# q-Gaussian Tsallis Line Shapes for Raman Spectroscopy

Amelia Carolina Sparavigna

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Amelia Carolina Sparavigna. q-Gaussian Tsallis Line Shapes for Raman Spectroscopy. SSRN Electronic Journal, 2023, 10.2139/ssrn.4445044. hal-04123119

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
q-Gaussian Tsallis Line Shapes for Raman Spectroscopy

Amelia Carolina Sparavigna

Department of Applied Science and Technology, Polytechnic University of Turin, Italy

Email: amelia.sparavigna@polito.it

**Abstract**

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 can be used for the analysis of Raman spectra, where Lorentzian and Gaussian profiles are among the line shapes most used to fit the spectral bands. In the first part of this research, we consider fitting simulations with q-Gaussian Tsallis lines and comparison to analyses made by means of Lorentzian and Gaussian lines. In the second part we show several examples of fitting experimental data. The proposed cases are demonstrating that q-Gaussians are properly fitting Raman bands.

**Keywords:** q-Gaussian distribution, Gaussian distribution, Cauchy distribution, Lorentzian distribution, Voigt distribution, Pseudo-Voigt function, Carbonaceous Materials, Biochar, Synthetic Organic Pigments, Graphite, Graphene, Single-layer graphene, Kerogen, Carbonaceous material in black schist, Silicate glasses, Raman spectroscopy, EPR spectroscopy, Tsallis line shape, IR spectroscopy.

SSRN, Submitted May 11, 2023, Posted May 13, 2023, Revised June 7, 2023

DOI: https://dx.doi.org/10.2139/ssrn.4445044

**1. Introduction**

The q-Gaussian functions are probability distributions proper of the Tsallis statistics (Tsallis, 1988, Hanel et al., 2009). These functions are based on a generalized form of the exponential function (see for instance the discussion in Sparavigna, 2022), where a continuous real parameter q is characterizing it. When q is going to 1, the q-exponential becomes the usual exponential function.

The q-Gaussian is Tsallis generalization of Gaussian distribution, being the distribution maximizing the q-entropy in the canonical ensemble (Tsallis, 1988). In the range of q-parameter from 1 to 3, we pass from the Gaussian to a heavy tailed distribution. The value q=2, (Naudts, 2009), corresponds to the Cauchy distribution, also known in physics as the Lorentzian distribution; then, the q-Gaussian function is a generalization of the Lorentzian distribution too. The change of q-parameter is therefore allowing the q-Gaussian function to pass from the Gaussian to the Lorentzian distribution.

The q-Gaussian had been proposed in 2003 by Howarth et al. as a line shape suitable for describing electron paramagnetic resonance (EPR) spectra, "and possibly nuclear magnetic resonance (NMR) spectra as well". In the article by Howarth et al., the q-Gaussian function is not mentioned in this manner, but as the "Tsallis lineshape". Howarth and coworkers stressed that the Tsallis line shape is generalizing Gaussian and Lorentzian functions, "widely used in simulations". The researchers compared the proposed Tsallis line shape with experimental EPR spectra, evidencing that the q-line shape "often provides a better approximation of the experimental spectrum". Yu. A. Koksharov, 2015, for the electron magnetic resonance study of nanoparticles, proposed to use the decomposition of
spectra with the Tsallis distributions (that is, the q-Gaussian functions). The method has been tested on a two-component spectrum of colloidal magnetite nanoparticles and on broad spectra of iron-containing nanoparticles. In an article of 2009, by Yuanlu Li, the “Tsallis distribution” is given as a function which is facilitating the generalization of Gaussian and Lorentzian line shapes by varying the parameter q “as a model of the individual band to correctly assign overlapping bands”. Yuanlu Li simulated the bands by computer and analyzed the experimental infrared spectrum of 1,2-bromofluoroethane.

A q-Gaussian representation of line shapes has been proposed for the Mössbauer spectroscopy by Ashok Razdan, 2007. “Mössbauer line shapes for systems like proteins and glasses show non-Lorentzian behaviour”, and Razdan uses q-Gaussians instead of Lorentzian functions.

The Raman spectroscopy bands are usually given as characterized by Lorentzian or Gaussian distributions, or by a linear combination (pseudo-Voigt distribution) or by the convolution of them (Voigt distribution) (Meier, 2005). In a previous discussion (Sparavigna, 2023a, 2023b), we have proposed the q-Gaussians for Raman spectroscopy showing that they are properly mimicking pseudo-Voigt, Voigt functions and the Egelstaff-Schofield spectral line shapes. We have also discussed the Raman spectroscopy, for what is regarding the D and G bands of carbon-based materials<fnref n="1" />. Here we consider a series of simulations to understand how the fitting by means of q-Gaussians is different from that obtained with Gaussian and Lorentzian lines. Then, we apply q-Gaussians to the fit of several experimental data. The fitting examples demonstrate that q-Gaussians are proper for Raman spectroscopy.

<figure><img src="image_2.png" /><figcaption>Fig. 1.1: q-Gaussian functions, for different q indices, from 1.1 (quasi-Gaussian) to 2.9 (over-Lorentzian). The blue curve is the Lorentzian line shape.</figcaption></figure>

For reader’s convenience, some literature proposed in Sparavigna, 2023a, is here reported in the final Discussions, from A to E.

## 2. The q-Gaussians

As given by Umarov et al., 2008, the q-Gaussian function is:

$$f(x) = C e_q(-\beta x^2) \quad (1),$$

where $e_q(.)$ is the q-exponential function and $C$ a scale constant. In the exponent, we use $\beta = 1/(2\sigma^2)$.

<footnote n="1">1 The G-band is the main Raman mode in graphite and graphene, linked to the planar configuration sp² bonded carbon, which is constituting the graphene layers (Application Note). “The band position is pretty much independent of excitation laser frequency” (Application Note). The D-band is the “disorder band” or the “defect band”. “This band is linked to the ring breathing mode from sp² carbon rings ... The band is typically very weak in graphite and is typically weak in graphene as well. If the D-band is significant, it indicates that there are a lot of defects in the material” (Application Note).</footnote>
The q-exponential has the expression:

$$exp_q(u) = [1 + (1 - q)u]^{1/(1-q)} \quad (2).$$

The plots in the Figures 1.1 and 2.1 are showing the behaviour of this exponential for different q values. Note that, for q less than one, the function is different from zero on a limited interval.

<figure><img src="image_3.png" /><figcaption>Fig.2.1: q- exponential functions, where the blue curve is representing a Lorentzian function (q=2). The pink curve corresponds to q=1.5 and light blue to q=1.01, practically a Gaussian function. The green curve is the q-Gaussian for q=0.75 and red curve for q=0.5. For q < 1, the function is different from zero in a limited interval. Being the line symmetric, only the right part of it is given in the figure.</figcaption></figure>

The Half Width at Half Maximum of q line shape is given by: $\sqrt{2} \sigma \sqrt{(1 - (1/2)^{1-q})/(1 - q)}$.

<figure><img src="image_4.png" /><figcaption>Fig.2.2: q- exponential functions for q=1.01 and q=2 on the left, on the right the Half Width at Half Maximum as a function of q.</figcaption></figure>

A recent publication (published on May 18, 2023) by V. Witkovský is telling that “The Tsallis q-Gaussian distribution is a flexible and versatile generalization of the standard Gaussian distribution that can effectively model input quantities in a wide range of applications and measurement models”. Mentioning Vignat and Plastino, 2009, Witkovský considers “the possible reasons why q-Gaussian distributions are frequently observed in various natural and artificial phenomena. They [Vignat and Plastino] argue that the detection of q-Gaussian behavior may be influenced by the normalization process performed by the measurement device”.

Plastino and Plastino, March 30, 2023, in their review on the connection between the q-Gaussian and the micro-canonical ensemble, are stressing q-Gaussians as “physically relevant mathematical objects”. “To speak about things, it is often useful to have names for them” (Plastino and Plastino are referring to the term “q-Gaussian” and to Garcia Marquez’s Macondo<fnref n="2" />), therefore “One of the tasks of scientists

<footnote n="2">² Macondo “is a place of solitude. ... Macondo, however, is less a paradise than a limbo. That is, the settlers become so much of the town’s raw, pre-human nature that they suffer a psycho-cultural condition in which they are not modern men, nor are they savages. They are simply, uniquely “different,” suspended in time. Common things become lost because the names describing their use are forgotten. Things from the civilized world or past history have either been forgotten or are deemed so new that it becomes necessary to point because things have not yet received a name. We may consider this observation a wry and perhaps fanciful</footnote>
is to identify and baptize things, including abstract mathematical objects, that deserve to have a name. The probability distributions, nowadays called q-exponentials and q-Gaussians (q-distributions, for short), constitute nice illustrations of physically relevant mathematical objects that, for a long time, lacked a deserved name" (Plastino & Plastino, 2023. "Since the very inception of statistical mechanics, the q-distributions are discernible in the scientific literature. These distributions are closely related to the micro-canonical ensemble, and are central to some standard derivations of the Gibbs canonical probability distribution from the micro-canonical one. The relevance of the q-distributions, however, remained unrecognized for more than a century. The unnoticed distributions remained unnamed" (Plastino & Plastino, 2023). As clearly shown by Plastino and Plastino, the q-exponentials "lacked a name" from Maxwell time to the mid 1990s.

Besides Plastino and Plastino, also Tsallis, 2023, considers the role of q-exponentials in Maxwell's work. "The unique character of this connection is particularly transparent in the classical non-relativistic regime: within that regime we can say that, to the extent that Nature prefers quadratic kinetic energies, it also prefers q-exponentials. In this regard, it is significative that q-exponentials are clearly discernible in a paper by Maxwell from 1879 [101,102] (see Equation (41) of [102]), which is one of the first papers ever discussing the micro-canonical ensemble, as we call it nowadays. Let us emphasize that Maxwell arrived at the q-exponentials without explicitly optimizing any entropic functional at all, just by assuming equal probabilities in the occupancy of the phase-space corresponding to a given total energy" (Tsallis, 2023). The Ref. [101] and [102] given by Tsallis are Maxwell's works (Niven, 2013, Maxwell, 1879).

### 3. GLS, GLP, Voigt and synthetic lines

In Jain et al., 2018, comparisons are given among Gaussian-Lorentzian sum (GLS or Pseudo-Voigt) functions, Gaussian-Lorentzian product (GLP) functions, and Voigt functions. The framework is the peak fitting of X-ray photoelectron spectroscopy (XPS). "Plots of the GLS show that it is a better mathematical representation of a function that is intermediate between a pure Gaussian and a pure Lorentzian". The GLS also represents a better approximation of the Voigt function. A reason is in the fact that the "Voigt function looks like Gaussian for small x (i.e., near line center), and like Lorentzian for large x (i.e., out in line wings)" (Townsend, 2008), depending on the choice of parameters.

$$V(x; \sigma, \gamma) \equiv \int_{-\infty}^{\infty} G(x'; \sigma) L(x - x'; \gamma) dx',$$

$$G(x; \sigma) \equiv \frac{e^{-x^2/(2\sigma^2)}}{\sigma\sqrt{2\pi}}, \quad L(x; \gamma) \equiv \frac{\gamma}{\pi(x^2 + \gamma^2)}.$$

<figure><img src="image_5.png" /><figcaption>Fig.3.1: Definition of the Voigt convolution and related functions (plot courtesy Shiyu Ji, under CC BY-SA 4.0 license.</figcaption></figure>

<footnote>one; but, upon reflection, we would have to concede that our lives are, in fact, full of encounters with things for which we lack identifying names — things, which to describe, it becomes necessary "to point." " Quoted from Carl Senna, Cliffs Notes on 100 Years of Solitude. 27 May 2023 <https://www.cliffsnotes.com/literature/o/100-years-of-solitude/about-100-hundred-years-of-solitude>.</footnote>
Regarding the GLP functions, “because of the more compact nature of the Gaussian, the GLP does not have significant wings” (Jain et al., 2018). GLS, that are also known as Pseudo-Voigt functions, have wings with different consistency, and the same is also true for the q-Gaussians, depending on the value of parameter q.

Jain et al. considered XPS. “In the theory of X-ray photoelectron spectroscopy, natural line shapes are generally assumed to be Lorentzian. There are, however, reasons why this line shape may not be observed experimentally” (Jain et al., 2018). After specific discussions, Jain and coworkers explain it is possible to argue that “when a set of photoelectrons, which may inherently have a Lorentzian line shape, is perturbed by a spectrometer and/or the broadening mechanisms mentioned above, one would expect the final signal to have at least some Gaussian character”. Then, “it is plausible that many of the components of XPS narrow scans will be best defined and fit by peaks that have both Gaussian and Lorentzian character”. According to the researchers, “mathematically, the ‘purest’ way to handle this problem is to use a Voigt function”, (Jain et al., 2018), but GLS, GLP and other functions are generally proposed by several fitting packages. The same is true for the Raman spectroscopy: the purest line shape is considered being the Voigt one (Meier, 2005), but pseudo-Voigt and other functions are used too (see Discussion C, Fitting the bands, and Discussion D, Broadening the bands).

As stressed by Jain et al., there is “an active area of research [aiming] to determine which synthetic function is most appropriate in different situations”. “However, ... the most important test of a synthetic line shape is not the theory behind it but rather its effectiveness in fitting real data”. For the Raman spectroscopy too, we can propose the investigation of new “synthetic” lines, and the q-Gaussian Tsallis function is one of them. Synthetic or real? We could tell that it is “real”, according to Maxwell.

## 4. Physical reasons

Besides what we have mentioned about articles by Witkovský, Plastino & Plastino and Tsallis, let us add other physical reasons for q-Gaussians based on Tsallis statistics, regarding fractal and granular matter. “The macroscopic stability of powder mixtures and other forms of granular or fibre matter might be associated with fractal sets ... Under these circumstances, there is no reason for having a relevant "thermodynamic" energy proportional to a standard power of the mass of the system. Consequently, nonextensive phenomena could be present” (Tsallis, 1995).

The Tsallis statistics, with the use of q-parameter to evaluate the degree of non-extensivity of the system, has been proposed for the Stimulated Raman Scattering (SRS) in a collisionless plasma by Sharifi & Parvazian (2015). SRS involves a resonant decay of laser electromagnetic waves into scattered electromagnetic waves and electron plasma waves (EPW). “Most of the studies for SRS are derived in the frame of a fluid description, a Maxwellian distribution function, or a relativistic Maxwellian distribution function. However, many space and laboratory plasmas show a non-Maxwellian behavior” (Sharifi & Parvazian, 2015). According to Sharifi and Parvazian new statistical approaches, based on the generalization of Boltzmann–Gibbs entropy, have been proposed, after the works by Alfred Rényi (1955) and Constantino Tsallis (1988). The statistics that Sharifi and Parvazian are using is the q-Gaussian velocity distribution, which is generalizing the Maxwellian distribution. As we can find mentioned in the article by Plastino and Plastino, Ray and Graben, 1991, pointed out that “the momentum distribution of small systems described by the micro-canonical ensemble is non-Maxwellian”, that is the Maxwellian distribution with Gaussian function is the limit of a q-Gaussian.

