Effect of TiO 2 on optical properties of glasses in the soda-lime-silicate system

Titania is widely considered as an alternative constituent for replacing heavy metal oxides in optical glasses. Its effect on optical properties, however, is complex. This is due to the dielectric properties of the prevalent ionic species, Ti, the potential co-existence of trivalent titanium, Ti, giving rise to intrinsic and extrinsic charge transfer reactions, and the existence of different coordination polyhedra, depending on matrix composition. Here, we present a systematic study of the optical properties of the soda-lime-silicate glass system as a function of TiO2 addition. We consider the silica-rich region of the SiO2-Na2O-CaO-TiO2 quaternary, which may be taken as model for a variety of technical glasses. Trends are described in the refractive index, the Abbe number, the optical bandgap and the Urbach energy. The addition of TiO2 increases the refractive index and the optical dispersion while it lowers the optical bandgap and the Urbach Energy. Results are discussed in relation to relevant literature data towards using titania silicate glasses as high-index replacements for heavy metal – containing oxide glasses. ©2016 Optical Society of America OCIS codes: (160.2750) Glass and other amorphous materials; (160.4670) Optical materials; (160.4760) Optical properties.


Introduction
Many applications of glass rely on their optical properties, where glasses are among the few solid materials which transmit visible light.This is because without the specific incorporation of localized electronic states with low activation energy, most inorganic glassy materials are electronic isolators which do not absorb light in the visible spectral range, i.e., without going into details, the electronic orbitals, atoms, molecules or chemical bonds in glassy materials do not resonate with visible photons [1].This in itself, however, is not a unique feature of glasses.More importantly, glassy materials present a degree of homogeneity which covers a very wide range of length scales, and glasses are usually ideally isotropic.These properties lead to the practical absence of visible light scattering, and also to the absence of birefringence.Unless glass is dedicatedly doped with specific coloring (or scattering) agents, glass-light interaction is, in a simplistic view, dominated alone by the polarization of dipoles with very low dipole interaction (symmetry) and, hence, low light refraction as compared to a hypothetic single crystal of identical composition.As long as the glass composition does not undergo dramatic changes (i.e., staying in the area of oxide or even silicate glasses), the resonance function of this polarization remains relatively similar in shape, and compositional variations act mainly on the position of the resonance frequency.This aspect has led to an empirical understanding where optical refraction and optical dispersion (Abbe number) are roughly correlated, leading to the so-called banana-shape of the data distribution in the Abbe diagram [1].Adding TiO 2 to a glass has been considered as a way to at least partially push these borders, and TiO 2 -doped optical glasses have consequently found applications which rely on the high field strength and polarizability of the relatively small tetravalent titanium ion when placed within in a large coordination polyhedron.
TiO 2 is a semiconductor with a typical band gap of 3.20 eV and 3.03 eV for the anatase and rutile structure, respectively [2].This means that doping a glass with TiO 2 changes the optical bandgap of the glass, so that the electronic resonance in the ultraviolet spectral range shifts to higher wavelength.The change of the UV absorption edge results in a decrease of the Abbe number in the visible spectral range, typically to below 50, i.e., to the region of socalled flint glasses.Typical titania containing optical glasses are light flints such as LF5HTi and LLF1HTi, but also deep flints (TF) such as high-TiO 2 containing fluorophosphate or fluorosilicate-based glasses [3].Several of the Kodak crown glasses developed by Morey also had high TiO 2 contents [4][5][6], other titania containing crown glasses are e.g.BK7HTi and FKSHTi.Titania was also used in combination with silica in the core of the first low-loss optical fiber (<20dB/km) [7], and it is extensively used in highly nonlinear optical glasses (e.g. the TiO 2 -Nb 2 O 5 -Na 2 O-SiO 2 [8].The low Abbe number at relatively high refractive index (the comparably strong dependence of refractive index on wavelength), has -besides optical applications -made TiO 2 an interesting component in stemware, glass jewelry, crystal ware and tableware, where the combination of these two properties leads to higher Fresnel reflection with high wavelength-dependence and, thus, more appealing appearance.However, while several studies have dealt with the effect of TiO 2 on the optical properties in the ternary system of TiO 2 -Na 2 O-SiO 2 [9][10][11][12][13][14][15], only little has been published in the scientific literature on the technically more relevant quaternary system of soda-lime-silicates doped with TiO 2 [16,17], as it is most often used in glass jewelry, crystal ware and tableware.In the present paper, we therefore provide a comprehensive study of the effect of TiO 2 on the optical properties in the soda-lime-silicate system.The glass series studied were composed according to the quaternary diagram in Fig. 1 while the Na 2 O content was kept constant.

