## 4 Theoretical Background and Simulations  
Diffuse reflectance stems from light scattering of single particles and the light-matter interaction between individual scatterers. Originally, scattering at single particles of arbitrary size was described by [PERSON] for spheres [118] yielding computationally expensive expressions that only nowadays can be calculated at relative ease [119]. A plane wave is scattered in a distinct fashion which differs significantly for varying material properties such as refractive index, absorption, or particle diameters. For example, the dependence of light scattering for different particle sizes is shown in Figure 13.  
Figure 12: Tauc plot of the non-contaminated rutile-TiO\({}_{2}\) (circles, cf. Figure 7a) and corresponding linear fit (solid line) for determination of the band gap \(E_{\mathrm{g}}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Transition Type** & \(n\) \\ \hline direct, allowed & 1/2 \\ direct, forbidden & 3/2 \\ indirect, allowed & 2 \\ indirect, forbidden & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Types of electronic dipole transitions and their corresponding exponent \(n\) to be used in Equations (4)–(6) [104; 105; 106; 107; 108; 109]. Mie theory, describing the general case of light scattering at spherical particles, can be approximated by less complex expressions for particles sizing towards either zero or infinity. For very small particle sizes in relation to the wavelength, i.e., \(d\ll\lambda\), the emission pattern is accurately described by Rayleigh scattering [120; 121]: Light in the perpendicular polarization, with respect to an incoming linearly polarized plane wave, is scattered isotropically whereas the parallel polarization is characterized by a dipole radiation pattern. Especially in the context of coherent processes in the visible (VIS) and near-infrared (NIR) range, the particle size should not exceed \(5\,\mathrm{nm}\) to \(10\,\mathrm{nm}\), although a large number of studies has used this treatment for larger particles as well [122] (sec. 2.5). In the opposite case, that is for particles sizing well above \(1\,\mathrm{\SIUnitSymbolMicro m}\), nearly all visible light is scattered into the forward direction with negligible scattering into other directions. This condition is the prerequisite for geometrical optics. In general, nanoscaled particles typically exhibit a rather homogeneous scattering pattern when illuminated with visible to infrared light. Nevertheless, for accurate calculations, Mie theory or derived approaches should be preferred, especially when more complex particles such as anisotropic materials [123; 124], multilayer systems [125; 126; 127; 128], or nonspherical objects or clusters [129; 130; 131; 132; 133; 134; 135] are considered. These theories work well in case of _independent scattering_, i.e., when radiative interparticle interactions can be neglected. Typical systems where this assumption holds true are diluted mixtures of colloidal suspensions, paints or pigments among several others [136]. With decreasing interparticle distances, that is when the interparticle clearance length \(c\) scaled by the wavelength \(\lambda\) approaches \(c/\lambda\leq 0.3\)[137], a considerable amount of scattered waves will interact with surrounding particles which is termed _dependent_ or _multiple scattering_. In regards to practical applications such as paint layers, films, powders, pellets, etc., significant effort has been made to incorporate dependent scattering into the theoretical description of light-matter interaction [136; 138; 139; 140]. Usually, light emitted from strongly packed particles is diffuse [1; 141] which was initially treated by [PERSON] and [PERSON] using a purely statistical approach [142]. The framework developed by latter accurately describes observed diffuse reflectance and hence plays a vital part in the evaluation of material characteristics today [1; 15].