Sharifi and Parvazian are mentioning the “nonextensive statistics”. “The Boltzmann–Gibbs–von Neumann–Shannon additive entropy”, and continuous and quantum analogues, “constitute the
grounding concept on which the Boltzmann-Gibbs statistical mechanics is constructed" (Tsallis, 2023).
"However, recent decades have seen a proliferation of natural, artificial, and social complex systems
which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988
into the nonextensive statistical mechanics—as currently referred to—grounded on the nonadditive
entropy" (see the function given by Tsallis, 2023). Let us stress that the non-extensivity is referring to
the use of a generalized additivity in the entropy expression.

Regarding the profile of Raman scattering and fluorescence, the collisional broadening and the Doppler broadening are relevant (Demtröder, 1985). The Doppler shift is usually given with the Maxwellian distribution (Nienhuis & Schuller, 1977). Then, a q-nonadditive generalization is possible in the case of Raman scattering too (for neutron scattering, see please De Abreu et al., 2022, Guedes et al., 2019). For the Doppler effect on light scattering, let me stress the discussion proposed by Chandrasekhara Venkata Raman, in his letter of 1931.

**5. Separating the overlapping peak signals**

Besides the choice of the synthetic line shape, for fitting the experimental data it is required to establish
the number and positions of the bands composing the Raman spectrum (see further information in the
Discussion A). This is fundamental for the problem of “separating overlapped peak signals (OPS)”. Li
et al., 2010, proposed it in the framework of Tsallis model and fractional-order differentiation. OPS “is
a predominant aspect of signals processing of modern times. The separation of received signal and
restoration of the desired ones have long been a problem in overlapped signals decomposition” (Li et
al., 2010). OPS pertains to a wide range of applications, such as in communications, thermal analysis,
nuclear-magnetic resonance, in spectroscopy in general, and in the X-ray diffraction, as told by Li and
coworkers. Here in the following, for the fitting simulations we consider OPS in the case of a
composition of three spectral bands. The aim is that of simulating the fitting with three bands D1, D3
and G, relevant for carbon-based materials.

In Sparavigna, 2023a, the q-Gaussians had been proposed for the analysis of biochar, a carbonaceous
material, considering data collected by Tagliaferro et al., 2020. In the Figure 6 of their paper, we can
find the Raman spectra for different biochar samples. Biochar is the solid residue of pyrolysis of biomass
obtained by thermochemical decomposition at moderate temperatures under oxygen-limiting conditions
(Bartoli & Giorcelli, 2022, Brassard et al., 2019, Han & Kim, 2008, Ok et al., 2015, 2018, Giorcelli et
al., 2021, Das et al., 2021, Yasim-Anuar et al., 2022). The spectrum analysis made by Tagliaferro and
coworkers is based on a synthetic line shape the authors proposed with the name “GauLor”. It is a
function producing a line shape with a “central” part which is Lorentzian and the “wings” that are
Gaussian functions. According to the definition given by W. Demtröder, 1985, the central part of the
line (kernel) is within the range determined by the Full Width at Half Maximum of the line, the wings
are outside of it; in the GauLor line shape the onset of the Gaussian wings is given by a frequency
threshold value determined by the best fit of all the Raman spectrum. Supposing the existence of a
threshold within the Raman scan range, the GauLor has the onset of the Gaussian tail which can be very
close to the center of the line or quite far from it.

The panels of the Figure 6 are primarily based on a four-band model D¹, D², G¹, G² (we will use bands G, D1, D3, D4, according to notation given by Sousa et al., 2020, and we will discuss this point in detail in Sect.16, for what is regarding the “G+D2” band). To fit Tagliaferro et al. data by means of q-Gaussians, parameters q, β, C have been used for the functions corresponding to the four bands. The parameters have been determined by minimizing the sum of the squares of deviations Δ² = Σ(f<sub>Data</sub> − f<sub>q-Gau</sub>)² (the sum is made on equally spaced data points). The resulting fit for one of the cases proposed by Tagliaferro and coworkers is here shown in the Figure 5.1 (Sparavigna, 2023a).
<figure><img src="image_6.png" /></figure>

Fig. 5.1: The plot here given is obtained by means of four q-Gaussians, fitting data from Tagliaferro et al., 2020, regarding a biochar sample. Green (fit) and red (data) curves are in good agreement. The positions of the centers of the bands are also marked in the plot. The bands are given according to Sousa et al., 2020, notation. In fact, the band here noted by “G”, it is a “G+D2” band, made by two merged bands.

As told by Morga and Bielowicz, 2022, in their research about the Raman spectroscopy of lignite, spectra of morphotypes they are considering show the occurrence of five bands, that is D2, G, D3, D1, and D4. Here in the following, the Figure 3 from Morga and Bielowicz.

<figure><img src="image_7.png" /></figure>

Fig.5.2: The position of the Raman bands in the spectra investigated by Rafal Morga and Barbara Bielowicz. Many thanks to these authors for providing their work under the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). For comparison with Fig. 5.1, please reflect vertically the plot. In Fig. 5.2, D2 is a shoulder of G. In the Fig. 5.1, D2 is merged into G, as we will discuss in detail.

Morga and Bielowicz specify that “G band (~1585 cm⁻¹) corresponds to the graphitic lattice vibration (E₂g mode) or aromatic ring breathing [34,63,64]. The D2 band (~1612 cm⁻¹) makes a *shoulder* on the G band, and it is also assigned to the E₂g mode [65,66]
occurrence of small aromatic systems ... organic molecules or functional groups forming “amorphous” carbon phase” (Morga & Bielowicz, 2022).

“Sometimes two [48] or three [34] bands are detected within the D3 band region. The D1 band (~1340 cm⁻¹) is related to vibration mode A₁g of graphitic lattice and assigned to in-plane defects, and the occurrence of heteroatoms [62,63,68,69] as well as aromatics with six or more fused rings [34]. The D4 band (~1180–1200 cm⁻¹) is attributed to sp³ or sp²–sp³ carbons ... [33,34,59,71,72,73]” (Morga and Bielowicz, 2022). Numbers in square brackets are referring to articles mentioned by Morga and Bielowicz, https://www.mdpi.com/1996-1073/15/16/6057.

In a curve fitting of Raman spectrum of biochar, proposed by Feng et al., 2017, the names of the bands are different and given as in the Figure 10 of their article. “As shown in Fig. 10, the biochar aromatic ring structure was characterized by Raman spectroscopy. The Raman spectrum of the char was divided into 10 small peaks [15,53,54], representing the typical structure of biochar. Among them, the D peak (1320 cm⁻¹) mainly represents a relatively large aromatic ring structure ... GR (1540 cm⁻¹), VL (1465 cm⁻¹) and VR (1380 cm⁻¹) mainly represent typical structures, such as smaller aromatic rings found in amorphous carbon” (Feng et al., 2017; the numbers in square brackets are referring to literature references provided by these authors). GR is D3 (the “right” shoulder of G in the Fig.4b).

Xu et al., 2021, are giving the D3 band at 1540 cm⁻¹, at the same position of the GR band by Feng et al.

<figure><img src="image_8.png" /><figcaption>Fig. 5.3 - In Feng et al., 2017, the Raman spectrum of the char is proposed divided into 10 small peaks. In this cartoon, eight of them are given. Among them we recognize the GR peak (1540 cm⁻¹), our D3, and the G peak. Note that G in Feng et al. is a component equivalent to our “G”, that is G+D2. GL is equivalent to a supplementary D band (D5, as given here and in Sousa et al., 2020).</figcaption></figure>

**6. Fitting simulations**

Looking at the Figure 5.1, it seems that Gaussian and Lorentzian functions could be suitable for fitting data as well as q-Gaussians, but it is not so. Then, let us simulate some fitting cases to have specific information about the role of the line shape. For simulations, let us consider just three peaks, as the peaks D1, D3 (that is, GR in Feng et al., 2017) and “G” (that is, G+D2, the band and its shoulder). The synthetic curve to be fitted is given in the Figure 6.1. The position of the peaks is maintained fixed as in the Figure 5.1.
<figure><img src="image_9.png" /><figcaption>Fig. 6.1: The green curve here given is the curve that we will use for the first fitting simulations. It is given by three q-Gaussians regarded as the three bands D1, D3 and G, of the Fig. 5.1.</figcaption></figure>

Therefore, we use three q-Gaussians with parameter q as in the Figure 6.1. Other parameters of the three q-Gaussians are: C=0.13956, σ=42.500 (D1), C=0.10971, σ=17.5 (G) and C=0.07600, σ=32.0 (D3). The curve provided by the sum of these three q-Gaussians (Fig.6.1) is our synthetic “experimental” data. This curve - from now on depicted in red - is given as a function of integers *n* (500 equally spaced data points, that is *n* from 0 to 500, x-axis). A convenient scale is used for y-axis.

The fit of this red curve will be given in green color. The fitting calculation is obtained by minimizing the sum of the squares of the deviations $\Delta^2 = \sum (f_{red} - f_{green})^2$ iteratively (gradient method). The first best fit is proposed in the Figure 6.2. In this case, the values of parameter q for the tested q-Gaussians are constrained to the discretized values 1.01, 1.2, 1.4, 1.6, 1.8, and 2.0. q=1.01 is representing the Gaussian-like q-function. The positions of the centers of these q-Gaussians are fixed as that of Figure 5.1. The components of the fit are given by the blue and magenta lines. Of course, the constrained q-Gaussian fit is providing a perfect fitting curve.

<figure><img src="image_10.png" /><figcaption>Fig. 6.2: Red curve is the curve to fit (the same as the green curve in Figure 6.1), and the green curve is the best fit (constrained q parameters). Curves are indistinguishable. The components are given by blue and magenta lines. Fitting parameters are: q=1.2, C=0.13966, σ=42.416 (D1), q=1.2, C=0.10966, σ=17.416 (G) and q=1.4, C=0.0763, σ=32.083 (D3). The sum of squares of deviations is $\Delta^2 = 4.1 \times 10^{-6}$. In the lower part of the image, the brown line is giving the difference between the red and the green line (misfit). The misfit scale is from -0.02 to 0.02. No appreciable difference is displayed.</figcaption></figure>
If we relax constraints about the values of the q-parameters, that is all values ranging from 1.01 to 2.0 are possible for the three bands, the best fit we obtain is given in the following Figure 6.3.

<figure><img src="image_11.png" /><figcaption>Fig. 6.3: Red curve is the curve to fit, and the green curve is the best fit (unconstrained q). Fitting parameters are: $q=1.1875$, $C=0.13900$, $\sigma=42.750$ (D1), $q=1.1875$, $C=0.1096$, $\sigma=17.583$ (G) and $q=1.4125$, $C=0.07633$, $\sigma=31.916$ (D3). The sum of the squares of the deviations is $\Delta^2 = 1.0 \times 10^{-5}$. As in the previous figure, the brown line represents the difference between red and green line (misfit).</figcaption></figure>

Comparing the results given in the Figures 6.2 and 6.3, we could conclude that an uncertainty of ±0.01 is given about the value of fitting parameter q.

Let us change the curve to be used for simulations. Again, we use three peaks, D1, D3 and G, as given in the Figure 6.4. The q-Gaussians parameters q are 1.2 (D1), 1.8 (G) and 1.4 (D3). Other parameters of the three q-Gaussians are: $C=0.13956$, $\sigma=42.500$ (D1), $C=0.10971$, $\sigma=17.5$ (G) and $C=0.07600$, $\sigma=32.0$ (D3). These parameters are the same of those for the simulation regarding the Figure 6.1.

<figure><img src="image_12.png" /><figcaption>Fig. 6.4: As in the Figure 6.1, the green curve here given is the curve that we will use for the new fitting simulations. The three q-Gaussians are corresponding to D1 ($q=1.2$), D3 ($q=1.4$) and G (1.8). Note the different right wing.</figcaption></figure>

Here in the following Figures the results of the best fit (constrained and unconstrained).
<figure><img src="image_13.png" /></figure>

Fig. 6.5: Red curve is the curve to fit, and the green curve is the best fit (constrained q parameters). Curves are indistinguishable. The components are given by blue and pink lines. Fitting parameters are: $q=1.2$, $C=0.13966$, $\sigma=42.416$ (D1), $q=1.8$, $C=0.10966$, $\sigma=17.416$ (G) and $q=1.4$, $C=0.0763$, $\sigma=32.083$ (D3). The sum of the squares of the deviations is $\Delta^2 = 3.1 \times 10^{-6}$.

<figure><img src="image_14.png" /></figure>

Fig. 6.6: Red curve is the curve to fit and the green curve is the best fit (unconstrained q). Fitting parameters are: $q=1.20$, $C=0.13900$, $\sigma=42.583$ (D1), $q=1.7625$, $C=0.1096$, $\sigma=17.583$ (G) and $q=1.4375$, $C=0.07633$, $\sigma=32.083$ (D3). The sum of the squares of the deviations is $\Delta^2 = 8.2 \times 10^{-6}$.

Comparing the results in the Figures 6.5 and 6.6, we could conclude that an uncertainty of ±0.05 is given about the value of parameter q obtained from fitting procedure.

## 7. Two components

Let us consider a simulation, where the D3 peak has a null contribution. How is reacting the unconstrained simulation? It reacts in a very positive manner.

<figure><img src="image_15.png" /></figure>

Fig. 7.1: Red curve is the curve to fit, obtained just with D1 and G components. The green curve is the best fit (unconstrained q). Fitting parameters are: $q=1.200$, $C=0.13966$, $\sigma=42.4166$ (D1), $q=1.800$, $C=0.10966$, $\sigma=17.5833$ (G) and $q=1.862$, $C=-3.333 \times 10^{-4}$, $\sigma=7.250$ (D3). The sum of the squares of the deviations is $\Delta^2 = 4.0 \times 10^{-6}$. The result regarding D3 tells us that it does not exist for sure.
8. Moving D3

In the previous simulations we have maintained fixed the positions of the centers of the bands. Now, let us relax this condition, and assume that the position of D3 can change. To understand the role of the position of the center of this peak, we use the fit with unconstrained q parameters.

<figure><img src="image_16.png" /><figcaption>Fig. 8.1: The position of the center of the peak D3 in the fitting is changed. The best fit is given at position 340 (399 in the Figures 6.1 and 6.4). However, the minimum is quite flat and therefore the position of the center can be assumed with an uncertainty of ±2 units.</figcaption></figure>

<figure><img src="image_17.png" /></figure>

To observe the effect of position of D3 center line, let us move it to the right and to the left of 5 units, maintaining fixed the positions of D1 and G. The effect of the shift of D3 in the fitting results regarding the two components G and D3 is shown in the three plots of Figure 8.2.

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 8.2: Changing the position of D3 band of -5 (top) and +5 (bottom) with respect to the reference position (middle), the values of the sum of the squares of deviations change according to the previous Figure 8.1. Note that the G band quite changes accordingly.</figcaption></figure>
## 9. Gaussian and Lorentzian functions

What happens if we use Gaussian and Lorentzian functions instead of the q-Gaussians? Results are given in the following figures. We use for the fit Gaussian-like functions, that is q-Gaussians with q=1.01, and Lorentzian functions (q=2).

<figure><img src="image_20.png" /></figure>

Fig. 9.1: Here we use three q-Gaussians, with the same q value equal to 1.01, for the fit of the red curve. The positions of the centers of these Gaussians have been fixed as that of the Figure 6.4. The green curve is the best fit. In the lower part of the image, the difference between the red curve and the green curve is given. The misfit scale is from -0.02 to 0.02. There is a great difference due to the wings of the Gaussians. $\Delta^2 = 7.83 \times 10^{-3}$. Fitting parameters are: C=0.14100, σ=45.250 (D1), C=0.096, σ=24.25 (G) and C=0.0776, σ= 39.41 (D3).