Experimental
Glass samples as shown in Fig. 1 were melted at 1450 °C for 18 h, followed by a homogenization step of 1 h and a conditioning step of 2 h at 1500 °C in Pt/Rh10 crucibles.The homogenization consisted of stirring with a Pt/Rh10 flag at 8 rpm (about 48-50 Nm).Employed raw materials are listed in Tab. 1.We did not visually observe any Rh or Pt dissolution in the glasses (which would generate a characteristic violet-brownish coloration).Annealing was performed for 1h at 580-550 °C before the glass was cooled with initially 0.5 °C/min until 430-400 °C, and subsequently more rapid cooling to room temperature (approximately 2 °C/min).In total, three glass series were prepared as stated above.The actual chemical compositions of the melted glasses were determined by laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) without internal standard.Data are given in Table 2.The laser ablation unit was a LSX-213G2 + , CETAC and the ICP-MS was an X-series II, Thermo Fischer Scientific.The chemical composition was determined from the average of five analyses per sample.The uncertainty was determined from the standard deviation; it was ≤0.3 for Na 2 O, ≤1.3 for SiO 2 , ≤0.3 for CaO and ≤0.2 for TiO 2 (all in mol.%).As can be seen from Tab. 2, the total oxygen content was kept roughly constant over all three glass series.Refractive indices were measured using a V-block Pulfrich refractometer PR2, Carl Zeiss, with 90° polished specimens.The refractive indices n C' at 643.8 nm (red Cd line), n d at 587.6 nm (yellow He line), n e at 546.1 nm (green Hg line), n F' at 480.0 nm (blue Cd line) and n g at 435.8 nm (violet Hg line) were recorded.The molar electronic polarizability (α m ) was calculated from the refractive index of the glass according to Eq. (1), also known as the Lorentz-Lorenz equation as originally derived by Lorentz [18] and Lorenz [19].Here, V m is the molar volume, N is the amount of polarizable ions per mole, normally assumed to be equal to Avogadro's number and n is the refractive index (in the present work n d has been used).The molar electronic polarizability per oxide atom was calculated by dividing α m with the total number of oxygen (-II) atoms in the glass [20].
The optical dispersion (D) was calculated according to Eq. ( 2) and the Abbe number (ν e ) was calculated according to Eq. (3).
' ' 1 An exemplary powder X-ray diffraction (XRD) pattern of specimen 1.4 was obtained on a Siemens Kristalloflex D500 diffractometer over a 2θ range of 15-75°, step size 0.02°.XRD was performed in order to prove the amorphous nature of the specimen.
UV-Vis-NIR spectra were recorded on an Agilent Cary 5000 double-beam spectrophotometer with integration sphere.The recorded data interval was 1 nm, a scan rate of 600 nm/min and a spectral bandwidth (SBW) of 2 nm was used.The source changeover was at 350 nm and the detector as well as grating changeover was at 800 nm.A masking aperture of 10 mm 2 was used for all specimens.
The optical absorption coefficient (A) was derived by normalizing the absorption to the thickness (t), according to Eq. ( 4), where ln (I 0 /I T ) is the absorbance and I 0 and I T are the intensities of the incident and transmitted light, respectively.
All specimens were polished to 10 mm ( ± <0.1 mm) thickness, which means that for the present study, the absorption is roughly the same as the absorption coefficient.Noteworthy, in this consideration, we neglect eventual scattering losses, assuming that within the experimental regime of interest, the glass samples are ideally homogeneous.
( ) The optical absorption of topologically disordered materials follows Tauc's rule, Eq. ( 5), as was first shown by Tauc et al. [21] and further developed by Mott and Davis [22].Here, hc/λ is the incident photon energy, E opt is the optical bandgap and r denotes the nature of the transition.It is common to use r = 2 which corresponds to indirectly allowed transitions, as suggested by Rao et al [23].Rearranging Eq. ( 5) into Eq.( 6) suggests plotting left side of Eq. ( 6) as function of hc/λ.By linear extrapolating, the value of E opt can subsequently be derived.This plot is also known as Tauc's plot.

