Line shape function Voigt

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. 

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. 

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 ... 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

Multiplying this expression by n yields the Voigt function that occurs in the description of spectral-line shapes resulting from combined Doppler and pressure broadening. We elaborate on these phenomena in Section I of Chapter 2. 

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[. 

The line-shape given by (9.6) is a Voigt function that can be evaluated numerically by a standard procedure [81]. For a single inhomogenously broadened line, the transition probability is set to 1 as in a simple two-level system. The relaxation times are given by ... 

Another approach for data processing involves simulation of pure spectra. These model spectra are then taken for a quantitative description of the mixture spectra. This procedure is referred to as indirect hard modelling (IHM). Obviously, changes in line shape, line width, and chemical shift may occur as function of concentration and due to system imperfections which are taken into account by IHM. The peaks are modelled by Voigt-functions with variable Gaussian to exponential ratio. The main advantage of IHM is that it allows a limited physical interpretation of the models. Further, unlike PLS based methods, IHM only requires reference spectra of the pure compounds, reducing the calibration effort drastically. 

The width of the Gaussian function, G(v), is the Doppler width, wj), and that of the Lorentzian function, L( v — v ), is the collision width, lUc. This generalized line shape, introduced by Voigt (1912), has no analytical expression, but it can easily be computed numerically using the convolution represented in Eq. (3.6.8). Its shape is shown in Fig. 3.6.1. An empirical expression relating wd and Wc to the total Voigt width, wm, has been introduced by Whiting (1968),... 

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. 

Fig. 9.1 (a) Gauss, Lorentz and Voigt (ABl/ABq = 1.0) lines as 1st derivatives adjusted to the same peak-peak amplitudes, (b) Integrated area of derivative lines with the same amplitude as function of the ABl/ABq ratio of a Voigt Une. For ABl/ABq 1 the line approaches a Lorentz shape with an area which is 3.51 times larger than that of a Gaussian with the same amplitude [13]... 

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). 

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