Let us relax the condition of the fixed position of the center of D3. We move it to the right, in the position 399+15. The result is given in the following Figure, and it is displaying a better approximation, but the relative weights of D3 and G are quite different.

<figure><img src="image_21.png" /></figure>

Fig. 9.2: Three q-Gaussians with the same q value equal to 1.01. The positions of the centers of D1 and G are fixed. The center of D3 is at 399+15. The green curve is the best fit. In the lower part of the image, the difference between the red curve and the green curve is given (misfit). $\Delta^2 = 2.43 \times 10^{-3}$. Fitting parameters are: C=0.14266, σ=45.333 (D1), C=0.049, σ=13.33 (G) and C=0.1213, σ= 39.16 (D3).

In the approximations given by the iterative approach, changing the positions of D1 and G does not produce a better result. However, let us stress the difference of relevance of D3 with respect to G, shown in the Fig. 9.2. In the following Figures, we continue with fitting simulations, introducing Lorentzian functions. Let us consider two Gaussians for D1 and G, and a Lorentzian function for D3. The positions of the centers of the peaks are the same as in the Figure 6.4.
<figure><img src="image_22.png" /></figure>

Fig. 9.3: Here we use for fitting two q-Gaussians with the same q value equal to 1.01 for D1 and G, and a Lorentzian function for D3. The positions of the centers of these Gaussians have been fixed as that of the Figure 6.4. The green curve is the best fit. In the lower part of the image, the difference between the red curve and the green curve is given. $\Delta^2 = 2.08 \times 10^{-3}$. Fitting parameters are: C=0.1300, σ=44.33 (D1), C=0.095, σ=19.5 (G) and C=0.095, σ= 30.66 (D3).

<figure><img src="image_23.png" /></figure>

Fig. 9.4: Let us consider, as in the previous figure, two q-Gaussians with the same q value equal to 1.01 for D1 and G, and a Lorentzian function for D3. The position of the center of G3 is shifted to the left of 4 units. The green curve is the best fit. In the lower part of the image, the difference between the red curve and the green curve is given. $\Delta^2 = 1.98 \times 10^{-3}$. Fitting parameters are: C=0.1310, σ=44.083 (D1), C=0.104, σ=20.41 (G) and C=0.089, σ= 29.75 (D3).

Comparing the two last figures, Fig. 9.3 and 9.4, we can see that a rather small shift of the center of D3 is producing an evident change in the G peak. The change of $\Delta^2$ is negligible. We can also compare the behavior of the D3 Lorentzian band (Fig. 9.3 and 9.4) to that of the D3 q-Gaussian band in Fig. 8.2: they are similar but in opposite trends.

For the fitting simulations, we limited the sampling to the range displayed by the x-axis of the figures. In the case of the use of Gaussian and Lorentzian functions, an expanded range is relevant for sure. However, if the curve to be fitted is composed by components which have a behavior intermediate between Gaussian and Lorentzian, it is better to use q-Gaussian functions.

## 10. Fitting data

Simulations tell that we can distinguish the results obtained by means of the q-Gaussians line shapes from those deduced using Gaussian and Lorentzian functions. We can also determine the value of q parameter for each component, and the position of its center. Of course, if the synthetic “experimental” curve that we use for the simulation is produced by pure Gaussian and Lorentzian functions, the fitting by means of q-Gaussians is suitable too, being Gaussian and Lorentzian functions two cases of q-Gaussians. Then, to obtain further information about the role of q-Gaussians in Raman fitting, we must pass to the analysis of true experimental data. This is the subject of the second part of the proposed research work.
# 11. Synthetic Organic Pigments

Let us start from a spectrum of SOPRANO database (https://soprano.kikirpa.be/index.php?lib=sop, Fremout & Saverwyns, 2012). The first Raman spectrum of this database that we consider is named PO72. It is a synthetic pigment benzimidazolone (Creative Common Attribution – NonCommercial - NoDerivatives 4.0 License), https://soprano.kikirpa.be/index.php?lib=sop&id=PO72_A_785_kikirpa, analyst Wim Fremout. The first peak considered is that at 1595 cm⁻¹. Here in the following Figures, the data and the best fit. The agreement is remarkable. The largest component has q=1.65.

<figure><img src="image_24.png" /></figure>

Fig. 11.1: Best fit (in green) of the spectral data (in red) regarding a peak of the SOPRANO PO72 spectrum. Arbitrary units on y-axis. Regarding the x-axis, data are given as a function of integer *n* (series of 200 equally spaced data points).

<figure><img src="image_25.png" /></figure>

Fig. 11.2: Data, best fit and components, $\Delta^2 = 2.43 \times 10^{-4}$, of the same peak given in Fig. 11.1. The components are three. From left to right, the fitting parameters are: $q=1.01, C=0.01223, \sigma=13.15$; $q=1.65, C=0.2673, \sigma=7.45$ (large peak); and $q=1.01, C=0.0067, \sigma=36.25$.

What happens if we use the Lorentzian function for the main peak? Here the result in the Fig. 11.3.

<figure><img src="image_26.png" /></figure>

Fig. 11.3: Data, best fit and components. $\Delta^2 = 1.93 \times 10^{-3}$. The components are from left to right Gaussian, Lorentzian and Gaussian. Note the difference in both kernel and wings. The fitting with the q-Gaussian function, given in Fig. 11.2, is perfect, not that obtained with a Lorentzian function.
Another spectrum that we consider is PBk31 (aniline black), available
https://soprano.kikirpa.be/index.php?lib=sop&id=PBk31_A_785_kikirpa. The Raman shift between
the two peaks is of 1579 cm⁻¹.

<figure><img src="image_27.png" /><figcaption>Fig. 11.4a: Data, best fit and components. $\Delta^2 = 2.33 \times 10^{-3}$. The components are two. From left to right, the fitting parameters are: q=1.60, C=0.260, σ=2.76 and q=1.01, C=0.158, σ=4.50.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 11.4b: Adding a component, we can improve the fit. $\Delta^2 = 1.74 \times 10^{-3}$. From left to right, the fitting q-parameters are: q=1.56, 1.45 and 1.01.</figcaption></figure>
Then, let us consider PBr42, Cromophtal Brown RBN, available at the link https://soprano.kikirpa.be/index.php?lib=sop&id=PBr42_A_785_kikirpa. The peaks of the Raman shift are at 955 and 983 cm⁻¹ respectively.

<figure><img src="image_29.png" /><figcaption>Fig. 11.5: Data, best fit and components. $\Delta^2 = 2.25 \times 10^{-3}$. The used components are two. From left to right, the fitting parameters are: q=1.875, C=0.341, $\sigma$=2.633 and q=1.125, C=0.059, $\sigma$=3.63. We could do the same as for the previous sample, that is adding further small components to improve the fit.</figcaption></figure>

## 12. Baseline or component?

Of the spectrum PV27, https://soprano.kikirpa.be/index.php?lib=sop&id=PV27_A_785_kikirpa, which is the violet Triarylcarbonium, we consider the two peaks at 1600 cm⁻¹. Here we can investigate a further problem, related to the role of the baseline. Let us start with the best fit with two components given in the following figure. As for the previously proposed fits, we used the baseline corrected data given by the web page.

<figure><img src="image_30.png" /><figcaption>Fig. 12.1: Data, best fit and components. $\Delta^2 = 1.28 \times 10^{-3}$. The components are two. From left to right, the fitting parameters are: q=1.162, C=0.0992, $\sigma$=16.916 and q=1.612, C=0.0984, $\sigma$=11.75..</figcaption></figure>
The agreement is good. However, we have a small difference in the part of the spectrum (saddle) between the two peaks and about the vertex of the first peak. To understand why there are these differences, let us consider the original data too. They are given in the following figure. We can use the original data and choose a local linear baseline (the green line), to have a different baseline correction.

<figure><img src="image_31.png" /><figcaption>Fig. 12.2 – On the left, the corrected and the original data as given by the web page of Soprano. In the middle, the original data and a local linear baseline here used for fitting, to obtain a different best fit. On the right, the best fit with two components, with the data corrected with the local baseline. We have a better agreement. Parameters *q* are slightly changed into 1.12 and 1.48 respectively.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>Fig. 12.3: Best fit with three components. The *q* parameters of the two large peaks are 1.02 and 1.6 respectively.</figcaption></figure>

In the Fig.12.1, we have proposed the best fit with two components. We can further investigate, for the same data, the role of a third component included in the fit. The result is given in the Fig.12.3. For the data here considered, and about the fitting with two or three components, a question can be posed: is the third component strictly necessary, or is it simply the result of an approximation in the choice of the baseline? We have used a local linear baseline, but a non-linear baseline could give us a better result, eliminating the necessity of the third component.
13. Other peaks

Of the spectrum PV27, https://soprano.kikirpa.be/index.php?lib=sop&id=PV27 A_785 kikirpa, let us
consider the fitting of other peaks (baseline corrected).

<figure><img src="image_33.png" /><figcaption>Fig. 13.1: Best fit with three components. The q parameter of the large peak (216 cm⁻¹) is 1.012 (a Gaussian).</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Fig. 13.2: Best fit with three components. The largest peak is at the Raman shift of 420 cm⁻¹. The q-parameters of the blue and magenta lines are 1.40 and 1.46 respectively.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>Fig. 13.3: Best fit with three components. The largest peak is at the Raman shift of 1180 cm⁻¹. The q-parameter of the largest blue peak is 1.30.</figcaption></figure>

Let us add other further examples. In the following figures, some bands of the sample PR2 (chemical class: Naphthol AS), available at the following link https://soprano.kikirpa.be/in-dex.php?lib=sop&id=PR2_A_785_kikirpa

<figure><img src="image_36.png" /><figcaption>Fig. 13.4: In red, experimental data PR2 from SOPRANO, in green, the q-Gaussian fitting the data (left). Components given with values of q-parameters (right). On the x-axis, data are given as a function of integer n (series of 340 data points).</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Fig.13.5: In red, experimental data PR2 from SOPRANO, in green, the q-Gaussian fitting the data (up). Components and values of q-parameters (down). On the x-axis, data are given as a function of integer *n* (series of 290 data points).</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>Fig.13.6: In red, experimental data PR2 from SOPRANO, in green, the q-Gaussian fitting the data (left). Components and values of q-parameters (right). On the x-axis, data are given as a function of integers *n* (series of 150 data points).</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>Fig. 13.7: In red, the experimental data PR2 from SOPRANO, in green, the q-Gaussian fitting the data (left). Components and values of q-parameters (right). On the x-axis, data are given as a function of integers *n* (series of 400 data points).</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>Fig. 13.8</figcaption></figure>

In the Figure 13.8, we assume just a part of the band given in Fig., to enhance the role of main component. We fit (green) the q-Gaussian over data (red). A convenient scale is used for y-axis. The value of q-parameter is slightly changing.
The following plots are regarding PR50, monoazo, available at link https://soprano.kikirpa.be/index.php?lib=sop&id=PR50_A_785_kikirpa&ds=baseline%20corrected

<figure><img src="image_41.png" /><figcaption>Fig.13.9: In red, the experimental data PR50 from SOPRANO, in green, the q-Gaussian fitting the data (left). Components and values of q-parameters (right). On the x-axis, data are given as a function of integers *n* (series of 500 data points).</figcaption></figure>

As we can see from the plot in 13.9, on the right, the three q-Gaussians are able to overlap with a proper fit of the two saddles, without the use of additional components. The same can be observed for the saddle in the following plot, where the spectrum of PBr22, nitro pigment, available at link https://soprano.kikirpa.be/index.php?lib=sop&id=PBr22_A_785_kikirpa has been considered.
<figure><img src="image_42.png" /><figcaption>Fig.13.10: In red, the experimental data PBr22 from SOPRANO, in green, the q-Gaussian fitting the data (left). Components and values of q-parameters (right). On the x-axis, data are given as a function of integers n (series of 400 data points).</figcaption></figure>

**14. Comparison with Voigt fitting**

In Raman spectroscopy Lorentzian and Gaussian profiles are the line shapes most used to fit the spectral bands. However, the real Raman lines seldom follow these specific cases (Kirillov, 2004a). To improve the analysis, the Voigt profiles are often used (Meier, 2005). Instead of the Voigt profile, here and in previously research, we proposed the use of q-Gaussian functions.

In the article by Robert Meier, 2005, where the author proposed "some crucial issues in curve-fitting vibrational spectra that do not always seem respected", we can find told that there is no problem in handling the isolated bands. "Problems arise when there is band overlap" (Meier, 2005). Problems exist also with isolated bands, since the profile is usually different from the Gaussian or the Lorentzian profile. The study of the isolated bands for what is regarding the behavior of q-Gaussian fitting is therefore fundamental. As previously told, to Gaussian or Lorentzian profiles, it is sometimes preferred the Voigt profile. Then, let us fit q-Gaussian functions over experimental data, compared with Voigt function fitting.

Thibault et al., 2002, studied the Raman linewidth of CO in Ar, using an inverse Raman spectrometer. In the Figure 1 of their article, the Raman spectrum of the Q(5) line is given. In the upper field of the figure, we can find Voigt profile fitted over the experimental spectrum. The lower field of the figure is giving the difference between experimental data and Voigt profile.

Let us consider the data in the Figure 1 by Thibault et al., 2002, and fit to a q-Gaussian function. The result is given in the following Figure 14.1.
<figure><img src="image_43.png" /><figcaption>Fig.14.1: In red, the experimental data from Thibault et al., 2002; in green, the q-Gaussian best fit (q-parameter, q=1.775). In the lower part of the figure, it is given the difference between the red and the green lines (misfit). The two black lines are giving the band of uncertainty of experimental data (the width of the band is 0.012 in the y-axis scale). Note that the difference is within the band. For what is regarding the x-axis, data are given as a function of integer *n* (series of 270 equally spaced data points).</figcaption></figure>

Where are the main differences between the use of q-Gaussian profile and Voigt profile? First, the q-Gaussian is a probability distribution having an analytic form. The Voigt function is the convolution of two functions, Gaussian and Lorentzian, with no associated analytic form. Then, the analytic q-Gaussian is characterized by the presence of the q-parameter. The width of its profile is provided by a single simple expression (see the Width at Half Maximum previously given). To characterize the width of a profile fitted to Voigt functions, two parameters are required, one related to the Gaussian the other to the Lorentzian component of a convolution, which needs a numerical approach.

Let us also consider the work by Huang et al., 2014, where the role of the substrate about the analysis of graphene Raman lineshapes is investigated. The researchers fitted “the bandwidths of G-band and 2D-band ... into the Voigt profile ... The bandwidths of Lorentzian parts were kept as constant whether it is the suspended and supported graphene. For the Gaussian part, the suspended graphene exhibits much greater Gaussian bandwidths than those of the supported graphene” (Huang et al., 2014).
"With data fitting into Voigt profile, one can find out the information behind the lineshapes. ... Under our analysis, details about the effects of charged impurities on the substrate can be realized. About the strain effect or doping effect on graphene, some possible broadening mechanisms may still be responsible for deforming it, so we considered the Gaussian profile necessary" (Huang et al., 2014).
"Based on the data fitting results, the analysis of measured point across the graphene surface, the bandwidths of Gaussian profiles and Lorentzian profiles given by Voigt fitting is presented in Figure 4a,b" (the figure is in Huang et al., 2014). In the mentioned article, it is also stressed that the "Lorentzian bandwidth is mainly contributed by the natural linewidth and partly from the uncertainty of data fitting ... and instrumental uncertainty ... The natural linewidth is just linked with the phonon lifetimes between interaction levels" (Huang et al., 2014).