( )
( ) The Urbach law is given by Eq. ( 7) [24].Plotting ln A(λ) as function of the incident photon energy hc/λ enables determination of the Urbach Energy (ΔE) from the inverse of the slope of the linear region [23].

( )
The cut-off wavelength was determined from the maximum of each absorption spectrum, in some cases, the first maximum when approaching from longer wavelengths.
Structural data of selected samples were determined using X-ray Absorption Fine Structure Spectroscopy (EXAFS) at the I811 Beamline of Max-II at Maxlab, Lund University in Sweden [25].The Max-II storage ring was operating at 1.5 GeV with positron currents in the range of 280-100 mA.The EXAFS Ti K-edge (4966 eV) was measured in fluorescence mode, collecting data up to 5700 eV.The incident ion chamber was filled with a mixture of about 85 vol.%He and 15 vol.%N 2 gas.The sample holder chamber was filled with pure He gas.A 6 μm metallic Ti foil reference sample was measured between each sample.The selected samples that were measured were 1.4, 2.7, 3.3, 3.5 and 3.7 (Table 2).They were ground and sieved to < 45μm powder.Each sample was then uniformly dispersed between two x-ray transparent Kapton adhesive tapes in a plastic sample cell.Five scans were collected for each sample and each scan was performed over 180 s.The scans were averaged in order to increase the signal-to-noise ratio.To reveal local structural information on the Ti environment, in particular, the Ti-O coordination number, the corrected EXAFS data were fit to a TiO 2 (rutile) model with a space group of P42/mnm using the ARTEMIS software as implemented in the IFEFFIT package [26,27].

Results and discussion
Besides a small amount of light scattering which was observed when pointing a laser through the glasses, especially in the first series, all samples were visually transparent as well as essentially colorless and homogeneous to the naked eye.Some very slight yellow coloration is discussed in section 3.2.The scattering intensity can be seen in the inset of Fig. 2, especially for the specimen 1.2 and 1.4.It is jointly attributed to the presence of tiny bubbles distributed in the glassy matrix, and to eventual liquid-liquid phase separation [28,29].XRD was employed to verify the presence of any crystalline components in the samples, revealing that at least sample 1.4 was fully amorphous within the detection limits of the laboratory XRD at the employed scanning rate (i.e., to a crystallite size of < 2nm).Therefore we anticipate all samples to be fully amorphous.Phase separation was not further investigated in this study as it is not relevant for applications where thin glasses are used, e.g., stemware or tableware.

