q-Gaussians and the shapes of Raman spectral lines


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, the 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 of q-Gaussian functions could be interesting for a specific application, that regarding the analysis of Raman spectra, where Lorentzian and Gaussian profiles are the most commonly used line shapes to fit the spectral bands. Therefore, we will discuss q-Gaussians with the aim of comparing the resulting fit analysis with data available in literature. As it will be clear from the discussion, in particular referring to (Meier, 2005), this is a very sensitive issue. Then, we will consider results given in a recently proposed analysis of carbon-based materials, obtained by means of mixed Gaussian and Lorentzian line shapes, defined as GauLors (Tagliaferro et al., 2020). In this paper, we will also provide a detailed discussion about pseudo-Voigt functions. We will show a successfully comparison of q-Gaussians with pseudo-Voigt functions. It is also considered the role of q-Gaussians in EPR spectroscopy (Howarth et al., 2003), where the q-Gaussian is given as the "Tsallis lineshape function".

Version notes

After the submission of the previous layout, I found reference to the "Tsallis line shape". In the article by Howarth et al., 2003, it had been proposed the use of such a line for EPR spectroscopy. Actually, a Tsallis line shape is a q-Gaussian. For this reason, I revised the text accordingly. In particular: 1) I included a section entitled "Tsallis line shape in EPR". 2) About pseudo-Voigt, I added discussion of the work by Claramunt et al., 2015. 3) Also a small discussion about Voigt function is added. 4) A section entitled "Bessel functions" has been added. Actually, the GauLor line shape function, proposed by Tagliaferro et al., 2020, is suitable for fitting a line shape function based on the modified Bessel function of the second kind.