q-Gaussian Tsallis Functions and Egelstaff-Schofield Spectral Line Shapes

q-Gaussian Tsallis Functions and Egelstaff-Schofield Spectral Line Shapes

Loading document ...
Loading page ...


Author(s): Amelia Carolina Sparavigna

Download Full PDF Read Complete Article

DOI: 10.18483/ijSci.2673 21 91 47-50 Volume 12 - Mar 2023


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.


q-Gaussian distribution, Gaussian distribution, Cauchy distribution, Lorentzian distribution, Voigt distribution, Egelstaff-Schofield line-shape, Raman spectroscopy, EPR spectroscopy


  1. Bunten, R. A. J., McGreevy, R. L., Mitchell, E. W. J., Raptis, C., & Walker, P. J. (1984). Collective modes in molten alkaline-earth chlorides. I. Light scattering. Journal of Physics C: Solid State Physics, 17(26), 4705.
  2. Burshtein, A. I., & Temkin, S. I. (1994). Spectroscopy of molecular rotation in gases and liquids. Cambridge University Press, Cambridge.
  3. Burshtein, A. I., Fedorenko, S. G., & Pusep, A. Yu. (1983). The lineshape of motion-averaged isotropic Raman spectra. Chem. Phys. Lett. 100, 155-158.
  4. Egelstaff, P. A., & Schofield, P. (1962). On the evaluation of the thermal neutron scattering law. Nuclear Science and Engineering, 12(2), 260-270.
  5. Fedorenko, S.G., Pusep, A.Yu., & Burshtein, A.I. (1987). The transformation of inhomogeneously broadened spectra due to frequency migration, Spectrochim. Acta A 43, 483-488.
  6. Ferrari, A. C., & Robertson, J. (2000). Interpretation of Raman spectra of disordered and amorphous carbon. Physical review B, 61(20), 14095.
  7. Ferrari, A. C., & Robertson, J. (Eds.) (2004). Raman spectroscopy in carbons: From nanotubes to diamond”. Philos. Trans. R. Soc. Ser. A 362, 2267.
  8. Ferrari, A. C. (2007). Raman spectroscopy of graphene and graphite: Disorder, electron–phonon coupling, doping and nonadiabatic effects. Solid state communications, 143(1-2), 47-57.
  9. Hanel, R., Thurner, S., & Tsallis, C. (2009). Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example. The European Physical Journal B, 72(2), 263.
  10. Howarth, D. F., Weil, J. A., & Zimpel, Z. (2003). Generalization of the lineshape useful in magnetic resonance spectroscopy. Journal of Magnetic Resonance, 161(2), 215-221.
  11. Keresztury, G., & Földes, E. (1990). On the Raman spectroscopic determination of phase distribution in polyethylene. Polymer testing, 9(5), 329-339.
  12. Kirillov, S. (2004). Novel approaches in spectroscopy of interparticle interactions. Vibrational line profiles and anomalous non-coincidence effects. In Novel Approaches to the Structure and Dynamics of Liquids: Experiments, Theories and Simulations; Springer: Berlin/Heidelberg, Germany, 2004; pp. 193–227
  13. Kirillov, S. A. (1999). Time-correlation functions from band-shape fits without Fourier transform. Chemical physics letters, 303(1-2), 37-42.
  14. Kirillov, S. A. (1993). Markovian frequency modulation in liquids. Analytical description and comparison with the stretched exponential approach. Chemical physics letters, 202(6), 459-463.
  15. Kubo, R., & Tomita, K. (1954). A general theory of magnetic resonance absorption. Journal of the Physical Society of Japan, 9(6), 888-919.
  16. Meier, R. J. (2005). On art and science in curve-fitting vibrational spectra. Vibrational spectroscopy, 2(39), 266-269.
  17. Naylor, C. C., Meier, R. J., Kip, B. J., Williams, K. P., Mason, S. M., Conroy, N., & Gerrard, D. L. (1995). Raman spectroscopy employed for the determination of the intermediate phase in polyethylene. Macromolecules, 28(8), 2969-2978.
  18. Naudts, J. (2009). The q-exponential family in statistical physics. Central European Journal of Physics, 7, 405-413.
  19. Rodrigues, P. S. S., & Giraldi, G. A. (2016). Fourier analysis and q-gaussian functions: Analytical and numerical results. arXiv preprint arXiv:1605.00452.
  20. Rothschild, W. G., Perrot, M., & Guillaume, F. (1986). Vibrational dephasing under fractional (“stretched”) exponential modulation. Chemical Physics Letters, 128(5-6), 591-594.
  21. Rotschild, W. G., Cavagnat, R. M., & Perrot, M. (1987). Vibrational dephasing under fractional (“stretched”) exponential modulation in a liquid crystal system. Chemical Physics, 118(1), 33-43.
  22. Sparavigna, A. C. (2021). Nozioni di q-calcolo nell'ambito del quantum calculus. Zenodo. https://doi.org/10.5281/zenodo.4982846
  23. Sparavigna, A. C. (2022). Entropies and Logarithms. Zenodo. DOI 10.5281/zenodo.7007520
  24. Sparavigna, A. C. (2023). q-Gaussian Tsallis Line Shapes and Raman Spectral Bands, International Journal of Sciences, 12(3), 27-40 DOI: 10.18483/ijSci.2671
  25. Svelto, O. (1998). Principles of Lasers, fourth ed., Plenum Press, New York, 1998, pp. 31–50.
  26. Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of statistical physics, 52, 479-487.
  27. Tsallis, C., Levy, S. V., Souza, A. M., & Maynard, R. (1995). Statistical-mechanical foundation of the ubiquity of Lévy distributions in nature. Physical Review Letters, 75(20), 3589.
  28. Umarov, S.,Tsallis, C., Steinberg, S. (2008). On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics. Milan J. Math. Birkhauser Verlag. 76: 307–328. doi:10.1007/s00032-008-0087-y. S2CID 55967725.

Cite this Article:

International Journal of Sciences is Open Access Journal.
This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License.
Author(s) retain the copyrights of this article, though, publication rights are with Alkhaer Publications.

Search Articles

Issue June 2023

Volume 12, June 2023

Table of Contents

World-wide Delivery is FREE

Share this Issue with Friends:

Submit your Paper