Refractive index and optical dispersion
The TiO 2 content has a large impact on the refractive index, as can be seen in Fig. 3. Overall, the data correspond fairly well to each other, with the exception of series 1 (where TiO 2 is systematically replacing CaO).Here, the effect of TiO 2 is absent, indicating that also CaO contributes to the refractive index, thus compensating the action of TiO 2 .As seen from the analyzed compositional data in Tab. 2, the TiO 2 content varies between 0 and 5.5 mol%, while CaO varies between 11.2 and 0.6 mol% in series 1, which means that the partial molar effect of TiO 2 on the refractive index is in fact larger than that of CaO.Series 1 is an excellent example on the possibility to reduce the Abbe number while keeping the refractive index constant (see Tab.   [9,11,14].The inset is a schematic of a V-block refractometer as it was used for measuring the refractive indices.Data of the present study are on n d (587.6 nm).For the cited data, Ref [14] is on n d , Ref [11] on n D (589 nm) and Ref [9] does not state the wavelength of consideration.
As has been discussed in detail by Duffy [20], the refractive index of a glass (i.e., a system of low symmetry) is derived from the molar electronic polarizability of the constituent ions through the Lorentz-Lorenz equation, c.f. Equation (1).In most cases (oxide glasses) the oxide ion polarizability has a much larger effect on the refractive index than the molar electronic polarizability of the cations except for those elements with unusually high polarizability such as, e.g., Pb 2+ and Ba 2+ .Duffy [20] showed that the effect of TiO 2 on the refractive index is similarly dominated by the molar electronic polarizability of Ti 4+ .The conclusion drawn by Duffy [20] reflects in the plot of the refractive index as a function of the molar electronic polarizability per oxide atom (Fig. 4), that is, through normalizing the oxide ion effect on the refractive index.The fact that Fig. 4 shows a fairly linear relationship confirms Duffy's reasoning that Ti 4+ has a large molar electronic polarizability relative to the other constituents of the present glasses (Duffy calculated it to be 1.6 Å 3 , comparable to Ba 2+ with a value of 1.55 Å 3 [30]).As can be seen in Fig. 4, some of the determined refractive indices are doubtful, e.g., some of the values in the series from Villegas [11] that has increasing molar electronic polarizability per oxide atom but not increasing refractive index.The optical dispersion is calculated from the difference between n F' and n C' , c.f. Eq. ( 2).For the visible spectral range, it is usually expressed as the Abbe number.Data for the optical dispersion and the Abbe number are shown in Tab. 3. As can be seen in Fig. 5, all series follow linear trends.With increasing TiO 2 content, the glasses change from crown (Abbe number > 50) to flint (Abbe number < 50).The TiO 2 content at the crossover is approximately 5.5-6 mol%.Abdel-Baki et al. [13] reported Abbe numbers in the series xTiO 2 -(60-x)SiO 2 -40Na 2 O (x = 0, 5, 15 and 20 wt%) ranging between 62.5 and 14.9 with TiO 2 content of 0 to 16 mol%.The value of 62.5 corresponds relatively well to the value of 58.5 found with no added TiO 2 ; however, with increasing TiO 2 content the reported values by Abdel-Baki et al. are considerably lower.In the absence of phase separation or other distorting effects, these lower values can be explained only with the considerably lower SiO 2 content (below 60.7 mol%) so that Ti 4+ is most likely acting as a network former.The considerably lower Abbe number also reflects the considerably higher estimate for the optical non-linearity of the glasses of Abdel-Baki et al [12], as given by the values of second-order index of refraction (n 2 ) [31] and the third order non-linear susceptibility (χ 3 ) [32].The estimated values of n 2 (10 −13 esu) range from 0.82 for the base glass to 6.78, and χ 3 (10 −13 esu) ranges from 0.03 to 0.26.On the other side, using the same approach for their estimation in the present study, we obtain ranges from 1.26 to 1.91 and from 0.05 to 0.08, respectively (the values were calculated according to semi-empirical relation of Boling et al. [33] for n 2 , see Eq. ( 8) and Eq. ( 9) given by Vogel et al. [32] for χ 3 , where K in Eq. ( 8) is an empirical constant equal to 391 [33]).
In Fig. 6, the data from series 1-3 are included into the Abbe diagram as it is commonly used for selecting optical glasses for specific device applications [34].Fig. 6.Representation of optical data of series 1-3 in the Abbe diagram, indicating the transition from flint to crown.Literature data on commercial optical glasses and indicated glass nomenclature have been adopted from [34] for referencing.

