Lorentzian linear combination

Additional information can be obtained by XPS (X-ray Photoelectron Spectroscopy). Figure 14.7 shows the N(ls) core level spectra of samples, degassed at 973 K and 1123 K. The most important assignments are presented in table 14.2. The band fitting was based on a 10/90 linear combination of Lorentzian and Gaussian curves. 

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... 

To describe symmetric lineshapes, Maltempo80 defined linear combinations of normalized Lorentzian and Gaussian lineshapes (hybrid lineshapes) as the function... 

This computational approach of finding the optimal alignment for the Kalman filter resolution of the overlapped shifted spectra by the simulated annealing algorithm has been tested on simulated overlapped spectra obtained by linear combination of Gaussian-Lorentzian curves, synthetically generated using the mathematical model described by Eqn. (9)... 

Figures. Synthetic Gaussian-Lorentaan curves. The values of the Gaussian-Lorentzian factor b in the Eqn.(9) were 0.3, 0.5, 0.7 for the curves 1, 2, and 3, rei )ectively. The FWHM of curves IB and 2B were doubled with respect to 1A and 2A. The peak maxima coordinates Xq were 150, 200, 250 for the curves 1, 2, and 3. ( — ) are the linear combination (unitary coefficients) of the three curves. (From Fresenius J Anal. Chan(1993) 345 490, with permission).

Least squares models, 39, 158 Linear combination, normalized, 65 Linear combination of variables, 64 Linear discriminant analysis, 134 Linear discriminant function, 132 Linear interpolation, 47 Linear regression, 156 Loadings, factor, 74 Lorentzian distribution, 14... 

The Voigt function is known to be a very good description of the diffraction peaks, however, it is a difficult function to program and pseudo-Voigt functions are often used [WER 74] these are linear combinations of Gaussian and Lorentzian functions ... 

This equation is expressed as a linear combination of Lorentzian contributions from the respective random motions, if the fourth term is assumed to be negligibly small. This indicates that Equation (3.35) is a model-free equation for three types of independently superposed random motions, which is in good accord with the results of the model-free treatments described in the following section, even though a specific structural model, shown in Fig. 3.6, was used for the derivation of Equation (3.35). 

In an attempt to resolve spectra of condensed systems into summations of frequency functions, several investigators " have reported that most often the frequency functions are neither Gaussian nor Lorentzian. Usually the Gaussian function does not provide absorbance in the wings of the peak, whereas the Lorentzian function often provides too much. They generally serve as opposite limits of the true function. Accordingly, linear combinations of Lorentzian and Gaussian frequency functions have been used to obtain empirical fits. 

Often, it is found that certain data profiles are best described by non-linear bases. A typical example is curves from e.g. infrared spectroscopy [20] which are well described by a linear combination of Lorentzian peaks. Sometimes it is also possible to use Gaussian peaks. The Lorentzian peak function is written as... 

Curve Fit is a function used to decompose the area of heavily overlapping bands into constituent components. The implemented procedure is based on the least-squares minimization algorithm. Each band is characterized by the parameters band position, intensity, and width. Furthermore, the type of the band shape is taken into account, whereby you can choose a Gaussian or Lorentzian function or a linear combination of both. 

The spectrum fitting is done by the least-squares method by using an appropriate model function S, e.g., a linear combination of Lorentzians. (See also Sect. 9.3.6 on Fitting nuclear spectra in Chap. 9, Vol. 1, on Stochastics and Nuclear Measurements. ) In this procedure, the sum... 

Usually, Raman spectra can be seen as the sum of single specific bands, each centered on a wavenumber v . Raman bands are also characterized by some spreading around the central wavenumber v, defined by the full width at half maximum (FWHM), w, of the band. Unlike intensity, FWHM does not generally change with the concentration of the related chemical species. Theoretically, Raman bands can be described by two kinds of disp>ersion function Gaussian or Lorentzian. The most general case is a linear combination of both functions. 

Fitting When dealing with real media, Raman spectra are the weighted sum of several contributions (bands), each of them described by a specific Gaussian-Lorentzian function. Thus, the whole experimental spectrum can be theoretically recomposed by a linear combination of Gaussian-Lorentzian functions, according to the formula ... 

The Mossbauer spectrum thus yields important information about lattice sites, internal magnetic fields, and chemical shifts, made possible by least-squares fits of the observed spectrum to linear and nonlinear combinations of Lorentzian waveforms (Fig. 11.92). 

---