We can use data from Figure 4a,b in Huang et al., 2014 and fit q-Gaussians over them. We find that the q-parameter of the Raman spectrum of the suspended graphene is lower than the q-parameter of the spectrum of the supported graphene.

<figure><img src="image_44.png" /><figcaption>Fig.14.2: In red, the experimental data from Huang et al., 2014, Fig.4a (supported graphene). In green, the q-Gaussian best fit (q-parameter, q=1.64). In the lower part of the figure, it is given the difference between the red and the green lines (misfit). The two black lines are giving the band of uncertainty of experimental data (the width of the band is 0.02 in the y-axis scale). Note that the difference is almost within the band. On the x-axis, data are given as a function of integer n (series of 270 data points).</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>Fig. 14.3: In red, the experimental data from Huang et al., 2014, Fig. 4b (suspended graphene). In green, the q-Gaussian best fit (q-parameter, q=1.48). In the lower part of the figure, it is given the difference between the red and the green lines (misfit). The two black lines are giving the band of uncertainty of experimental data (the width of the band is maintained equal to 0.02, in the y-axis scale, as in the Figure 2). Note that the difference is almost within the band. On the x-axis, data are given as a function of integer n (series of 270 data points).</figcaption></figure>

We can decompose the two q-Gaussians given above, in Figs. 14.2 and 14.3, in the sum of a Gaussian (q=1.01) and a Lorentzian (q=2.0). Note that q-value 1.01 is giving a q-Gaussian which is a function numerically indistinguishable from the corresponding Gaussian function. The two decompositions are given in the following figure.

<figure><img src="image_46.png" /><figcaption>Fig. 14.4: Decomposition of the two q-Gaussians in the sum of a Gaussian and a Lorentzian function. In red, the q-Gaussian, in green the best fit obtained with the sum. Note please that the q-Gaussian is not a pseudo-Voigt function (that is a linear combination Gaussian and Lorentzian functions). This explains the negligible numerical difference between q-Gaussian and best fit.</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>Fig. 14.5: Decomposition of q-Gaussians with different q-parameters (σ is the same) in the sum of a Gaussian and a Lorentzian function. In red, the q-Gaussian, in green the best fit obtained by means of the sum. The Gaussian component is represented by the blue profile and the Lorentzian one by the magenta profile. The areas under the Gaussian and Lorentzian functions can be used as the relative weight of the components.</figcaption></figure>

As shown in the Figure 14.5, the q-parameter can be related to the relative weight of the Gaussian and Lorentzian components. The simplest evaluation could be made by integration, that is, by the area under the two curves, blue and magenta in the Figure 14.5.

## 15. Biochar

Let us consider the Raman spectra of biochar obtained by Tagliaferro et al. and proposed in their Fig.6. The first analysis that we will do is that regarding the spectrum of the sample OSR 2200. Tagliaferro and coworkers have used a standardized biochar, produced by pyrolysis at 550 °C, named OSR550 with features given by Specification Sheet at http://www.charchive.org/record.php?record_id=98. This standardized biochar had been carbonized at different temperatures, from 900 to 2200 °C. The Raman spectra were recorded using laser at 514 nm.
<figure><img src="image_48.png" /></figure>

Fig. 15.1: On the left, data (red) and best fit (green) of the Raman spectrum of OSR2200 (data from Tagliaferro et al., 2020, Fig.6f, Raman shift from 1200 to 1700 cm⁻¹). On the right, data, best fit and components of the same Raman spectrum. Here we have used four components. From left to right, the components have q-parameters 1.6, 1.01, 1.8 and 1.2. Note that the D2 shoulder is split from G. Let us stress that the second component, between the two peaks, is very low and rather flat. Is it a real component or the consequence of the chosen baseline?

In the Figure 15.1, we considered four components. One of them is very low and flat. So, we can pose again the question already considered for the Fig. 12.1-12.3. Can we consider it a true component or the result of the baseline approximation? Then, let us use just three components. The result is given in the following figure.

<figure><img src="image_49.png" /></figure>

Fig. 15.2: On the left, data and best fit of the Raman spectrum of OSR2200 (data from Tagliaferro et al., 2020, Fig.6f). On the right, data, best fit and components of the same Raman spectrum of OSR2200. Here we have used three components. From left to right, the components have q-parameters 1.65, 1.86 and 1.2.
In the following plots, the other samples from the Figure 6 of Tagliaferro et al. are considered.

<figure><img src="image_50.png" /></figure>

Fig. 15.3: On the left, data (red) and best fit (green) of the Raman spectrum of OSR1500 (data from Tagliaferro et al., 2020, Fig.6e, Raman shift from 1200 to 1800 cm⁻¹). On the right: data, best fit and components of the same Raman spectrum. We have used four components. D2 is split from G. From left to right, the components have q-parameters 1.68, 1.01, 1.86 and 1.01.

Also in this case, let us consider just three components, removing that between the two peaks. In this case the best fit reacts producing two Lorentzian functions (q=2.0) and a Gaussian (q=1.01), as shown in the next figures. The calculus is constrained to q values between 1.01 and 2.0.

<figure><img src="image_51.png" /></figure>

Fig. 15.4: On the left, data and best fit of the Raman spectrum of OSR1500 (data from Tagliaferro et al., 2020, Fig.6e). On the right: data, best fit and components of the same Raman spectrum. We have used three components. From left to right, the components have q-parameters 2.0, 2.0 and 1.01.
Here in the following the other data.

<figure><img src="image_52.png" /><figcaption>Fig. 15.5: Data, best fit and components of the Raman spectrum of OSR1300 (data from Tagliaferro et al., 2020, Fig.6d, Raman shift from 950 to 1800 cm⁻¹). We have used four components (in fact, we have component G which is G+D2). From left to right, the components have q-parameters 1.01, 1.71, 1.30 and 1.425.</figcaption></figure>

<figure><img src="image_53.png" /><figcaption>Fig. 15.6: Data, best fit and components of the Raman spectrum of OSR1100 (data from Tagliaferro et al., 2020, Fig.6c, Raman shift from 980 to 1800 cm⁻¹). The center position of D3 (blue) is at about 1540 cm⁻¹, G (magenta) is at about 1595 cm⁻¹. We have used four components. From left to right, the components have q-parameters 1.01, 1.55, 1.16 and 1.225. OSR 1100 data have been already proposed in the Fig.5.1. In that case, the fit had been made with constrained values of q-parameters (1.01, 1.2, 1.4, 1.6, 1.8 and 2). The constraint influenced the resulting D4.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>Fig. 15.7: Data, best fit and components of the Raman spectrum of OSR900 (data from Tagliaferro et al., 2020, Fig.6b, Raman shift from 950 to 1800 cm⁻¹). We have used four components. From left to right, the components have q-parameters 1.01, 1.45, 1.01 and 1.125.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>Fig. 15.8: Data, best fit and components of the Raman spectrum of OSR550 (data from Tagliaferro et al., 2020, Fig.6a, Raman shift from 950 to 1800 cm⁻¹). We have used five components. From left to right, the components have q-parameters 1.01, 1.56, 1.34, 1.01 and 1.01 (smallest component).</figcaption></figure>

From the Figures given above, we can observe a consistent behavior of bands D1, D3 and G. Let us now go from Fig. 15.8 up to Fig. 15.1, we can note the decrease of the D3 band and the increase of the G band. This is evidencing the role of carbonization at different temperatures.
<figure><img src="image_56.png" /><figcaption>Fig.15.9: The role of carbonization. Note that in plots 1500 and 2200, the D2 shoulder turns out to be separated from G in an evident manner. As given in the following figure, the G peak has a q-parameter close to 2, indicating a "purer" component.</figcaption></figure>
<figure><img src="image_57.png" /><figcaption>Fig.15.10: The values of the q-parameter for the three main components D1, D3 and G. Here also we can note the role of carbonization. For samples OSR-550, 900, 1100 and 1300, D2 band is merged into G. For OSR 1500 and 2200, G and D2 are separated. This is evident in the values of related q parameters.</figcaption></figure>

Figure 15.10 is giving the values of q-parameters for components D1, D3 and G. What is interesting is the behavior of q for the G band, with its value increasing from 1.01 toward 2. Let us note that the quasi-Lorentzian value is observed for the samples OSR1500 and OSR2200, where we can find the separation of what is known as the D2 component from the G component. As we have seen before, the D2 component is considered a “shoulder” of G. In the samples OSR550, 900, 1100 and 1300, it happens what stressed by Beyssac et al., 2002a: “In poorly organised carbon it is impossible to separate the two components [that is, G and D2] and only one broad band occurs near 1600 cm⁻¹” (Beyssac et al., 2002a).

Comparing the results obtained by means of the q-Gaussians and the plots given by Tagliaferro et al. in their Figure 6, we can see some differences. In particular, the results regarding sample OSR550 are different in the behavior of D3 and G. Here, we have considered the large peak on the right in Figure 15.1 (2200 °C) and Figure 15.3 (1500 °C) as made by the G component, according to the assignment of components given by Beyssac et al., 2002a. The notation by Beyssac et al. has been used also for the D3 component, which is consequently between D1 and G. As previously told, in the Figures 15.1 and 15.3, we have also the D2 component, according to the same notation.

Tagliaferro et al. are using a notation where D¹ and D² are the names of the first and second component of D peak, G¹ and G² the first and second component of G peak, ordered according to the increase of the Raman shift (this is what we can obtain from the Figure 7 in Tagliaferro et al. about the positions of the centers of components). Then, in the Figure 6, in all the panels of Tagliaferro et al., we have that D1 is corresponding to D². For what is regarding our D3, it is corresponding to G¹, in panels (b), (c) and (d). Then, in these panels, we have that our G, that is G+D2, is corresponding to G². In the panels (e) and (f) of the Figure 6 by Tagliaferro and coworkers, we have G corresponding to G¹, and D2 corresponding to G². Moreover, in these panels (e) and (f), D3 is considered as a further band D¹'. These correspondences can be evidenced by means of the following figures, where the Raman spectra of biochar, in the range from 1200 cm⁻¹ to 1700 cm⁻¹, as given by Tagliaferro et al., 2020, in their Figure 6.
<figure><img src="image_58.png" /></figure>

Fig.15.12: Raman spectra of biochar, in the range from 1200 cm⁻¹ to 1700 cm⁻¹, as given by Tagliaferro et al., 2020, in their Figure 6. Many thanks to these authors for providing their work under CC BY 4.0 license.

<figure><img src="image_59.png" /></figure>

Fig.15.13: Raman spectra of biochar OSR2200, in the range from 1200 cm⁻¹ to 1700 cm⁻¹, superimposed to the other spectra from Tagliaferro et al., 2020, in their Figure 6. Many thanks to these authors for providing their work under CC BY 4.0 license.
<figure><img src="image_60.png" /></figure>

Fig.15.14: Positions of the centers of the bands, according to Tagliaferro et al., 2020, in their Figure 7. Many thanks to these authors for providing their work under CC BY 4.0 license.

For OSR550, by the comparison given in Fig.15.13 and from the Fig.7 by Tagliaferro et al. here proposed in Fig.15.14, we find the center of $G^1$ at 1580 cm$^{-1}$, at the same position of the center of $G^1$ for OSR1500. For the other samples, the position of $G^1$ is different, shifted to lower wavenumbers.

In the previously given discussion, we have tried to compare the components. In any case, differences of the two approaches, with q-Gaussians and GauLors, are not surprising since the functions used to describe the line shapes are different. In the article by Yuan and Mayanovic, 2017, which is an empirical study on Raman peak fitting, it is told that "the fitted peak position, intensity, area ... values of the measured Raman bands can vary significantly depending upon which peak profile function is used in the fitting, and the most appropriate fitting profile should be selected depending upon the nature of the Raman bands" (Yuan & Mayanovic, 2017). Yuan and Mayanovic, in their study about the "peak fitting", considered "Gaussian, Lorentzian, Gaussian-Lorentzian [pseudo-Voigt], Voigtian, Pearson type IV, and beta profiles". As previously told, Tagliaferro et al. used their synthetic line shape named "GauLor"; of its framework, we discussed in Sparavigna, 2023c.

## 16. Assignment of bands

The methodology by Tagliaferro et al., besides considering a new synthetic line shape, is assuming the number of components as low as possible. In our analysis of the experimental data provided by Tagliaferro and coworkers, we assumed the same number of components. In fact, using the q-Gaussians we obtained the proper fitting without introducing further components.

At the beginning of our research work, we started with simulations regarding three components, using letters G, D1, D3, according to that given by Sousa et al., 2020, and Beyssac et al., 2002a. For comparing the q-Gaussian results and those given by Tagliaferro et al. in their Figure 6, besides what we have previously told, the following Figure 16.1 can be useful too.

<figure><img src="image_61.png" /><figcaption>GauLors</figcaption></figure>

<figure><img src="image_62.png" /></figure>

Fig.16.1(a): Components in the present research work (left) and in the paper by Tagliaferro et al. (right). The right panel has been elaborated from one of the plots in the Figure 6 of Tagliaferro et al., 2020. Many thanks to these authors for providing their work under CC BY 4.0 license.
In Sousa et al., 2020, a study on ancient biochar, the analogous number of components is equal to five; as previously told, in Sparavigna, 2023a, it has been considered that two components, those indicated by G and D2 in Sousa et al., can be merged into a component defined as G+D2, or simply G (in agreement with Beyssac et al., 2002a).

In Xu et al., 2021, it is told that Ferralis et al., 2016, “studied the fossil organic matter ... They curve-fitted the Raman spectra with eight bands including 1080 cm⁻¹ (R1), 1180 cm⁻¹ (S), 1270 cm⁻¹, 1350 cm⁻¹ (D), 1450 cm⁻¹ (V), 1540 cm⁻¹ (V2), 1590 cm⁻¹ (G+D2), and 1680 cm⁻¹ (R2). They set up a very good correlation between the ratio of the integrated intensities of the 1270 cm⁻¹ band and (G + D2) bands and elemental H/C ratio” (Xu et al., 2021). In Ferralis et al., 2016, it is told that “G and D2 have physically different origin within an aromatic or graphitic cluster, their presence is strictly related to the aromatic component only. Ideally, given D2 variability both in intensity and width as function of defects (such as physical defects in the aromatic structure or presence of heteroatoms), G alone would serve as the ideal reference band for aromatic. Yet, for smaller aromatic clusters in low maturity OM [organic matter], the blueshift of the G peak towards higher wavenumbers due to confinement effects leads to the G and D2 peaks merging into a single broad peak” (Ferralis et al., 2016). (see Refs. [4, 33, 74] in Ferralis et al.). Therefore, within the work by Ferralis et al., the researchers “use the sum of the integrated intensity of the fitted G and D2 peaks when the peak can be discriminated or, otherwise, the overall fit of the G+D2 band”.