Optical absorption and absorption edge
The optical clarity of the glass specimens is displayed in Fig. 7 as the absorption in the visible range, 380 to 780 nm, and the average transmission of the same range as a function of the TiO 2 content in the inset.As can be seen in Fig. 7, the absorption spectra confirm that the glasses do not contain any significant amount of Ti 3+ (which would give rise to blue to violet coloration [35]).The presence of Ti 3+ could be determined by, e.g., electron spin resonance (ESR) and the Ti 3+ /Ti 4+ ratio by e.g.x-ray photoelectron spectroscopy (XPS).From Fig. 7, it can be seen that there is some apparent difference in the absorbance, and that the average T% becomes lower with increasing TiO 2 content.The latter observation is due to increased Fresnel reflection.As mentioned previously, a slight yellowish coloration was observed by the naked eye which is attributed to the Fe 2+ -Ti 4+ charge transfer reaction [36,37].It increases as the TiO 2 (and also the impurity Fe 2 O 3 ) content increase, as has been shown by Chavoutier et al. [38].The optical band-gap of TiO 2 is low in comparison to all other oxide constituents of the present glass (depending on configuration, i.e., anatase ~3.2 eV [39], as compared to amorphous SiO 2 , ~9.3 eV [40]).Therefore, when TiO 2 is added to a glass composition that mainly consists of SiO 2 , it changes the absorption edge.Here, the absorption edge shifts to higher wavelength, see Figs.The optical bandgap data as a function of TiO 2 content are given in Fig. 9 along with literature data [13,16,17].There is a clear trend, although it is not linear.The determined optical bandgap data in the present work is considerably higher than those reported in the literature for both the TiO 2 -Na 2 O-CaO-SiO 2 and the TiO 2 -Na 2 O-SiO 2 glass systems.Higazy et al. [16,17] studied the optical bandgap in TiO 2 doped soda-lime-silicate glass as a function of temperature and as a function of the TiO 2 content.The data provided by Higazy et al. at 298 K is recalled in Fig. 9, following a rather linear trend with increasing TiO 2 content.Furthermore, it was found in this previous study that the optical bandgap as a function of temperature also red-shifts linearly [17].The optical bandgap value of Higazy et al. [16,17] is somewhat lower than the values in the present work.However, as Higazy et al. do not reveal any other component amounts besides the TiO 2 content, it is difficult to deduce the origin of the lower values.In fact, the optical bandgap of fused silica is reported by Wemple [41] to decrease with the addition of other oxides, an effect which is increasing in the order CaO<BaO<Na 2 O<PbO.Optical bandgap data in the TiO 2 -Na 2 O-SiO 2 glass system have been reported by Abdel-Baki et al. [13] (shown in Fig. 9) as well as Hogarth and Khan [10], both somewhat lower than the values we report in the present work and without any obvious trend with the TiO 2 content.Abdel-Baki et al. [13] explained the higher values (see Fig. 9) of the optical bandgap to be due to the higher glass stability in the terms of the average bond strength of the glass.However, using Sun's [42] calculated single bond strength values to determine the average bond strength of the glasses according to Dimitrov and Komatsu [14], one can see that the average bond strength decreases in the series of Abdel-Baki et al. [13].In line with this information, Abdel-Baki et al. [13], in other words, suggest that the structure somehow rearranges as the TiO 2 content increases since the calculated bond strength values of Sun [42] do not take into account variations in cation coordination.The reviews on the structure of silicate glasses (including TiO 2 containing glasses) by Henderson [43] and also by Farges et al. [44][45][46][47] suggest that there are changes in the coordination environment of Ti 4+ with increasing TiO 2 content.The published information of the Ti 4+ coordination in the literature, though, seems contradictory in that some authors suggest the presence of 4-and 6coordinated Ti 4+ while others suggest also 5-coordinated entities [43].In conclusion, the low optical bandgap values which are found in the literature can be explained by the larger amount of network modifier species variations in the coordination environment of the titania species and the eventual presence of Ti 3+ (which may trigger Ti 3+ -O 2-charge transfer).Plotting the optical bandgap as a function of the SiO 2 content for the present data confirms that these arguments where the decrease in the SiO 2 content lowers the optical bandgap (Series 2 in Tab. 4).
