. Irrespective of the preparation method, the integrated area (cm- ) of the composite band at 980-1070 cm- was proportional to the V-content up to 3 atoms nm-2. An analysis of spectra by the curve-fitting procedure showed the presence of several V=0 modes. The relative intensity of the various peaks contributing to the composite band depended only on the V-content and did not depend on the method used for preparing the catalysts. Samples with V > 3 atoms nm-2 R-spectra features similar to those of pure V2O5 (spectrum 8 in Fig. 4, a). 

In order to extract the contributions and dynamics of the ketyl radical and fluoranil anion from the TR spectra obtained with the 416 nm probe wavelength, a deconvolution of the Raman bands were done using a fitting procedure employing a Lorentzian lineshape for the Raman bands of the two intermediates. Figure 3.20 shows a comparison of the best-fit (lines) to the experimental TR spectra (dots) in the left-side spectra and the deconvolution extracted from this best fit for the ketyl radical spectra... 

Fig. 22. FTIR (upper panel), isotropic, and anisotropic Raman spectra (Aexc = 457 nm) of trialanine measured at the indicated pD. The solid lines and the band profiles arise from the fitting procedure described in the reference. From Schweitzer-Stenner et al., (2001)./. Am. Chem. Soc. 123, 9628-9633, 2001, Reprinted with permission from American Chemical Society.

In principle, valence band XPS spectra reveal all the electronic states involved in bonding, and are one of the few ways of extracting an experimental band structure. In practice, however, their analysis has been limited to a qualitative comparison with the calculated density of states. When appropriate correction factors are applied, it is possible to fit these valence band spectra to component peaks that represent the atomic orbital contributions, in analogy to the projected density of states. This type of fitting procedure requires an appreciation of the restraints that must be applied to limit the number of component peaks, their breadth and splitting, and their line-shapes. 

These factors are used in the equations given in Table I. The computation requires only that the variance ratios be accurately known. The absolute precision of the method may change from day to day without affecting the validity of either the least-squares curve-of-best fit procedure or the confidence band calculations. (It is not practical to regularly monitor local variances, and errors may develop in variance ratios. Eowever, the error due to incorrect ratios will almost always be much less than the error due to assuming constant variance. Even guessed values of, say, S a concentration are likely to yield more precise data.)... 

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

The fitting procedure in the OH bending region revealed that the band positions were relatively stable as the temperature increased. Moreover, the band near 1730 cm did not diminish in intensity as the temperature was raised which indicates that it too is a structural band as are the 1930 and 1823 cm bands. The areas of the OH bending bands for the unmilled and 10 min milled samples are shown in Figure 7. 

Due to the difference in the number of probe photons required to ionize the clusters from each state, careful study of the ion signal probe power dependence made it possible to determine the origin of each of the observed components. The X, X2 component is the result of the two photon excitation of the S02 F band. The X3 decay component is due to the one photon excitation of the coupled 1A2, Bi states. The plateau is believed to be due to ion-state fragmentation of larger clusters and does not seem to influence the values of the time constants obtained from the fitting procedure. 

Figure 22. Transient TPI spectra of different vibrational bands of the Na3 C state compared with the fit function f(t) = Noe /T. The listed lifetimes t of the different vibrational bands were obtained by a least-squares fit procedure [22].

The widths of the amide I bands which are associated with the different structures are large compared to the separation of their peak maxima, which is why the amide I absorption of a complex protein consists of several overlapping bands. Deconvolution and derivative techniques have been employed to detect the individual components (Kaup-pinen et al., 1981 Susi and Byler, 1983 Mantsch et al., 1986). Quantitative estimates for different conformations have been obtained by ulterior curve fitting procedures (Byler... 

Due to the rotational structure as well as the so-called hot hand absorptions (Sec. 2.5.3), the contour of a rovibrational band depends on the temperature. Today it is possible to determine molecular constants such as moments of inertia, Coriolis coupling constants, centrifugal distortion constants, and anharmonicity coefficients by FTIR as precisely as possible in order to calculate the intensity and shape of an absorption band. In such a simulation process the temperature may be used as a parameter. The results can be compared to the experimental spectra and the temperature may be deduced by fitting the calculated to the observed bands. This is possible with IR as well as with Raman bands. A review of curve fitting procedures and their limitations has been given by Maddams (1980). 

The shoulder observed in Fig. 5(a) at 2078 cm is assigned to the free CN ejected. The CN anion is only seen as a shoulder, as its extinction coefficient is much weaker than when it is bound within a complex. We determined an integrated absorption coefficient of 840 moF dm cm" for the CN band at 2080 cm" and an integrated absorption coefficient for Fe(CN)g of 97,000 mol dm cm . The integrated absorption coefficient for Fe(CN) was determined as 15,800 moF dm cm . These determinations of integrated absorption coefficients enable us, after deconvolution of the peaks observed thanks to a fitting procedure, to know exactly the concentrations of the different species detected. 

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