Lorenzian distributions

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

The Pearson VII model contains four adjustable parameters and is particularly well suited for the curve fitting of large spectral windows containing numerous spectral features. The adjustable parameters a, p, q and v° correspond to the amplitude, line width, shape factor and band center respectively. As q —the band reduces to a Lorenzian distribution and as q approaches ca. 50, a more-or-less Gaussian distribution is obtained. If there are b bands in a data set and... 

The resulting data were then treated by use of standard XPS methods. An asymmetric background correction using the Shirley function (described in detail by Castle and Salvi 16)), was used for all spectra, and peaks were fit by using a mixed Gaussian-Lorenzian distribution. Quantitation was performed with the equation (A/Sj) + EA/S, where A, is the peak area and 5/ is the atomic sensitivity factor for the element i being determined 17). Atomic sensitivity factors are empirical constants determined on standards of the elements 18). This yields an atom percentage (atom %) for each element at the surface of the material. 

Tweezer stiffness can therefore be calibrated by measuring the mean squared Brownian motion. In addition, if the power spectrum of the movement is measured we find that, because of viscous damping, thermal noise shows a Lorenzian distribution with a roll-off given by... 

In general, it may be expected that the sites for Fe2+ and Fe3+ in these noncrystalline ion exchange resins will have a large distribution of chemical environments. This expectation should be reflected as a significant broadening of the Mbssbauer resonance, as experimentally observed by Johansson 188). In addition, this broadening should result in a non-Lorenzian spectral line shape. Indeed, a computer analysis of the spectra showed that Gaussian peaks provided a better fit of the data than did Lorenzian peaks. In this case then, the linewidth and peak shape provide information about the distribution of support interactions for the various resonant atoms in the sample. 

If the molecules studied do not display any motions, the spectral distribution will display a Lorenzian-type profile. In this case, the probability of the electronic transition ) — f is identical for all molecules that belong to the ) level. 

Thermal motion induces different displacement speeds for the molecules and thus different transition probabilities. These change from one molecule to another and from one population of molecules to another. In this case, the spectral distribution will be Gaussian. The full width at half maximum of a Gaussian spectrum is greater than that of a Lorenzian spectrum. 

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