The Urbach tail is the tail of the conduction band and the valence band, originating from localized states in disordered solids as originally introduced by Franz Urbach in 1953 [24].The width of the Urbach tail is frequently called the Urbach energy and can be determined from the exponential part of the absorption edge, see Figs. 8(a)-8(c) and Eq. ( 6).The glasses investigated in the present work were found to follow Urbach's rule, i.e., the absorption edge has an exponential part.In Fig. 10, the Urbach Energy shows a clear trend with increasing TiO 2 content.Samples from the 2nd Series deviate a little from series 1-3, indicating that CaO might affect the Urbach energy, as well.Kumar et al [48] also found that the absorption edge obeys the Urbach rule at low TiO 2 additions (0.1 wt%), however, no Urbach energy data were given.The Urbach energy has not frequently been determined, however, Abdel-Baki et al. [13], determined the Urbach energy values for some glasses in the TiO 2 -Na 2 O-SiO 2 ternary.However, except for the Ti-free specimen, we do not find any collation between these data and our own analyses.
Notable work has been performed on the optical properties of oxide glasses, culminating in the pioneering work by Duffy and Ingram [49,50].The understanding of optical properties and its relation to composition has been extensively reviewed by Dimitrov and Komatsu [51], and it has been shown that many of the properties relate to each other, at least theoretically.The determined values of the optical bandgap and the Urbach energy in the soda-lime-silicate glass system upon addition of TiO 2 derive from the molar electron polarizability.Its values, determined according to the Lorentz-Lorenz equation, Eq. ( 1), are given in Tab. 4 and visualized in Fig. 11.The relation is not very surprising as the molar electronic polarizability is linearly related to the TiO 2 content.As has been described by Henderson [43] and further by Farges et al. [44][45][46][47], Ti 4+ can be coordinated differently in different glass hosts and the study by Ponader et al. [52] showed that the coordination of Ti 4+ changes upon changes in the Na 2 O/TiO 2 ratio, and also upon changes in the Na 2 O/SiO 2 ratio, in ternary TiO 2 -Na 2 O-SiO 2 glasses.The glass series in the present work have a fairly constant Na 2 O/SiO 2 ratio, however, the ratio of the Na 2 O/TiO 2 ratio changes.As can be seen in Figs.12(a)-12(b), plotting the optical bandgap and the Urbach Energy as a function of Na 2 O/TiO 2 reveals distinct regimes.In accordance with the reasoning of Ponader et al. [52] and Duffy [20], the change in the Na 2 O/TiO 2 ratio leads to a change in the coordination number.This subsequently reflects in the optical bandgap and in the Urbach energy, which also depend on the coordination of Ti 4+ .The data of Figs.12(a)-12(b) suggest that there is a coordination change of Ti 4+ when the Na 2 O/TiO 2 ratio is about 3, which corresponds to a TiO 2 content of about 5-6 mol%.However, the different regimes of the optical bandgap and Urbach Energy do not reveal any direct information of the coordination states of Ti 4+ .Ponader et al. [52] suggest that upon decrease in the Na 2 O/TiO 2 , the proportion of six-coordinated relative to the proportion five-coordinated Ti 4+ is increased.Duffy [20] on the other hand suggests that upon decreasing the Na 2 O/TiO 2 ratio, the Ti 4+ is continuously transformed into the four-coordinated state (from the state of fivefold-coordination).Fargin et al. [53] showed from ab-initio calculations based on the Lines theory [31] that the single oxide molar electronic polarizability is higher for [TiO 6 ] −8 than for [TiO 5 ] −6 .The results by Fargin et al. indicate that there is a change in optical properties with the coordination change of Ti 4+ .
To clarify this controversy, EXAFS analyses were performed on the present samples (see Fig. 13 and Table 5).These clearly indicated that as the Na 2 O/TiO 2 ratio is below 3, the average coordination number is closer to 4, and as the ratio is above 3, the average CN is closer to 5.This suggests that the conclusions which were drawn by Duffy [20] are correct.Here, one should also bear in mind that the previous studies were performed in the ternary TiO 2 -Na 2 O-SiO 2 glass system, as compared to the presently studied quaternary glass system.
13. Fourier transforms of k3-weighted EXAFS spectra (solid lines) and fits (dashed lines) as well as the R-factor that represents the goodness of fit [54], the spectra have been phase shift corrected.The data derived by EXAFS, together with the information given in Figs.12(a)-12(b) enable a clear understanding of the absorption edge and its relation to the coordination of Ti 4+ .Similar to Figs. 12(a)-12(b) one can find different regimes when plotting the refractive index, the molar volume and the Abbe number as a function of Na 2 O/TiO 2 ratio.Our data show that the single oxide polarizability and, evidently, its impact on the refractive index depend on the size of the polyhedron, i.e., on the Ti-O distance.This experimentally verifies the Lines theory [31] and the ab-initio calculations made by Fargin et al. [53].