Ferralis et al. used peak fitting with Horiba LabSpec LabSpec 5, Horiba Scientific. The peaks are D5, D1, D3, G+D2 peaks. “If D2 is discernible from G, a separate peak is fit (at 1600 cm⁻¹)”. “Peak fitting is carried out using PseudoVoigt profiles”.

Actually, Ferralis and coworkers are using multiple bands. As we have seen in Morga and Bielowicz, 2022, two or three D3 bands can be used for fitting Raman spectra.

<figure><img src="image_63.png" /><figcaption>Fig. 16.1(b) - In Ferralis et al., 2016, a multiple D3 band for the Raman spectrum of kerogen type III, from the Penn State coal bank, has been considered. In the case of a large saddle between the G and D regions, two D3 bands are useful. Note that, in the Fig.16.1(a), the saddle is narrow.</figcaption></figure>

Ferralis et al. also add that “The peak shape itself is parameterized between the two extremes (fully Gaussian or fully Lorentzian). Its determination through the fit allows for the identification of the nature of the statistical distribution of the vibrational frequency of a particular peak. A fully Gaussian peak will follow a normal distribution, which is expected for vibrations generated by a normal distribution of bonds with different local geometries and chemical environments. A fully Lorentzian peak fit represents a uniform individual vibration of a specific peak, with no statistical distribution around its mean” (Ferralis et al., 2016, available at http://nrs.harvard.edu/urn-3:HUL.InstRepos:33973837).
Of course, q-Gaussians are proper for providing the statistical distribution of each peak. The q-Gaussian are statistical distributions by themselves.

Let us return to the work by Sousa et al., 2020. The illustration proposed by these researchers, available at the following link [Dynamic of the structural alteration of biochar in ancient Anthrosol over a long timescale by Raman spectroscopy | PLOS ONE](https://www.plosone.org/) is giving the components (D4,D1,D3,G,D2), plus component D5 added to further improving the fit. In the figure by Sousa and coworkers, it is plotted a line representing the best fit with such components, but not the experimental data. About the fitting procedure, Sousa et al. are telling the following: "The baseline of the spectra was extracted via a linear function. The spectrum fit was performed with 6 curves: two Gaussian (D1 and G) and four Lorentzian functions (D2, D3, D4, and D5). In general, the deconvolution of the spectrum can be performed by simple adjustment using only two Lorentzian or two Gaussian functions, ... However, when the primary aim of the research is to identify the subtle nuances of the spectrum, a 5-band [that is with G, D1, D3, D3, and D4] adjustment is required" (Sousa et al., 2020).

"The most prominent features in the Raman spectra of graphite materials are the so-called G and D bands. The G band is the signature of graphite materials associated with the vibration of the carbon atoms tangentially to the plane of the graphene. This band can be observed at approximately 1580 cm⁻¹, and it indicates the presence of organized sp2 domains" (Sousa et al., 2020). "The D band is related to defects associated with the breaking of the hexagonal symmetry of the carbon atoms in the graphene sheets and can be observed at approximately 1350 cm⁻¹. The relationship between the intensities of the D and G bands can provide an estimate of the crystallite size and also related to the number of defects present in the material. However, for highly disordered materials, other bands are induced by other types of defects in the crystalline lattice and appear as first-order peaks at approximately 1150 (D4), 1530 (D3) and 1600 cm⁻¹ (D2)" (Sousa et al., 2020, mentioning Hammes et al., 2008).

With the Figure 2 in the paper by Sousa et al., 2022, we can perform a further test with the q-Gaussians. Here in the following Figure 40, the result of fitting with five q-Gaussians.

<figure><img src="image_64.png" /><figcaption>Fig. 16.2: Data (red) of the fitted spectrum proposed by Sousa et al., 2022, in their Figure 2 (Raman shift from 1000 to 1950 cm⁻¹). The green line is representing the best fit with q-Gaussians. We have used five components. From left to right, using Sousa et al. notation, the components have q-parameters 1.4 (D4), 1.64 (D1), 1.46 (D3), 1.02 (G+D2) and 1.01 (D5). D5 is adjusting the fit (according to Sousa et al.).</figcaption></figure>
The position of the center of D3 component is shifted to the right. Of course, this is due to the use of q-Gaussians. Therefore, the name D3 for the q-Gaussian component is assumed for comparison with the D3 components given by Sousa et al. and other researchers, such as Beyssac and coworkers.

Different notations exist, such as that by Claramunt et al., 2015, who proposed the study of Raman spectra of graphene oxide and thermally reduced graphene oxide. In all their recorded spectra, the five bands are given as D, D', G, D", and D*, between 1000 and 1800 cm⁻¹. Comparing Sousa et al., 2020, to Claramunt et al., 2015, we have that the notation D means D1, D' is D2, G remains the same, D" is D3, and D* corresponds to D4.

<figure><img src="image_65.png" /><figcaption>Fig. 16.1(c) - In Claramunt et al., 2015, we can find a different notation as shown in the cartoon on the left. D2 is D', D3 is D'' and D4 is D*.</figcaption></figure>

According to Claramunt et al., 2015, Raman peaks in crystalline graphene are the G at ~1585 cm⁻¹, and the bands at ~2700 cm⁻¹, that is the first- and second-order Raman mode E₂g. Changes in the width of the bands are observed for graphene oxide GO, which is the subject of the research by Claramunt et al.; the changes had been considered as coming from defects related to oxidation. The consequent defect concentration produces “an intense peak centered at ~1350 cm⁻¹, called the D band”. “The D band is related to the A1g breathing mode, and it is observed because graphite oxidation and the subsequent reduction of GO seriously alter the basal plane structure of graphene” (Claramunt et al., 2015). “Some weak peaks centered between 1100 and 1800 cm⁻¹ were observed in GO flakes and powders, ... while they have not been detected in pristine graphene or graphite”. Further discussions are given for D' (our D2) and D* (our D4), and for “a broad peak between 1500 and 1550 cm⁻¹”, that Claramunt et al. are referring as D", that is our D3. “In fact, some examples of these bands can be found in the Raman spectral analysis of carbon-based materials such as soot and carbon black” (Claramunt et al., 2015).

“Previous work related the D* with disordered graphitic lattice provided by sp²–sp³ bonds at the edges of networks [Sadezky et al., 2005], while the D" band allocation is controversial. Some authors attributed the D" band to contributions from the phonon density of states in finite-size crystals of graphite. [Nemanich and Solin, 1979] S. Vollebregt et al. related the band to amorphous lattices since they observed a decrease of the intensity of the D" peak with the increase of the crystallinity.” (see further discussion and references in Claramunt et al., 2015).

For what is regarding the component D2, let us consider also how it is represented by Beyssac et al., 2002a, in Figure 5 of their work. The assignment is close to that here used. So let us repropose our Figure 15.3 and indicate the components according to Beyssac and coworkers.
<figure><img src="image_66.png" /><figcaption>Fig.16.3: The name of the components according to Beyssac et al., 2002a, given to the plot in the Fig. 15.3.</figcaption></figure>

Here in the following it is given what we can find told by Beyssac and coworkers regarding carbonaceous material (CM). “The Raman spectrum of CM is composed of first-order (1100–1800 cm⁻¹) and second-order (2500–3100 cm⁻¹) regions ... . In the first-order region [see the Fig. 5 in Beyssac et al.], the graphite band (G band) occurs at 1580 cm⁻¹ and corresponds to the E₂G₂ vibration mode of a crystal with D⁴₆h symmetry, i.e. to in-plane vibration of aromatic carbons in the graphitic structure” (Beyssac et al., 2002a). For what is regarding the position of the G band, it “was originally estimated around 1575 cm⁻¹ by Tuinstra & Koenig (1970). More recent studies have shown that the heating of CM induced by the laser microprobe causes a shift of the E₂G₂ mode downward to 1565 cm⁻¹” (see Beyssac et al., 2002a, Kagi et al., 1994; Everall et al., 1991). “In finite crystals, relaxation of wave-vector selection rules results in the appearance of other first-order bands around 1350 cm⁻¹ and 1620 cm⁻¹ (Tuinstra & Koenig, 1970). These bands, as the wide band around 1500 cm⁻¹ in very poorly ordered carbons, are assigned to defects in the graphitic structure” (Beyssac et al., 2002a).

“The band occurring at 1350 cm⁻¹ (D1 band) is intense and very wide in poorly ordered carbons. During the graphitization process, its relative area decreases with the stiffening of aromatic planes. Consequently this band has been attributed to in-plane defects like heteroatoms (O, H, N for instance) or structural defects (Bény-Bassez & Rouzaud, 1985). The 1620 cm⁻¹ band (D2 band) appears as a shoulder on the G band. In poorly organised carbon it is impossible to separate the two components and only one broad band occurs near 1600 cm⁻¹. The 1500 cm⁻¹ band (D3 band) is present in poorly ordered carbons as a very wide band. Bény-Bassez & Rouzaud (1985) attributed this band to out-plane defects like tetrahedral carbons by suggesting that this kind of defect is released early in the graphitization process. This component is also due to a small crystallite size (Nemanich & Solin, 1979)” (see please all the references given by Beyssac and coworkers in their article).

Let us consider the problem of the bandshift induced by laser with more detail as given by Everall, et al., 1991. The peak is shown in the Figure 1 of their work. It is depicting the “laser-induced bandshift in single (40 µm diameter) graphite grains: (a) E₂g mode, 1 mW laser beam power at sample. (b) E₂g mode, 6 mW laser beam power. The original band position was restored on returning the laser power to 1 mW”. Everall and coworkers told that “the likely explanation for these effects is that localised sample heating occurs and distorts the vibrational spectrum, although a simple interpretation in terms of expansion of the graphite lattice parallel to the layers cannot hold, since the in-plane coefficient of thermal expansion is negative below 400 °C.”

The bandshift of the G peak has nothing to do with the position of G3 band.
Let us consider the profiles given in the Figure 16.2 and in the Figure 16.3 to compare once more the name of components. The profile in Fig.16.2 is related to the work by Sousa et al. fitted by means of five q-Gaussians; let us add to Fig.16.2 the Figure 15.8, obtained by fitting with five q-Gaussians the data OSR550 from Tagliaferro et al. To these two plots, let us give also the plot provided by Tagliaferro and coworkers in their Fig.6, panel (a), all in the following Fig. 16.4.

<figure><img src="image_67.png" /><figcaption>Fig.16.4: Panels are reproducing Fig.16.2 (left, Sousa et al. data), Fig.15.8 (middle) and panel (a) of Fig.6 from Tagliaferro et al. (right). Many thanks to these authors for providing their work under the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). A very small component D5 is also present in the plot by Tagliaferro et al.</figcaption></figure>

<figure><img src="image_68.png" /><figcaption>Fig.16.5: Panels are reproducing Fig.16.3 (left) and panel (e) of Fig.6, sample ORS1500 from Tagliaferro et al. (right). Many thanks to these authors for providing their work under the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).</figcaption></figure>

From Figs. 16.4 and 16.5, it is evident that components are strongly depending on the line shape and relative position, particularly D3. Tagliaferro et al. used their synthetic line shape named “GauLor”, which is a symmetric function with a central part which is Lorentzian and tails which are Gaussian functions. The onset of the tail happens at a frequency threshold, determined by the best fit. Supposing the existence of the threshold within the Raman scan range, the GauLor has the onset of the Gaussian tail which can be very close to the center of the line or quite far from it. At the threshold, the Lorentzian and Gaussian functions and their derivatives are continuous.
To understand the behavior of GauLor synthetic line shape, let us consider data OSR550 and band D1 (our D1 band, which is D² for Tagliaferro et al.), as in the Fig.16.4 (let us stress, *only* the D1 band). Then, we choose the GauLor which is fitting the data near the peak. Let us plot three different GauLor functions, having the same peak, but different thresholds. They are shown in the following Fig. 16.6. The thick blue line, with the threshold for the onset of Gaussian tail at *t*<sub>B</sub>, is the best one (the thick red line has a tail which is too low and the green tail which is too high), for fitting the left wing of D1.

<figure><img src="image_69.png" /><figcaption>Fig. 16.6: The blue line is the best GauLor. The thick red line is another GauLor (the thin red line is representing the data to be fitted).</figcaption></figure>

It is clear, from Fig.16.6, that the fitting of D1 is almost cancelling any contribution from D4. This behavior does not mean that the resulting GauLor is more representative of a related physical phenomenon. Let us consider further threshold values as in the Fig. 16.7, and the comparison with q-Gaussians.

<figure><img src="image_70.png" /><figcaption>Fig.16.7: GauLor functions (left side, with threshold values 0, 1, 2, 3, 4, 5, 6 and infinity) and q-Gaussians (right side, with different q-parameter values 1.01, 1.1, 1.25, 1.5, 1.75, 1.9 and 2.0). For GauLors, the value of Γ is fixed and, for each GauLor function, the corresponding values of σ are given according to Eq. 3 and 4 in Tagliaferro et al., 2020. For the q-Gaussians, the value of σ is the same and equal to that of GauLor with threshold equal to zero (Gaussian). Note the different behavior of the functions; q-Gaussian is intermediate between pure Gaussian and pure Lorentzian, GauLor is a piecewise continuous function made of two parts, Lorentzian and Gaussian.</figcaption></figure>
# 17. Four-band model of deconvolution

As we can find in Lünsdorf et al., 2017, the Raman spectrum of carbonaceous materials (CMs) is made of two major "intensity accumulations in the regions" of 1000–1500 cm⁻¹ and 1500–1700 cm⁻¹. These major regions are defined as D- and G-band regions. Assuming D and G regions of the Raman spectrum primarily composed by four bands, we have in our model D1 and D4 for the D region, and, for the G region, D3 and G (that is, "G+D2", if the D2 shoulder is not observed). In Morga and Bielowicz, 2022, we can find D3 in the G band, close to the region boundary, but D2 split from G. In Lünsdorf et al., 2017, we find the subdivision in D4, D1 and D3 of the D region, and G and D2 of the G region. In the article by Tagliaferro et al., 2020, we find D¹ and D², and G¹ and G², in the D and G regions, respectively. That is, the model is a four-band model.

According to Tagliaferro et al., 2020, “G¹ peak appears to be proportional to reduction of interlayers spacing of aromatic layers and increment of graphitic crystallites showing an origin related to the ordering of the carbonaceous material”. For what is regarding G², the “area percentage trend was more complex showing a constant increment of both area percentage and band width of up to 1300 °C and a decrement afterwards. We hypothesized G² was related with bond angle disorder as suggest by Shimodaira et al. [53]” (the Ref. 53 mentioned by Tagliaferro et al. is Shimodaira & Masui, 2002).

The article by Shimodaira and Masui is based on a pure four-band model; it does not involve any further component (only a D' band, which is our D2 band, is mentioned in the Introduction for the polycrystalline graphite). In the following cartoon, we can see how the D and the G region are proposed. Note that G1 in Shimodaira and Masui is corresponding to G² in Tagliaferro et al., and that G2 is corresponding to G¹. The same for the D region.

