Line shape Voigt profile

The line shapes are described by Voigt functions, which reflect the Lorentzian line profiles due to natural line width and Gaussian profiles due to Doppler broadening. The instrumental broadening by the rocking curve of the crystal, de-focusing and the finite resolution of the detector is described well by a Voigt profile shape too [3[. 

Similar to a Voigt profile, this line shape expression is a convolution of Equation (11) and a Gaussian distribution to account for inhomogeneous broadening [12,31]. This profile is the best profile currently employed for the analysis of spectra exhibiting significant overlap of modes of different phases. 

ESR lines in solution can almost always be approximated by a Lorentz function. In the solid state the line-shape can in general be reproduced by a Gauss curve. In some instances a so-called Voigt profile can give a better approximation to the experimental line-shape. A Voigt line is a convolution of a Lorentz and a Gauss line. The shape is determined by the ratio ABi/ABg of the respective line-widths. The shapes of the 1st derivative lines of these types are given in Fig. 9.1. 

The line-shape of an experimental spectrum can in principle be determined by the procedure illustrated in Fig, 9.2. The 2nd derivative of the resonance line is then recorded. For a Gauss line the ratio hi/h2 between the minimum and maximum amplitudes of the 2nd derivative (Fig. 9.2(a)) equals 2.24 [18], while for a Lorentz shape it approaches the value 4. The hi/h2 ratio for a Voigt profile varies... 

Note that observed Une shapes may not be purely Lorentzian or Gaussian when more than one broadening mechanism contributes in the interaction. The combinations of Lorentzian and Gaussian line-shape functions can normally be approximated by a so-called Voigt profile. 

Fig.3.9. Voigt profile as a convolution of Lorentzian line shapes L o) ) — coi) with coi o(l + %/c)...

Fig. 3.6.1 Collision, Voigt, and Doppler line shapes. All three are shown with the same maximum amplitude, and with the same width at half maximum amplitude. The Voigt line shape is one of a continuum of profiles between Gaussian and Lorentzian, depending on the degree of collision broadening.

The Voigt line shape closely describes line profiles measured undermost laboratory and atmospheric conditions. 

Fig. 13 Isotopic line splitting of the V3 stretching vibration in single crystalline (see also Fig. 12(a)), after [108, 109], The origin of each absorption band is indicated by an isotopomer present in crystals of natural composition. While the absorption could be fitted by a Lorentzian band profile, the remaining peaks were dominated by the Gaussian contribution in the Voigt band shapes (solid lines below the spectrum). The sum result of fitting the isotopic absorption bands is inserted in the measured spectrum as a solid line...

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Figure 2.3 shows Gauss and Lorentz profiles of equal area and FWHM as well as the resulting Voigt distribution. While the Lorentz portion dominates at the line wings, the Gauss portion determines the shape in the line core. 

Figure 2.12 Calculated transmittance profiles for Voigt-shaped absorption lines with maximnm absorbance valnes of (A) 0.03, (B) 0.3, and (C) 3, assuming 3 % stray light level...

Figure 2.13 Calculated relationship between wavelength-integrated absorbance and peak (maximum) absorbance in the line center for Voigt-shaped absorption lines and a rectangular instrument profile curve parameter AAinsu, assumed stray light level 3 %...

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

Fig.8.7. Comparison of the shapes of the Lorentzian distribution T, the Gaussian distribution and the Voigt line profile for which T = A(ln 2) Profiles are normalized to the same peak intensity and have the same half width.

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