q-Gaussian Tsallis Functions and Egelstaff-Schofield Spectral Line Shapes
Amelia Carolina Sparavigna
Abstract
In this article we will discuss the Egelstaff-Schofield line shapes, as used in Raman spectroscopy, and their fit by means of q-Gaussian Tsallis functions. q-Gaussians are probability distributions having their origin in the framework of Tsallis statistics. A continuous real parameter q is characterizing them so that, in the range 1 < q < 3, q-functions pass from the usual Gaussian form, for q close to 1, to that of a heavy tailed distribution, at q close to 3. The value q=2 corresponds to the Cauchy-Lorentzian distribution. This behavior allows the q-Gaussian function to properly mimicking the Egelstaff-Schofield line shape, which has been introduced to fit the bands of first-order Raman scattering in ionic liquids. This line shape is based on a modified Bessel function of the second kind. Moreover, since the Fourier transform of the Egelstaff-Schofield line shape is given by a simple analytical expression, we can use this expression as an easy substitute for the Fourier transform of the q-Gaussian function.