<figure><img src="image_71.png" /><figcaption>Fig. 17.1 - In Shimodaira and Masui, 2002, the Raman spectrum is decomposed in four bands of Gaussian shape. The cartoon here proposed is giving them as in the article by Shimodaira and Masui (left), Raman shift wavenumber decreasing, and with Raman shift wavenumber increasing (right). Only four Gaussian bands are used (G1, G2, D1, D2).</figcaption></figure>

From the abstract of Shimodaira and Masui's article: "Raman spectra of activated carbon materials have been investigated by a peak-deconvolution technique. It has been found as a result of our fitting trials that four Gaussians and a linear background are necessary and enough to reproduce the spectral data throughout our experiment, with a pair of relatively sharp peaks at about 1600 cm⁻¹ and about 1350
$cm^{-1}$, namely G1 and D1, and a pair of relatively broad peaks around 1560 $cm^{-1}$ and 1340 $cm^{-1}$, namely G2 and D2. From the characteristic behavior of these paired peaks, it has been concluded that the structure of activated carbon materials is schematically classified into two parts".

In the Introduction of their article, Shimodaira and Mausi remember that, in the polycrystalline graphite, "two sharp Raman peaks, namely G(graphite)- and D(disorder)-band, explicitly appear at about 1580 and 1360 $cm^{-1}$, respectively. In addition, a small feature near 1620 $cm^{-1}$, called D' band, is also observed". "In sputtered or evaporated amorphous carbon films (a-C) regardless of whether hydrogenated or not, a relatively broad band around 1550 $cm^{-1}$ overlapped with a broader band around 1400 $cm^{-1}$ is observed" (Shimodaira & Masui, 2002). "The Raman spectra are deconvoluted into two peaks by using Gaussian line shapes, also called G and D band in a-C, and the peak position, the band width, and the relative intensity ratio of the D band to the G band, $Id/Ig$, have been examined as a function of deposition parameters, ..." (Shimodaira & Masui, 2002). For their measurements, Shimodaira and Masui used a Ramanor T64000 (Jobin/Yvon), with 514.5 nm Ar line as an excitation source. The laser was focused with a power less than 5 mW, "in order to prevent thermal degradation".

The researchers conclude that "In general, the following features are observed:" 1) The G1 peak position is very stable at about 1605 $cm^{-1}$, 2) The D1 peak position is also stable at about 1351 $cm^{-1}$, 3) "In the G2 and D2 peaks, both position and FWHM are widely scattered". "Thus, it has been demonstrated that the Raman spectra in various activated carbons and precursors can be accurately deconvoluted into four Gaussians". "By using this technique, it has been found that the activated carbon materials schematically consist of two structural parts: ... This peak deconvolution technique will be successfully applied to the further study not only of activated carbon materials with physical and/or chemical treatments, but of other carbon materials" (Shimodaira & Masui, 2002).

## 18. The power of laser beam (about G+D2)

In Childress and Jacobsen, 2017, we can find an investigation of high-pressure high-temperature Raman spectroscopy of kerogen<fnref n="3" />. "For blue-green laser excitation, [Childress & Jacobsen, 2017] find optical irradiance exceeding ~3 kW/cm² induces changes in spectral features of the primary graphitic (G-band) and two main disordered modes (D1 and D2) that might otherwise be mistaken for thermal maturation".

In the Figure 2 (b) of Childress and Jacobsen, 2017, we can see the first-order Raman spectra of kerogen at various laser power settings measured at the sample position. It is evident the splitting of G and D2 bands as the power of the laser beam increases.

In the Fig.1(a), Childress and Jacobsen are proposing the deconvolution of the first-order region of the Raman spectrum. They note that, for the proposed spectrum "the G band (1580 $cm^{-1}$) associated with C–C in-plane stretching of aromatic layers (e.g., Ferrari and Robertson 2000) overlaps with the D2 (1620 $cm^{-1}$) disordered carbon band. Also present in the first-order region are the D1 (1350 $cm^{-1}$) and D3 (1500 $cm^{-1}$) bands (Beyssac et al. 2002)." (Childress & Jacobsen, 2017).

For what is regarding the D bands, they "are attributed to various graphitization disorder effects, including reduced graphite lattice symmetry near edges, edge vibrations, heteroatoms, and defects in or between the aromatic plane" (Childress & Jacobsen, 2017, Bény-Bassez & Rouzaud, 1985, Wang et al., 1990, Beyssac et al., 2002b).

<footnote n="3">³ Kerogen is the "naturally occurring, solid, insoluble organic matter that occurs in source rocks and can yield oil upon heating. Kerogen is the portion of naturally occurring organic matter that is nonextractable using organic solvents. ... ". See more at https://glossary.slb.com/en/Terms/k/kerogen.aspx</footnote>
<figure><img src="image_72.png" /><figcaption>Fig. 18.1 - In Childress and Jacobsen, 2017, Fig. 2(b), the Raman spectra of kerogen is proposed according to different powers of the laser beam. The cartoon here on the left is giving three of these spectra. We can see that an increase of the power of laser is producing the splitting of G and D2 bands.</figcaption></figure>

Using the data given in the Figure 1 by Childress and Jacobsen, for kerogen from the Eel River, California, U.S.A. with 458 nm excitation and 0.05 mW laser power, we can obtain a comparison of q-Gaussians and Voigt decompositions. Childress and Jacobsen use Voigt functions.

<figure><img src="image_73.png" /><figcaption>Fig. 18.2: In red, the experimental data from Childress and Jacobsen, 2017, and in green, the q-Gaussian best fit. In blue and magenta colors, the q-Gaussian components. In the lower part of the figure, it is given the difference between the red and the green lines (misfit). q-Gaussians are giving a misfit which is quite lower than that displayed by the fitting with Voigt functions, proposed by Childress and Jacobsen, 2017. The q-parameters are 1.39, 1.92, 1.98, 2.02 and 1.78 (from left to right).</figcaption></figure>
In the case of the Raman spectrum given in the Fig.18.2, the use of G+D2 band is giving the following result (Fig. 18.3). This is resembling the Shimodaira and Masui approach, here with q-Gaussians.

<figure><img src="image_74.png" /><figcaption>Fig.18.3: In red, the experimental data from Childress and Jacobsen, 2017; in green, the q-Gaussian best fit. In the lower part of the figure, it is given the difference between the red and the green lines (misfit). q-. The q-parameters are 1.31, 1.85, 2.04 and 1.18 (from left to right).</figcaption></figure>

In the Figs. 18.2 and 18.3, we can find values of q-parameter larger than 2. Probably, this is due to baseline effects. However, since the Lorentzian line shape is coming from the Lorentz model, we could imagine different models which are involving over-Lorentzian line shapes (Kirillov, 2004b). Moreover, the natural signal of the line is convoluted with instrumentation and electronic amplification. So, are we sure that the convolution is limited between Gaussian and Lorentzian lines?

In the Figure 18.3 we can note that the use of the band G+D2 is producing a shift of the D3 band to greater wavenumbers. Therefore, besides the line shape used, also the number of components is relevant.

## 19. Nanotubes

The methodology by Tagliaferro et al. had been used for fitting the Raman spectra of oxidized carbon nanotubes by Bartoli et al., 2022. Here in the following Figures we show some of the Raman spectra proposed by Bartoli and coworkers in the Supplementary Information of their article, Figure S3, panels (a), (b) and (d). The fitting is here made by means of q-Gaussians.
<figure><img src="image_75.png" /></figure>

Fig. 19.1: Data (between 1000 and 2000 cm⁻¹), best fit and components of the Raman spectrum (data from Bartoli et al., 2022, Supplementary Information, Fig. S3 (a)). We have used two q-Gaussian components. From left to right, the components have q-parameters 2.0 and 1.92. Between the two peaks, is there a third component?

<figure><img src="image_76.png" /></figure>

Fig. 19.2: Data (between 1000 and 2000 cm⁻¹), best fit and components of the Raman spectrum (data from Bartoli et al., 2022, Supplementary Information, Fig. S3 (a)). Here we use three q-Gaussian components. From left to right, the components have q-parameters 2.0, 1.35 and 1.90 (largest peak).
<figure><img src="image_77.png" /><figcaption>Fig. 19.3: Data (red, between 2000 and 3500 cm⁻¹), best fit (green) and components of the Raman spectrum (data from Bartoli et al., 2022, Supplementary Information, Fig. S3 (a)). We have used four q-Gaussian components. From left to right, the components have q-parameters 1.86, 1.84 (large peak), 1.82 and 2.0.</figcaption></figure>

<figure><img src="image_78.png" /><figcaption>Fig. 19.4: Data (red, between 1000 and 2000 cm⁻¹), best fit (green) and components of the Raman spectrum (data from Bartoli et al., 2022, Supplementary Information, Fig. S3 (b)). Here we have used two q-Gaussian components. From left to right, the components have q-parameters 1.98 and 1.67.</figcaption></figure>
<figure><img src="image_79.png" /><figcaption>Fig. 19.5: As in the previous figure, data (red, between 1000 and 2000 cm⁻¹), best fit (green) and components of the Raman spectrum (data from Bartoli et al., 2022, Supplementary Information, Fig. S3 (b)). Here we have used three q-Gaussian components. From left to right, the components have q-parameters 1.86, 1.01 and 1.60.</figcaption></figure>

<figure><img src="image_80.png" /><figcaption>Fig. 19.6: Data (red, between 1000 and 2000 cm⁻¹), best fit (green) and components of the Raman spectrum (data from Bartoli et al., 2022, Supplementary Information, Fig. S3 (d)). Here we have used four q-Gaussian components. From left to right, the components have q-parameters 1.01, 1.96, 2.0 and 1.51.</figcaption></figure>

Bartoli and coworkers collected Raman spectra by means of a Renishaw inVia instrument with green laser line (514 nm).

A general good agreement with the fitting proposed by Bartoli et al., 2022, is observed. However, comparing to plots in Fig.S3, panels (b) and (d), Supplementary Information, available at the following link https://pubs.acs.org/doi/suppl/10.1021/acs.jpcc.2c04693/suppl_file/jp2c04693 si 001.pdf, we can tell that q-Gaussians are giving better wings.
<figure><img src="image_81.png" /></figure>

Therefore, comparing to GauLors fitting, we can tell that the q-Gaussians, besides perfectly describing the central part of the lines, are also more properly fitting the tails (at least as in the previously proposed cases). As we can see from Fig.16.7 (the comparison of GauLors and q-Gaussians), the definition of the piecewise synthetic line is strongly conditioning the resulting shape of the function, especially for the wings, where “wings” are considered as defined by W. Demtröder, 1985.

<caption>Fig. 19.7: Kernel and wings of a line (Demtröder, 1985).</caption>

**20. Laser induced bandshift**

We have previously mentioned the bandshift induced by laser and mentioned the work by Everall, et al., 1991. The Figure 1 of the work by Everall and coworkers is giving two peaks showing the bandshift. One peak is recorded with the 1 mW laser beam power, the other with 6 mW beam power. The sample is made of graphite grains (40 µm diameter). The shift is probably caused – Everall and coworkers told - by the distortion of the vibrational spectrum. Using the data in this Figure, we can investigate if the value of the q-parameter is different for the two Raman spectra recorded under different beam powers.

In the following figure, on the left, it is given the analysis of obtained with 1 mW beam power, on the right that for the 6 mW case. Only one component is used.

<figure><img src="image_82.png" /><figcaption>Fig. 20.1: Data (red) and best fit (green) of the Raman spectra in the Figure 1 by Everall et al., 1991. On the left, the peak is given at 1575 cm⁻¹, the beam power was of 1mW; on the right (1565 cm⁻¹) the power was of 6 mW. The q-parameters of the best fit are different. On the left, q=1.78, on the right q=1.48.</figcaption></figure>
**21. Carbonaceous material in black schistes**

The Fig. 5 proposed by Beyssac et al., 2002a, has been considered for the assignment of the band. We can consider it for another example of fitting by means of q-Gaussians (the deconvolution in the Figure 5 has been made by Beyssac and coworkers with Voigt functions). The data are regarding sample SL5, from black schistes near Chisone River (Piemonte, Italy).

As told in the abstract, “Metasedimentary rocks generally contain carbonaceous material (CM) deriving from the evolution of organic matter originally present in the host sedimentary rock. During metamorphic processes, this organic matter is progressively transformed into graphite s.s. and the degree of organisation of CM is known as a reliable indicator of metamorphic grade” (Beyssac et al., 2002a).

<figure><img src="image_83.png" /><figcaption>Fig. 21.1: Data (red, from 1100 to 1800 cm⁻¹), best fit (green) and components of the Raman spectrum (data from Beyssac et al., 2002a). Here we have used four q-Gaussian components. From left to right, the components have q-parameters 1.94, 1.01, 1.91 and 1.01.</figcaption></figure>

**22. Saint-Sauveur Graphite**

At the link https://rruff.info/Graphite/R120025 we can find the Raman spectrum of graphite (courtesy RRUFF Project). The RRUFF Project is giving a complete set of spectral data from well characterized minerals. The data collection provides a standard for “the identification of minerals both on earth and for planetary exploration” (Lafuente et al., 2015).

Given data details for RRUFF R120025 are Ideal Chemistry: C, Locality: Saint-Sauveur Graphite occurrence, Saint-Sauveur, Les Pays-d'en-Haut RCM, Laurentides, Québec, Canada, Source: Donald Doell. Owner: RRUFF. Description: Semi-metallic, dark gray flakes. Status: The identification of this mineral is not yet confirmed. The data that we consider are related to the broad scan with spectral artifacts, RRUFF ID: R120025, Wavelength: 532 nm, Sample Description: Unoriented Raman on the primary sample, Instrument settings: Thermo Almega XR 532nm @ 100% of 150mW.
<figure><img src="image_84.png" /><figcaption>Fig. 22.1: Data (red, from 1250 to 1700 cm⁻¹), best fit (green) and components of the Raman spectrum (data from RRUFF R120025). Here we have used just four q-Gaussian components. The use of further components can further improve the fit.</figcaption></figure>

## 23.1-layer graphene

In Graft et al., 2007, we find the Raman investigation of single- and few-layer graphene. In the Fig.5 of their article, the peaks for an increasing number of graphene, fitted with Lorentzian functions, are given. “The single peak position for the single layer graphene is at 2678.8 ± 1.0 cm⁻¹.” In the following plot, let us consider the single-layer peak; the fit is obtained with a q-Gaussian with q=1.38.

<figure><img src="image_85.png" /><figcaption>Fig. 23.1: Data (red) of single layer graphene from Graft et al., 2007. In green the q-Gaussian best fit. The fit is better than that shown in Graft et al. and obtained with a Lorentzian function.</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>Fig.24.1: In red, the experimental data from Ghazvini et al., 2016). In green, the q-Gaussian best fit.
On the right, the two components are given with the related q-parameters. On the x-axis, data are
given as a function of integer *n* (series of 250 data points). In the case of the upper image, we can see
that the best fit is giving a value of q-parameter which is slightly larger than 2 (this is probably due to
a small shift of the baseline).</figcaption></figure>
## 24. Zinc and zinc acetate

We have proposed many fitting examples about carbonaceous materials. Let us consider different cases. In Ghazvini et al., 2016, the researchers are reporting on the electrodeposition of zinc from the ionic liquid 1-ethyl-3-methylimidazolium acetate with various concentrations of zinc acetate. Raman spectroscopy was used to study the coordination of Zn2+ ions with OAc⁻ (Ghazvini et al., 2016). In the Figure 6 of Ghazvini et al., 2016, we can find the Raman spectra of [EMIm]OAc at different concentrations of Zn(OAc)₂ in the wavenumber range between 880 and 940 cm⁻¹. In the Figure 7, the researchers provide the deconvolution of some of these Raman spectra by two Voigt functions. Here, let us consider the data from the Figure 7 and proposed a deconvolution obtained by means of two q-Gaussians. In the previous page, the results of q-Gaussians fit oved data are (Fig.24.1).

