Lorentz band shape

To obtain the absorbances at 910 and 967 cm. 1, it was necessary to correct the observed band intensities for the overlapping of adjacent bands. The band at 910 cm."1 for the vinyl group was corrected for the absorbance from the wing of the 967-cm."1 frarw-vinylene band,. and the latter band was corrected for the vinyl band at 995 cm. 1. The Lorentz band shape equation was used to calculate the absorbance in the wings, and in the thicker specimens, successive approximations were necessary. This treatment gave the four equations below, which yielded the concentrations of trans and vinyl groups for the emulsion and sodium polybutadienes listed in Table I. Implicit in these equations is the assumption that the absorptivities are independent of concentration. 

If the absorption band in question is overlapped by neighboring bands, it is clear that the determination of the true values of Dn (v0) and Da(v0) can be ambiguous. In such cases it is necessary to resolve the band of interest from the complex absorption of which it is a part. This is usually done by a graphical analysis, which assumes symmetrical bands of a Lorentz shape centered at the absorption maxima and simple summation of these to give the observed spectrum. The results of such a resolution are of course subject to the uncertainty of the true band shape and width, so that an indeterminate, if nevertheless small, error can be introduced in this manner. 

The default setting is Lorentz, i.e. a pure Lorentzian function. A single click on the upper arrow key switches immediately to a pure Gaussian function. The next click on the same arrow sets the peak to Baseline. If, beginning again with the Lorentzian type, the down arrow is clicked on instead, the band shape changes to 100% Lorentz + Gauss. In principle this band... 

The experimental structure below 1 eV in Figures 1.10 and 1.11 corresponds to the free-electron behavior typically observed in metals. The lowest interband absorption starts to occur above 1.5 eV. In the non-noble (Cu) and noble metals (Ag, Au), Cooper et al. [18] indicated that the sharp rise in 2 at the lowest interband absorption edge is due to the fact that the transitions are from a very flat lower band to the Fermi surface and are not of the critical-point type. More strictly, these transitions occur between occupied states in band 5 and unoccupied states in band 6 as these cross the Fermi surface (5 —> 6 (Ep)) at the L point in the Brillouin zone. Here, the bands are numbered starting from the lowest band at a given k. Relatively poor agreement between the Lorentz line shape and the experiment observed at -1.5 eV in Figiues 1.10 and 1.11 may reflect this fact... 

Now let s consider the part played by the slit width in determining the shape and intensity of an absorption band, i,e, its effect. We shall assume that the sample band is of the Lorentz curve shape and has a half band width of 8 cm (Ai ). We shall also assume that the slit width is 1 cm (Ay/) and follows a triangular function. In this case the band as seen by the instrument will have a half band width (Ayi/ ) approximately equal to Ay, and the band will be of the Lorentz shape. Here the spectrophotometer accurately sees the band shape (see Fig. 9). 

Figure 1 Experimental (A,C) and calculated (B,D) absorption (A, B) and VCD (C, D) spectra of (1R, 4/ )-(+)-camphor. The resolution of the experimental spectra was 4 cm-. Calculated spectra were obtained using DFT, B3PW91 and 6-31G. Band shapes are Lorentz-ian (half width at half height 4 cm- ). Fundamental modes are numbered.

The complex spectral structure from 750 to 650 cm."1 was resolved mathematically by Binder into a series of overlapping bands of the theoretical Lorentz shape (3). It was shown that only the band at 740 cm."1 originated in the cis-olefin group. However, measurements on... 

The interpretation of band progressions by the time dependent procedure is therefore identical with the Franck-Condon analysis and, in the low temperature limit, to the method of molecular distributions as well. The line shape function obtained on the basis of Eq. (52) (for E = hv) differs under this condition from that of Eq. (12) only in the line shape function of each vibrational member in the progression which in Eq. (52) is the delta function and in Eq. (12) has a Lorentz type distribution. 

For liquids and solutions, p and a in the preceding equations are replaced by M (molar concentration) and e (molar absorption coefficient), respectively. However, the extrapolation method just described is not applicable, since experimental errors in determining B values become too large at low concentration or at small cell length. The true integrated absorption coefficient of a liquid can be calculated if we assume that the shape of an absorption band is represented by the Lorentz equation and that the sht function is triangular [96]. 

The right-hand side of this equation represents a curve centering about v = 0, which is shown in Figure D.2b. This curve is called the Lorentz profile which is known to fit approximately the shapes of infrared absorption bands. The integrated area under this curve is 3t, and its full width at half maximum (FWHM) is 2a. 

As the line shape function f v), a Gauss, Lorentz or Voigt function can be chosen. The mathematical expressions for these functions have v as the variable and have two parameters the band center Vq (wavenumber at which the function reaches its maximum value) and the curve width r (half width at half height). Sometimes A (half width at height 1/e) is used instead of F. The relation between both quantities is A = (In F. All these line shape functions are symmetric around the band center vo- The expressions for the functions are ... 

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