Conclusions
We provide a comprehensive study of the effect of TiO 2 on optical properties of glasses of the technically relevant soda-lime-silicate system.Compositional trends are obtained for the refractive index, the optical dispersion (Abbe number), the overall transmission, the optical bandgap and the Urbach energy.It is revealed how these trends are deriving from both the Ti 4+ polarizability and the chemically induced change in the coordination environment of Ti 4+ ion species.Our findings also experimentally verify that the single oxide polarizability depends not only on the properties of the cation but also on the cage size of the cation polyhedron, i.e., the M-O distance and the number of M-O bonds per unit volume.

Fig. 2 .
Fig. 2. Exemplary XRD pattern of specimen 1.4, showing that it is X-ray amorphous.The inset shows light scattering as a laser goes through.
3).Refractive indices at other wavelengths, given in Tab. 3, show similar behavior as a function of the TiO 2 content.

Fig. 3 .
Fig.3.Refractive index as a function of TiO 2 content.Literature data taken from Refs[9,11,14].The inset is a schematic of a V-block refractometer as it was used for measuring the refractive indices.Data of the present study are on n d (587.6 nm).For the cited data, Ref[14] is on n d , Ref[11] on n D (589 nm) and Ref[9] does not state the wavelength of consideration.

Fig. 5 .
Fig. 5. Abbe number and optical dispersion as a function of TiO 2 content.( ) 13 5/ 4 2 1 10 e d n n e u K V s −

Fig. 7 .
Fig. 7. Absorption coefficient spectra in the visible range (380 to 780 nm) and the average transmission in the same range as a function of TiO 2 content in the figure inset.
8(a)-8(c).The exact location of the absorption edge is usually determined in different ways such as by determining the cut-off wavelength or by determining the optical bandgap; data on both properties are given in Tab. 4. The cut-off wavelength ranges from 254 nm for zero TiO 2 content to 319 nm for 12 mol% TiO 2 content.The optical bandgap is a more conventional way to determine the absorption edge.It was determined by extrapolating the linear region of the Tauc's plot, a guide for the eye is given in the insets of Figs.8(a)-8(c).The determined optical bandgaps ranges from 3.86 eV for zero TiO 2 content to 3.55 eV for 12 mol% TiO 2 content.

Fig. 9 .
Fig. 9. Optical bandgap as a function of TiO 2 the lines are guides for the eye.Data from the present work and literature are taken from ref [13, 16, 17].The inset is a figure explaining indirect optical transitions, i.e. r = 2 in Eqs.(2) and 3.

Fig. 10 .
Fig. 10.Urbach energy as a function of TiO 2 The dashed line is a guide for the eye.The inset shows a schematic image of an optical bandgap with an Urbach tail.Literature data on ternary TiO 2 -SiO 2 -Na 2 O are taken from [13].

Fig. 11 .
Fig. 11.Optical Bandgap and Urbach energy as a function of molar electronic polarizability.

Fig. 12 .
Fig. 12. Optical bandgap (a) and Urbach energy (b) as a function of the Na 2 O/TiO 2 ratio.The dashed lines are guides to the eye.The different linear regimes indicate the Ti 4+ coordination change.

Table 5 . Average coordination number (CN) of Ti 4+ to oxygen as determined by the fitting of the EXAFS spectra and Ti-O distances (R) and Debye-Waller factors (σ 2 ) of the 1st and 2nd shell.
Na 2 O/TiO 2 ratio Avg.CN Ti-O Avg, distance Ti-O 1st shell