## 25. Lithium Silicate Glasses

The concluding examples are from the article by Seuthe et al., 2017, where we can find microRaman spectra for a study about relaxation phenomena in lithium silicate glasses, subjected to irradiation with femtosecond (fs) laser pulses and subsequent thermal annealing. The spectroscopy analyses evidenced different behaviors in the glass network. The Fig.1 of the article by Seuthe and coworkers is giving the normalized baseline-corrected Raman spectra of the glasses LS, in the panel (a) and LNMS in the panel (b), see https://www.nature.com/articles/srep43815/figures/1. LS means Lithium Silicate glass and LNMS a multicomponent glass. The compositions are: LS (24mol% Li₂O and 76mol% SiO₂) and LNMS (13mol% Li₂O, 15mol% Na₂O, 9mol% MgO and 63 mol% SiO₂). The Fig.1 by Seuthe et al. is giving the spectrum deconvolution in “several individual bands with Gaussian shape”. We have therefore the occasion to compare a deconvolution with Gaussians and a deconvolution with q-Gaussians. These deconvolutions for LS and LNMS are given in the following plots. The names of the bands are J, K, M, N and O.

<figure><img src="image_87.png" /><figcaption>Fig. 25.1: Data (red, from 850 to 1300 cm⁻¹), best fit (green) and components of the Raman spectrum of LS (data from Seuthe et al., 2017). Here we have used five q-Gaussian components. The component J is very small. From left to right, the components have q-parameters 1.40, 1.01, 1.187, 1.475 and 1.01.</figcaption></figure>
<figure><img src="image_88.png" /><figcaption>Fig. 25.2: Data (red, from 850 to 1300 $cm^{-1}$), best fit (green) and components of the Raman spectrum of LNMS (data from Seuthe et al., 2017). Here we have used five q-Gaussian components. Components J and O are very small. From left to right, the components have q-parameters 1.40, 1.225, 1.225, 1.163 and 1.01.</figcaption></figure>

Remarks

1) In Raman spectroscopy, the overlapped signals decomposition is sometimes told “deconvolution”. However, deconvolution is the operation inverse to convolution. The convolution is a mathematical operation on two functions (f and g) that produces a third function $f * g$ is given. Therefore, the Raman spectra are “decomposed”, not “deconvoluted”, in a linear combination of functions. For instance, in the Fig.1 of Delarue et al., 2016, we can find the “decomposition” of a Raman spectrum of a carbonaceous material. “Raman spectra were decomposed into a combination of five Lorentzian/Gaussian bands (namely, D1, D2, D3, D4, and G; Fig. 1)” (Delarue et al., 2016). The decomposition is like that given by Morga and Bielowicz.

2) The decomposition is giving the fit of Raman spectrum according to a linear combination of a finite number of functions, each characterized by a specific position of its center. The functions, shape-bell functions, can be symmetric or not. The model of decomposition is therefore characterized by the choice of the line shape and by the choice of the number of components.

3) For every line shape considered (Gaussian, Lorentzian, Voigt, pseudo-Voigt, q-Gaussian, GauLor or others), the number of components is fundamental for the result. Let us stress that, the position of some peaks, such as D3 for instance in carbonaceous materials, depend on the chosen model. We have seen that the use of band G+D2 or of the two separated bands G and D2, is influencing the position of D3.

4) Single peaks, that is peaks which are enough separated from other peaks, can give us information about the line-shape preferred by the spectrum. This is the reason why we have considered several isolated peaks to investigate the q-Gaussian line shape.

5) The Voigt convolution is usually given as the proper distribution for Raman bands. It is not an analytical function and therefore requires numerical approximations. The q-Gaussian is analytical and provides two parameters useful to characterize the band, that is the q-parameter and the related FWHM parameter.
"... to the extent that Nature prefers quadratic kinetic energies, it also prefers q-exponentials."

Constantino Tsallis, 2023

**Discussion A – Raman spectroscopy and carbon-based materials**

The aim of this research was that of testing the possibility of using q-Gaussians in Raman spectroscopy. Raman spectroscopy is based on the interaction of light with a substance, evidencing chemical bonds in it. In this manner, information about chemical bonds, polymorphism and crystalline features can be deduced. The spectrum is characterized by bands, so that the related combinations are relevant for specific classes of materials. Several Raman band correlation tables exist, usually ranging from 100 $cm^{-1}$ to 3300 $cm^{-1}$. We concentrated the discussion on a limited part of the spectrum and considered mainly carbonaceous materials.

As told by Lünsdorf et al., 2017, the Raman spectrum of carbonaceous materials (CMs) is made of two major "intensity accumulations in the regions" of 1000–1500 $cm^{-1}$ and 1500–1700 $cm^{-1}$; these regions are given by Lünsdorf and co-workers as D- and G-band regions. The D band "is often decorated by additional shoulders on the low and high-wavenumber side (indicated by the asterisks). Due to the additional shoulders, the D-band-region is generally curve-fitted by three functions, commonly labelled D1, D3 and D4" (Lünsdorf and coworkers are mentioning Sadezky et al. 2005, Lahfid et al. 2010). For the shapes of these regions, see please the Figure 4 of the given reference (Lünsdorf et al., 2017), available at the following link https://www.researchgate.net/figure/Nomenclature-of-CM-Raman-spectra-a-A-Raman-spectrum-of-CM-from-very-low-grade_fig3_318160214. For the position of the "shoulder" D3, we must stress once more that it is depending on the line shape which is used and the number of bands considered. With q-Gaussians and GauLors, D3 passes from being the left shoulder of D to be the right shoulder of G.

"The intensity distribution in these regions [D and G] depends on the maturity stage or degree of graphitization and used excitation wavelength" (Lünsdorf et al., 2017, and references therein).

In the upper part of the Figure 4 by Lünsdorf et al., 2017, "none graphitized CM", we can see how the spectrum has been fitted. Lünsdorf and coworkers say that these regions are "usually curve-fitted by five Lorentz functions (Lahfid et al., 2010) or a mixture of four Lorentz and one Gauss functions (Sadezky et al., 2005). For the graphitic CM, we can see that the regions reduced to three bands (see the lower part of the Figure 4 in Lünsdorf et al., 2017). In this case, each band is usually described by a Voigt function. Lünsdorf and coworkers use a software that simultaneously evaluates the background and the Raman signal, respectively simulated by a fifth-order polynomial and pseudo-Voigt functions.

In Sadezky et al., 2005, the aim of researchers was that of optimizing a Raman microscope system using three different laser excitation wavelengths (514, 633, and 780 nm). The band combination that we can find in their work is made by four Lorentzian bands (G, D1, D2, D4) at about 1580, 1350, 1620, and 1200 $cm^{-1}$, and a Gaussian band (D3) at about 1500 $cm^{-1}$ in the framework of first-order spectrum of diesel soot (see the Figure 7 of Sadezky and coworkers' article).

The second-order spectra were fitted considering Lorentzian-shaped bands at about 2450, 2700, 2900, and 3100 $cm^{-1}$. In the Table 1 of Sadezky et al., 2005, first-order Raman bands and vibrational modes in soot and graphite are given (on the right).
<table><thead><tr><th rowspan="2">Band</th><th colspan="3">Raman-shift</th><th rowspan="2">Vibration mode</th></tr><tr><th>Soot</th><th>Disordered graphite</th><th>High-ordered graphite</th></tr></thead><tbody><tr><td>G</td><td>~ 1580</td><td>~ 1580</td><td>~ 1580</td><td>Ideal graphitic lattice</td></tr><tr><td>D1</td><td>~ 1350</td><td>~ 1350</td><td>-</td><td>Disordered graphitic lattice</td></tr><tr><td>D2</td><td>~ 1620</td><td>~ 1620</td><td>-</td><td>Disordered graphitic lattice</td></tr><tr><td>D3</td><td>~ 1500</td><td>-</td><td>-</td><td>Amorphous carbon (Gaussian or Lorentzian line shape)</td></tr><tr><td>D4</td><td>~ 1200</td><td>-</td><td>-</td><td>Disordered graphitic lattice</td></tr><tr><td></td><td>(cm<sup>-1</sup>)</td><td></td><td></td><td>(Lorentzian line-shape unless mentioned otherwise)</td></tr></tbody></table>

The Table I of Sadezky et al., 2005.

A discussion about Raman spectroscopy for investigating the role of thermochemical processes of coal and biomass is given by Xu et al., 2020. Xu and coworkers consider the Raman spectroscopy as a versatile tool for studying biomass. In the Table 2 of the given reference, we can find the description of the Raman bands for samples from graphite to amorphous carbon. We see the D4 band at ~1200 cm⁻¹, D1 ~1350 cm⁻¹, D3 ~1540 cm⁻¹, G ~1590 cm⁻¹, and D2 at ~1620 cm⁻¹. We can find also the different “names” attributed to the bands.

In the Table 3 we can also find the curve-fitting methods for the first-order Raman spectra as:

1) two bands with components about 1350 (D1), 1590 (G); 2) three bands with components about 1350 (D1), 1500 (D3), 1590 (G), or about 1350 (D1), 1590 (G), 1620 (D2), or 1200 (D4), 1350 (D1), 1590 (G); 3) four bands with components about 1200 (D4), 1350 (D1), 1500 (D3), 1590 (G); 4) five bands about 1200 (D4), 1350 (D1), 1500 (D3), 1590 (G), 1620 (D2), or about 1080, 1200 (D4), 1350 (D1), 1500 (D3), 1590 (G); or 1200 (D4), 1350 (D1), 1500 (D3), 1585 (G), 1750. And more bands are also mentioned. Xu et al., 2020, are giving related references for the models proposed in their Table 3.

In Xu et al., 2020, we can find an interesting discussion about the Raman spectra of raw coal. “In raw coals, there are structures including alkanes, oxygen-functional groups, small aromatic rings, and large aromatic rings system connected by cross-linked structures, small graphite microcrystalline, or even graphite crystalline” (Xu et al., 2020). “In the Raman spectrum of raw coal, the D and G bands were primary, and the bands at 860 cm⁻¹, 1180 cm⁻¹, 1230 cm⁻¹, 1450 cm⁻¹, 1540 cm⁻¹, 1620 cm⁻¹, and 1680 cm⁻¹ were found or thought to exist when the Raman spectra were resolved in many studies” (see Xu et al. and references therein).

As an example of four-band fitting, Xu et al. are referring the work by Han et al., 2017, about the role of structure defects in anthracite. The Fig. 4 of this reference is showing the Raman spectra of collected samples (a) and the fitting on one of them in (b); the fit is made by means of bands G, D, D3 and D4. Two major bands respectively at 1350 and 1600 cm⁻¹ are identified. A minor band at about 1230 cm⁻¹ is also visible. The researchers used PeakFit, version 4.12, software (Han et al., 2017). The resulting four bands are: G band (1597 cm⁻¹, Lorentzian profile), D band (1338 cm⁻¹, Lorentzian profile), D3 band (1499 cm⁻¹,
Gaussian profile), and D4 band (1237 cm⁻¹). A discussion about the corresponding vibrational modes is given too.

In the analysis process of Raman spectra, components are added to improve "the accuracy of the fit", as told by Sousa et al., 2020, who studied the structural alteration of biochar. Of Sousa and coworkers study, we have discussed when we discussed, in Sect. 14, the assignment of the bands. We have also mentioned Claramunt et al., 2015, who proposed the study of Raman spectra of graphene oxide and thermally reduced graphene oxide, reporting, in all the recorded spectra, five bands given as D, D', G, D", and D*, between 1000 and 1800 cm⁻¹. As previously told, comparing Sousa et al., 2020, to Claramunt et al., 2015, we have that the notation D means D1, D' is D2, G remains the same, D" is D3, and D* corresponds to D4.

For the fit of the first-order spectra of the graphene oxide derivatives, Claramunt and co-workers used five functions: two Gaussian and three pseudo-Voigt peaks. It is told that "data were fitted to a sum of five functions using the Origin 8.0 software" (Claramunt et al., 2015). "During the analysis it was found that Gaussian functions better fit D* and D" bands while, for D, G and D' bands as pseudo-Voigt functions render the best fit" (Claramunt et al., 2015). The pseudo-Voigt functions given by software Origin are linear combinations of Gaussian and Lorentzian functions.

Yin et al., 2018, have studied the char structure evolution during pyrolysis. In the Figure 4 of their article, we can see that the researchers used four bands (three Lorentzian, G, D1, D4 and one Gaussian, D3). In the article, references about the fit of Raman spectra are given. For chars, (pine needle, PN, and corn stalk, CS), the resulting positions of peaks are: "The band positions of the D1, D3, D4, and G peaks" for the CS char sample "are 1347, 1516, 1212, and 1594 cm⁻¹, respectively". "The band positions of the D1, D3, D4, and G peaks" for PN char samples "are 1343, 1514, 1207, 1599 cm⁻¹, respectively" (Yin et al., 2018). Yin and co-workers are also giving discussion about the origin of the bands, so that we can tell that D1 band, about 1350 cm⁻¹, "is usually related to disordered graphite structures, sp² hybrid carbon atoms, aromatic rings, edge carbon atoms, and in-plane vibrations with structural defects" (Yin et al., 2018, and references therein). The D3 band, about 1500 cm⁻¹, is coming from the "amorphous sp²-bonded carbon, fragments, or functional groups in the disordered structure" (Yin et al., 2018, and references therein), and so on for the other bands. The G band is told representing the "stretching vibration of aromatic rings and the crystal structure of sp² carbon atoms" (Yin et al., 2018).

In the article by Xu et al., 2020, it is told that Ferrari and Robertson, 2000, "pointed out that the Raman spectrum was the state's vibrational density modified by a coupling coefficient, incorporating various resonances. *Therefore, there was no special reason for choosing a particular function to fit the spectrum*" (Xu et al., 2020). And this is a very interesting observation. "Generally, the symmetric-line fit with two Gaussian lines or two Lorentzian lines were adopted in this method. The most widely used alternative to a Gaussian fit was a Breit-Wigner-Fano (BWF) line for the G band and a Lorentzian fit for the D band" (Xu et al., 2020). The line BWF is asymmetric. For the use of BWF in the Raman spectra of graphene, see please (Hasdeo et al., 2014).

"Other researchers also found that the combination of a Gaussian fit for the D band and a Lorentzian fit for the G band is more suitable for the highly disordered" carbonaceous materials (Xu et al., 2020). We have another interesting further observation proposed by the researchers: "When curve-fitting the Raman spectrum with the two bands, [~1350 cm⁻¹, ~1590 cm⁻¹] no variable parameter is needed to be fixed" and this is a relevant feature "to establish a common approach" (Xu et al., 2020). But, by using just two bands it is difficult to have an "ideal curve-fitting", and for this reason the multiband fitting has been developed (Xu et al., 2020). Then, it is possible to find used three bands, with an additional band "originally included to improve fitting precision in the deconvolution of Raman spectra. ... In these methods, the Gaussian line was preferred for the band at about 1500 and 1200 cm⁻¹, while the Lorentzian line was more used for the band at about 1620 cm⁻¹ ... Subsequently, some authors integrated the above methods and developed the four bands method and five bands methods" (Xu et al., 2020). And then, see please again the Table 2 by Xu and coworkers and references given therein.
In Ferrari, (2007), we can find Raman spectroscopy applied to graphene and graphite. "The toll for the simplicity of Raman measurements is paid when it comes to spectral interpretation. The Raman spectra of all carbon systems show only a few prominent features, no matter the final structure, be it a conjugated polymer or a fullerene" (Ferrari, 2007, mentioning Ferrari & Robertson, 2004). The spectra appear quite simple with "a couple of very intense bands in the 1000–2000 $cm^{-1}$ region and few other second-order modulations. However, their shape, intensity and positions allow to distinguish a hard amorphous carbon, from a metallic nanotube, giving as much information as that obtained by a combination of other lengthy and destructive approaches" (Andrea C. Ferrari, 2007, mentioning Ferrari & Robertson, 2004). In the article by Ferrari, it follows a detailed discussion about the main features of the bands, with physical interpretation.

