## 3 Results and Discussion
For the same reason, the differences in direct and indirect band gaps for a given material fall within the 95% CI, except for TiO\({}_{2}\)/Cu (0.05%)Zr (1%), which shows a difference of 0.60 \(\pm\) 0.25 eV. Although a decrease in the value of \(E_{g}\) for direct as well as indirect band gaps was expected after the TiO\({}_{2}\) modification, which leads to a decrease in the material's grain size [20,21], these changes were not significant at 95% CI. Such insignificant differences can be attributed to insufficient measurement sensitivity, which results in larger standard deviations of the determined band gap values. Consequently, the differences in band gaps of the investigated materials could not be confirmed by UV-Vis spectrometry transmission mode measurements. For this reason, the BDS technique, which is known for its high sensitivity and is less affected by light scattering within the sample, was applied for determination of the band gaps and relevant thermal and transport parameters in the characterized samples. The BDS signal results from the conversion of absorbed energy into heat. The heat generated in the BDS experiment depends on the power density of internal heat sources, which, in turn, are governed by intra-band transitions in the bulk, non-radiative inter-band transitions and non-radiative surface recombination [22]. Thus, only indirect band gaps can be determined in these experiments, since the indirect band gap is related to the non-radiative transitions between the energy levels with the presence of phonons in semiconductor material. Direct band gaps, on the other hand, are related to the radiative transitions with the emission of photons, which do not generate heat, and therefore, no BDS signal can be observed. In the BDS measurements, the amplitude and phase of the signal were recorded as a function of modulation frequency of the pump beam (Figure 9) for unmodified and Zr- and/or Cu-modified TiO\({}_{2}\) samples. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample** & **Direct Band Gap, eV** & **Indirect Band Gap, eV** \\ \hline TiO\({}_{2}\) & 3.00 \(\pm\) 0.20 & 3.35 \(\pm\) 0.25 \\ TiO\({}_{2}\)/Cu (0.05\%) & 2.85 \(\pm\) 0.20 & 3.20 \(\pm\) 0.25 \\ TiO\({}_{2}\)/Zr (1\%) & 2.75 \(\pm\) 0.20 & 3.30 \(\pm\) 0.25 \\ TiO2/Cu (0.05\%) Zr (1\%) & 2.55 \(\pm\) 0.15 & 3.15 \(\pm\) 0.20 \\ \multicolumn{3}{c}{95\% CI} \\ \multicolumn{3}{c}{TiO\({}_{2}\)} & [2.66; 3.34] \\ TiO\({}_{2}\)/Cu (0.05\%) & [2.51; 3.19] & [2.93; 3.62] \\ TiO\({}_{2}\)/Zr (1\%) & [2.41; 3.09] & [2.88; 3.72] \\ TiO\({}_{2}\)/Cu (0.05\%) Zr (1\%) & [2.30; 2.80] & [2.81; 3.49] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the direct and indirect band gaps and 95% CI of the unmodified and Cu- or/and Zr-modified TiO\({}_{2}\) obtained by UV–Vis spectrophotometry and Kubelka–Munk theory. The BDS signal falls off with the increase in the modulation frequency. Furthermore, both the amplitude and phase of the BDS signal clearly differ between samples of different compositions since they have different thermal (thermal conductivity and diffusivity), transport (carrier lifetime) and optical (band gap) properties. In fact, besides the band gap, BDS makes it possible to also determine the thermal and transport properties of materials. This can be accomplished by the multiparameter fitting process, in which the best fit of the theoretical model [21; 22; 23; 24] to the experimental data is achieved by the least-squares procedure. The applied theoretical model interrelating the thermal, optical and transport properties of the material with the amplitude and phase of the BDS signal is presented in detail in the literature [21; 22]. The best theoretical fits to the experimental data [21; 22] were obtained for the values of the thermal diffusivities and conductivities, band gaps and carrier lifetimes presented in Table 2.  
It can be seen from the results presented in Table 2 that modification of the TiO\({}_{2}\) surface leads to changes in the thermal, transport and optical properties of the material.