## 4 Theoretical Background and Simulations
### Monte Carlo Simulation
Once again, due to the statistical nature of the interaction mechanism, a significant number of light rays is required for accurate predictions. With parameters resembling those of the experiment, that is a \(4.5\,\mathrm{mm}\) thick TiO\({}_{2}\)-rutile sample with a volume filling fraction \(f_{V}\approx 55\%\), a particle density of \(4.23\,\mathrm{mg\,mm^{-3}}\) and a particle size distribution as shown in Figure 15, no transmittance through the sample is observed for 25 million nearly collinear rays (beam divergence \(\leq 1^{\circ}\)). The surrounding material has been assumed as air with a density of \(0.119\,\mathrm{mg\,mm^{-3}}\) and a refractive index \(n_{\mathrm{air}}=1.0003-0i\). This underlines the validity of the K-M relation for pressed powders: a thickness of a few millimeters is sufficient for canceling transmission. Consequently, this geometry is therefore sufficient for the determination of \(R_{\infty}\). In fact, light rays may be scattered several hundred to thousands times inside the sample relatively close to the surface before either exiting through the entrance plane or being absorbed at a particle (cf. the inset of Figure 17). The interplay between scattering and absorption is decisive in the development of the overall diffuse reflectance: A higher scattering coefficient increases the number of scattering events and, thus, the total absorption probability (that is a product between absorption probability at a single particle times the number of events), as does a growing absorption coefficient. As a result, these two quantities appear coupled when performing diffuse reflectance measurements. Taking scattering and absorption coefficients into account allows to calculate the diffuse reflectance as a function of wavelength. For rutile-phase TiO\({}_{2}\), the simulated diffuse reflectance spectrum shows significant resemblance to data coming from both literature [188; 189] and present measurements as it is shown in Figure 18.  
Figure 18: Monte Carlo ray tracing simulation results (circles) for a rutile powder sample (experimental data: solid blue line) and calculated imaginary refractive indexes \(m^{\prime\prime}\) (filled squares, red, solid line) and absorption coefficient \(\alpha\) (open squares, red, dashed line). Throughout the visible range above 450 nm, the absorption remains nearly fully wavelength-independent and only appears to grow in the infrared spectral region. However, this is likely subject to measurement artifacts as no significant absorption between 730 nm and 4000 nm has been reported [161]. For wavelengths below 430 nm, the diffuse reflectance declines significantly which can be attributed to a growing contribution from absorption induced by the material's band edge. Compared to literature data measured on thin films [161], the calculated absorption coefficient \(\alpha\) is in remarkable agreement, especially for wavelengths below 390 nm. The striking similarity of \(\alpha\) reveals agreement between thin film and diffuse reflectance measurements in this specific case. Therefore, DRS paired with Monte Carlo simulations opens up the possibility to quantitatively determine the absorption coefficient. This result has a tremendous impact on material science: A number of materials do not exist as bulk media or with adequate quality for alternative optical characterization methods such that DRS hereby provides ready and exclusive access. Both simulations, ab initio and Monte Carlo, yield similar absorption spectra based on identical diffuse reflectance data, particle size distributions and real refractive index. However, ab initio calculations tend to overestimate \(\alpha\) by an order of magnitude due to the exclusion of multiple scattering events.