Function Voigt

It is often convenient modeling the peak shape assuming some analytical functions [25]. The most commonly used functions are, at present, the Voigt and pseudo-Voigt functions, a combination of a Gaussian (G) and a Lore-ntzian (L) function centered at 20(y. An expression for Gaussian and Lorentzian contributions is ... 

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

So far no hypotheses are required concerning the true shape of the peak profile. Flowever, in order to avoid or reduce the difficulties related to the overlapping of the peaks, the experimental noise, the resolution of the data and the separation peak-background, the approach most frequently used fits by means of a least squared method the diffraction peaks using some suitable functions that allow the analytical Fourier transform, as, for example, Voigt or pseudo-Voigt functions (4) which are the more often used. 

Another typical problem met in this kind of analysis is known as the hook effect . It is due to an overestimation of the background line to the detriment of the peak tails. As a consequence, the low order Fourier coefficients of the profile are underestimated. In the fitting procedure by pseudo-Voigt functions, this problem occurs if the Gauss content is so high that the second derivative of the Fourier coefficients is negative this is obviously physically impossible because it represents a probability density. 

Voigt function to the 540 cm peak. It is very close to the expected value = 0.34 for a two-body Fe-NO oscillator. The calculated value = 0.39 (B3LYP functional) is in agreement with the experimental one. 

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. 

Lorentzians, Gaussians, and combinations of both like pseudo-Voigt functions 38Frequently the effect of instrumental broadening is tacitly considered as already eliminated. 

The complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

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 function that we obtain is the Voigt function encountered in Section III.C of Chapter 1. With definition of variables and scaling appropriate to the case of combined Doppler, natural, and collision broadening, the Voigt function appears as the probability per unit frequency of observing light at a frequency v ... 

By using the same definitions, the profile for combined Doppler, natural, and collision broadening may be expressed in terms of the Voigt function ... 

The Voigt function can be expressed in terms of the real part of the complex error function,68-71... 

For a Voigt function that is almost Lorentzian, the extent of Gaussian broadening can be visualized by plotting the dispersion of the lineshape, D f) against the absorption, A(f).76,77 For a pure Lorentzian lineshape, a circle is obtained. Hence, the extent of the departure from this circular shape indicates the extent of the Gaussian broadening. 

Using a Taylor series expansion in the TD it can be shown that, to a good approximation,78 the Voigt function can be written as V, a linear combination of Lorentzian, g (f) and Gaussian gad) functions having the same width, W = Wi = Wq, namely... 

Fig. 11. The peak area residual between the Voigt function and the approximated Voigt function is shown as a function of the Voigt parameter, a. This diagram was obtained by replotting the information in S. Bruce, J. Higinbotham, I. Marshall and P. H. Beswick,...

Fitting in the frequency domain is readily visualized graphically and the well-established Levenberg-Marquardt method85 is straightforward to implement. This method is applicable to any lineshape function. For example, Marshall et al 2 have used V ( f), an approximation to the Voigt function, when fitting... 

Fig. 11. (Upper) Splitting of pHe+ states due to magnetic interactions, and observable laser transitions between the F+ and F states according to Bakalov and Korobov [33]. (Lower) Observed hyperfine splitting of the unfavoured laser transition (n, L) = (38,34) —> (37, 35) [16]. The laser bandwidth is 1.2 GHz. The solid line is the result of a fit of two Voigt functions (a Gaussian fixed to the laser bandwidth convoluted with a Lorentzian to describe the intrinsic line width) to the spectrum. The intrinsic width of each lines was found to 0.4 0.1 GHz. From Widmann et al. [16]...

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 most notable exception is the shape of peaks in neutron powder diffraction (apart from the time-of-flight data), which is typically close to the pure Gaussian distribution. Peak shapes in TOP experiments are usually deseribed by a convolution of exponential and pseudo-Voigt functions. 

The mixing coefficient, p for pseudo-Voigt function and the exponent, p for Pearson-VII function, generally vary for a particular powder diffraction pattern. Their behavior is typically modeled with a different empirical parabolic function of tan0 and 20, respectively, as follows from equations... 

In the modified pseudo-Voigt functions described above (Eqs. 2.54 to 2.58), both the Gaussian to Lorentzian mixing parameter (q, Eq. 2.54) and their individual contributions to the total peak width H, Eq. 2.55) are tabulated. This feature may be used to lower the number of free parameters and to obtain more realistic peak shape parameters that are due to the physical state of the specimen. Either or both may be achieved by using one of the following approaches ... 

From this point of view, some applications of the modified pseudo-Voigt function (e.g. third and fourth peak shape functions employed in GSAS) are in a way similar to the fundamental parameters approach as they use instrumental parameters to describe certain aspects of peak shape. 

In some advanced implementations of the modified pseudo-Voigt function, an asymmetric peak can be constructed as a convolution of a symmetric peak shape and a certain asymmetrization function, which can be either empirical or based on the real instrumental parameters. For example, as described in section 2.9.1, and using the Simpson s multi-term integration rule this convolution can be approximated using a sum of several (usually 3 or 5) symmetric Bragg peak profiles ... 

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