## 3 Results and Discussion
Furthermore, the size of particles in the sample (\(\sim\)25 nm) was much smaller than the wavelength of incident light. Due to this, there were no secondary contributions of reflection, refraction and diffraction to the diffuse reflectance signal. Figure 6: Photoabsorption curves for TiO\({}_{2}\) material showing the presence of direct (1—rapid increase in \(\alpha\) with the increase in the incident photon energy) and indirect band gap (2—slow increase in \(\alpha\) with the increase in the incident photon energy). Figure 7: Diffuse reflectance spectra of unmodified and Cu- or/and Zr-modified TiO\({}_{2}\) recorded on a UV–Vis spectrophotometer. As shown in Figure 7, the TiO\({}_{2}\)-based materials showed strong absorption in the ultraviolet region, with the absorption edge located at around 400 nm. The values of direct and indirect band gaps were obtained from the diffuse reflectance spectra of unmodified and Cu- or/and Zr-modified TiO\({}_{2}\) recorded on a UV-Vis spectrophotometer (Figure 7) on the basis of the Kubelka-Munk theory [17]. The indirect relationship between the band gap \(E_{g}\) and the linear absorption coefficient \(\alpha\) was obtained from a plot of \((\alpha hv)^{n}\) versus the photon energy \(hv\) using the following equation [17]:  
\[(\alpha hv)^{n}\propto hv-E_{g} \tag{2}\]  
where \(E_{g}\) is the band gap of allowed transitions (eV), \(h\) is [PERSON]'s constant (6.63 \(\times\) 10\({}^{-34}\) Js), \(v\) is the frequency of the light (s\({}^{-1}\)) and \(n\) is the number characterizing the transition process (\(n\) = 0.5 corresponds to the indirectly allowed transition and \(n\) = 2 corresponds to the directly allowed transition). In order to determine the values of the band gaps for examined TiO\({}_{2}\), as well as Cu- and/or Zr-modified TiO\({}_{2}\) samples, the graph of \([\alpha hv]^{n}\) as a function of photon energy E = \(hv\) was plotted (Figure 8). Extrapolation of the linear parts of the graphs gives the values of band gaps as the intercept to the abscissa axis, for which \([\alpha hv]^{n}\) = 0. The determined values of both direct and indirect band gaps for the examined samples are presented in Table 1, along with their 95% confidence intervals (CI). The fitting accuracy of the determined parameters (SDs) was estimated in both surface and frequency scan method by calculating the square root of the covariance matrix [13; 19]:  
\[S_{p}=\sqrt{cov(P)} \tag{3}\]  
where \(P\) is the fitted parameters matrix and  
Figure 3:0  
where \(\sigma_{r}\) is the variance on residuals:  
Figure 3:1  
where \(N\) is the number of points in the dataset, \(J_{f}\) denotes the Jacobian matrix and \(P^{\prime}\) is the matrix of fitted parameters for which the minimum of error function was reached. Figure 3:2 Extrapolation of the direct (**a**) and indirect (**b**) band gaps on the basis of \([\alpha hv]^{n}\) versus hv dependence. To determine the source of uncertainty, the repeatability of the method was also tested. For that purpose, the measurements were repeated three times and the desired parameters found by the use of the multiparameter fitting procedure described above. To determine if significant differences exist between the obtained parameters, the \(p\)-values were calculated for all examined parameters at 95% CI. For all the calculated cases, the \(p\)-value was smaller than 0.05, which means that there were not significant differences between parameters' values in the single determination. Thus, the source of determination uncertainty results from the optical approach, but not from batch variations. With consistently higher values for the indirect band gap in all materials, a trend of decrease in band gaps was observed for Cu- and/or Zr-modified materials with respect to pure TiO\({}_{2}\). However, due to the relatively high standard deviations (5.9-7.3% RSD for direct band gap and 6.3-7.8% RSD for indirect band gap), no difference can be confirmed for the examined materials at 95% CI.