After the given literature, we can tell that Lorentzian, Gaussian, Voigt, pseudo-Voigt (linear combination of Gaussian and Lorentzian functions) and Breit–Wigner–Fano (BWF) line are mainly used. However, the problem is not only given by the choice of line shape.

## Discussion B - Biasing factors

In the article by Lünsdorf et al., 2014, we can find mentioned the "bias". The article is a discussion about the problem of comparing the results of Raman spectroscopy of carbonaceous materials, that is, the RSCM parameters obtained in different research works. The comparability "is low as there are at least three major sources of biasing factors. These sources are the spectral curve-fitting procedure, the sample characteristics itself and the experimental design including the used Raman system" (Lünsdorf et al., 2014). Moreover, the researchers have also shown that the "curve-fitting is strongly influenced by individual operator-bias and the degrees of freedom in the model, implying the need for a standardised curve-fitting procedure" (Lünsdorf et al., 2014). And we have also another fundamental fact: "Due to the diversity of components (optics, light detection device, gratings, etc.) and their combinations within the Raman systems, different Raman instruments generally give differing results" (Lünsdorf et al., 2014).

The consequence given by Lünsdorf and co-workers is that to estimate comparable RSCM data, "every Raman instrument needs its own calibration". Therefore, it is necessary to have a series of reference materials.

The Raman spectroscopy is a condensed matter of "several biasing factors", arranged in three categories: 1) the first is a bias regarding the curve-fitting, 2) the second is related to the intrinsic nature of CM and 3) the third bias is related to the experimental set-up and related Raman instrument which is used. In the first category we can find the different baseline correction methods, the different line shape functions (Gaussian, Lorentzian, Voigt, etc.), and the different number of bands used as a model. In the second category, we find the sample preparation and heterogeneity. In the third category, we can find for instance the excitation wavelength, the used spectral grating and light detector device.

In Lünsdorf et al., 2014, the above-mentioned issues are analysed. Let us stress that in the article we find also the "Influence of operator’s personal fitting strategy". The researchers used "The spectral processing … performed once with Lorentzian functions only and once with Voigt functions only" (Lünsdorf et al., 2014). In the study by Lünsdorf and co-workers, software Fityk has been used for peak- and curve-fitting (Wojdyr, 2010); "the position and shape of the components are detected automatically by the software. If this is not successful, the components are located manually" (Lünsdorf et al., 2014).

## Discussion C - Fitting the bands

After the discussion about the D and G bands, it is better to add some deeper points, that we can borrow from the article by Robert Meier, 2005, where the author proposed "some crucial issues in curve-fitting vibrational spectra that do not always seem respected". Here in the following some points.
1) In the case of isolated bands, no problem. "Problems arise when there is band overlap" (Meier, 2005). The fitting procedure is used to determine the bands. Curve fitting can also be considered as "modelling" because we need a model for the line shapes.

2) "Curve fitting is finding the best fit to an overlapping band profile starting from a prescribed set of individual bands" (Meier, 2005). And this is what we have found in the previously reported literature.

3) The curve fitting is sometimes also called "deconvolution". However, according to Meier, in the vibrational spectroscopy "deconvolution is generally the process in which instrumental effects are removed from a spectrum" (Meier, 2005). We have that the spectrum is the convolution of the physical line shapes and the broadening caused by instrumentation.

4) A Gauss-Lorentz sum function $S(\omega) = \alpha G(\omega) + (1 - \alpha) L(\omega)$ (linear combination of functions), "is not uncommon" (see Meier, 2005, and references therein), "but rarely justified". $\omega$ is the angular frequency. We will discuss the Gaussian-Lorentzian model in a specific section.

5) The line-profile in Raman spectra "is Lorentzian in nature", and Meier is mentioning that there are different manners to arrive at a Lorentzian line. One way refers to a hydrodynamic theory, but more recently we find NMR, optical and mass spectroscopy (Meier, 2005, referring to Marschal & Verdun, 1990). As a simple example, Meier is giving the damped harmonic oscillator.

6) The "shape of individual vibrational bands is thus in essence Lorentzian" (Meier, 2005), but we can have a Gaussian line broadening produced by instrumentation. "Broadening with a Gaussian line profile implies that one has to multiply the Lorentzian profile with a Gaussian profile, leading to the so-called Voigt profile" (Meier, 2005). Then, we arrive to the convolution of a Gaussian G and a Lorentzian L function.

The Voigt profile (by Woldemar Voigt) is the following probability distribution function:

$$V(x; \sigma, \gamma)=\int_{-\infty}^{\infty} G(x'; \sigma)L(x-x'; \gamma)dx' \quad (1)$$

7) "The main difference between a Gauss profile and a Lorentz profile is that the latter one has very long tails" (Meier, 2005). And then a "lot of the overall intensity" is buried in tails, "which we generally do not consider in a spectrum fit" (Meier, 2005). Really important observation. "The Voigt profile is obviously 'a situation in between'." Very interesting indeed: a q-Gaussian is 'a situation in between' too. However, see please the detailed discussion given by Meier, 2005.

8) First important remark. Broadening effects lead to convolution. "The regularly used sum of a Lorentz and a Gauss profile has no physical basis, unless ..." see please detailed discussion in Meier, 2005.

9) To fit the spectrum the number of bands is not crucial, "as even two clearly overlapping bands may cause a challenge comparable to an envelope comprising a large number of bands" (Meier, 2005). Taming the bands is a difficult issue. The curve fitting is accomplished by the selection of the wavelength range where there are the bands relevant for "quantifying certain functional groups or constituents of the sample involved" (Meier, 2005). After the spectral range is defined, the base-line determined and then the line shape chosen.

10) Second remark, about the subtraction of background. It is easily performed by software packages. However, there is a problem, because the Lorentz profile has a fat tail and in it we find some intensity of the line; "it is preferred to fit both the envelope and the background simultaneously" (Meier, 2005).

11) About the best fit: "mathematically the best fit (according to least squares fitting) is not a priori the best fit according to the physics of the system. Any good fit has small residuals, but even then various different
fits may have acceptably small residuals" (Meier, 2005). An excellent visual result coming from a very low value of the least square fitting does not ensure the physics of the fitting.

12) Let us conclude with the third remark given by Meier, 2005. Have "proper pre-selection of the number of bands, their width and position are highly important".

## Discussion D - Broadening the bands

We have mentioned, about Lünsdorf et al., 2017, the Voigt profile used for Raman bands, and we have also found this profile in the article by Meier, 2005. Let us read his specific words. Meier considers the Lorentzian essence of the line shape, experiencing a "Gaussian line broadening due to instrumental effects or sample characteristics". However, we have also the *Doppler broadening* which is occurring with Gaussian character. "The same holds for the broadening due to the use of a monochromator with rectangular entrance and exit slits. ... Broadening with a Gaussian line profile implies that one must multiply the Lorentzian profile with a Gaussian profile, leading to the so-called Voigt profile" (Meier, 2005). Once again, let us remember that the Voigt profile is the convolution of a Gaussian and a Lorentzian profile.

Linked to the Doppler broadening, we have also the Dicke narrowing (Ciuryło et al., 2001). In the case that the mean free path of an atom is quite smaller than the wavelength of the radiative transition, a narrowing of the line is produced. The atom is changing its speed and direction several times during photonic emission or absorption. In average on the different Doppler states, we find that the atomic line width is narrower than the Doppler width.

From book by W. Demtröder, 1985, entitled "Laser spectroscopy: Basic concepts and instrumentation", let us consider width and profile of a spectral line. The spectral lines are never monochromatic. Even in the case of high-resolution interferometers, we observe a spectral distribution $I(\nu)$ of absorbed or emitted intensity, around the central frequency $v_0$ which is corresponding to an atomic or molecular transition. The central frequency is given by the energy difference between energy levels. The function $I(\nu)$ in the vicinity of $v_0$ is called the line profile. The frequency interval between the two frequencies $v_1$ and $v_2$, for which $I(v_1) = I(v_2) = I(v_0)/2$ is the Full Width at Half Maximum of the line (FWHM), that is the linewidth or half-width of spectral line. The spectral region within the half-width is the "kernel" of the line, the regions outside ($v < v_1$ and $v > v_2$) are the wings (or also the tails).

Let us now consider the natural line shape. An excited atom can emit energy as spontaneous radiation. To have the spectral distribution of a spontaneous emission, according to the difference between energy levels, the simplest model for the excited atomic electron is a classical one with the electron considered as a damped harmonic oscillator (Lorentz model). Being damped, the amplitude $x(t)$ of the oscillation turns out with a frequency lower than the frequency of the undamped case, and the amplitude is decreasing with time. Because the amplitude $x(t)$ is decreasing gradually with time, the frequency of the emitted radiation is not monochromatic, but possesses a frequency distribution related to the function $x(t)$ by a Fourier transformation. The exponentially decreasing amplitude is therefore corresponding to a Fourier transform with a Lorentzian line shape.

In the book by W. Demtröder, 1985, we can also find the spectral profile of an absorption line, derived from atoms at rest. However, being atoms and molecules in thermal motion, we have the additional effect previously mentioned, that is the broadening of the line profile known as Doppler broadening. Then, the Lorentzian-line profile is the natural line, but Doppler effect changes the line.

Let us consider an excited molecule with a velocity in the rest frame of the observer. The central frequency of a molecular emission line has a value in the coordinate system of the molecule, but it is Doppler shifted in the frame of the observer. We know that at thermal equilibrium, the molecules of a gas, for instance, are following a Maxwellian velocity distribution. Since the emitted or absorbed radiant power is proportional to the density of molecules emitting or absorbing in the frequency interval $dv$, the intensity profile of a
Doppler-broadened spectral line becomes a Gaussian profile with a full half-width, which is called the Doppler width. But in Demtröder, 1985, it is stressed that "More detailed consideration shows that a Doppler-broadened spectral line cannot be strictly represented by a pure Gaussian profile". The reason is that not all molecules with a definite velocity component are emitting or absorbing at the same frequency. Let us remember that the response of these molecules is represented by a Lorentzian profile, so we arrive at the convolution of Lorentzian and Gaussian profiles, that is at the Voigt profile. To the Doppler effect, we have also to add the broadening due to elastic and inelastic collisions (see details in Demtröder, 1985).

For what concerns the Raman Scattering, it can be considered as an inelastic collision of an incident photon with a molecule in the initial energy level $E_i$. After the collision a photon with lower energy is detected and the molecule is found in a higher-energy level $E_f$ (Stokes radiation). The energy difference $E_f - E_i$ may appear as vibrational, rotational or electronic energy of the molecule. If the photon is scattered by a vibrationally excited molecule, it may gain energy and the scattered photon has a higher frequency. This scattering is called anti-Stokes radiation (see please the detailed discussion in Demtröder, 1985).

## Discussion E - Tsallis line shape in EPR

In Howarth et al, 2003, it is given a generalization of the line shape, considered as useful in electron paramagnetic resonance (EPR) and possible also in nuclear magnetic resonance (NMR) spectroscopy. And this line shape is defined as "Tsallis lineshape", however it is a q-Gaussian. In the introduction, Howarth and co-workers provide useful information. Here some remarks from their article.

1) Since the initial measurements, the line shapes have been considered as Lorentzian or Gaussian. These functions are simple, with a well-established theoretical basis for their use.

2) A Lorentzian line shape is justified by the linear response theory (Howarth et al. are mentioning R. Kubo, in J. Phys. Soc. Jpn. 9 (1954) 888, but it is Kubo & Tomita, 1954). Following Kubo and Tomita model, it is assumed the magnetization of the sample as oscillating harmonically in external magnetic field. In the presence of a dissipative phenomenon, due to energy transport from the spin system to a heat sink, "the linear response to the electromagnetic (microwave) radiation" produces an exponential decay of the magnetization, and consequently a Lorentzian absorption line "in a continuous-wave EPR (or NMR) absorption measurement" (Howarth et al, 2003).

3) Gaussian line shape is coming from "the inhomogeneity of the resonance field present at each paramagnetic centre. Thus, besides inhomogeneity of the external magnetic field, local stresses, and material imperfections cause all the paramagnetic centres ... to experience a slightly different interaction with neighbouring atoms" (Howarth and co-workers are mentioning Van Vleck, 1948).

4) Then we find statistics. Interactions are giving the "local magnetic field" acting on each centre, and the related resonance frequency. These frequencies can be described statistically. Quoting the central-limit theorem, we arrive to the Gaussian line shape (Howarth et al. mentioning O. Svelto, 1998).

5) In experiments, neither Gaussian nor Lorentzian line shapes are giving satisfactory results. "The effect of rapid changes of the individual centres magnetization vectors can lead to a dynamic averaging of the individual distributions, leading again to a *Lorentzian line-shape at the centre while the wings of the line remain Gaussian*. This effect is known as motional narrowing" (see Howarth et al. and the works by Anderson and Weiss mentioned by them). Let us stress that the motional narrowing is regarding the natural line shape, but we have also the effect of the instrumentation and of electronic amplification (see the discussion in Seshadri and Jones, 1963).

6) However, and here we find the proposal by Howarth et al., the "inhomogeneous broadening does not necessarily result in a Gaussian distribution"; then we arrive to different line shapes. We find the Voigt line shape function, where Lorentzian and Gaussian shapes are two special cases. Then we have also other generalizations (see references given by Howarth et al.). "A natural extension of the line-shape function is
a Levy distribution", but "the major disadvantage of the Levy distributions is that ... there is no analytical expression for these line-shape functions" (Howarth et al., 2003).

7) Extensions of Gaussian and Lorentzian functions are "the distributions introduced by Tsallis". In this manner we arrive to the Tsallis line shape functions (that is q-Gaussians).

8) "In addition to the linewidth parameter, the shape of the line is controlled by one more parameter (q) that determines the contribution of the line wings to the integral intensity" (Howarth et al., 2003).

Then, let us stress that the q-Gaussians, that we are here considering for the Raman line shapes, are the "Tsallis line shapes" proposed in the framework of Kubo-Tomita model.

In R. S. Johal, 1999, we can find an interpretation of Tsallis statistics. In the work by Johal, it is told that P. A. Alemany, 1997, conjectured that the Tsallis formalism is related to a scale invariant statistical mechanics. Therefore, the Tsallis distribution can be obtained "by assuming an ensemble which has replicas of the same system at different scales" (Johal, 1999). Physical examples are the poly-dispersed systems like "polymers, commercial surfactants, colloidal suspensions and critical spin systems on hierarchical lattices". Johal is presenting the Tsallis distribution in the framework of colloidal polydispersity.

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