## 3 Results and Discussions
### Single Particle Characterization
The function \(x(R_{\rm eq})\) together with the size distribution for the nanoparticle ensemble already displayed in Figure 1 are depicted in Figure 2.0  
Figure 2.1 **(a)** Optical extinction spectrum of the gold nanoparticle ensemble depicted in Figure 1(a) (solid line) and the modeled spectrum (squares); **(b)** Calculated function \(\hat{f}(a/b)\). To compare the size of the nanoparticles extracted by AFM and optical spectroscopy to the TEM data, the mean lateral diameter must be obtained. For this purpose, the most probable axial ratio has to be known. It can be obtained either by numerical integration of Equation 9 and calculating the maximum of \(f_{a/b}(x)\), or it can be obtained directly from Figure 2.2 The dotted arrow correlates the maximum of \(f_{R}(x)\) to the corresponding axial ratio, as indicated in Figure 2.3 For the investigated nanoparticle ensemble, the most probable axial ratio amounts to \(x(\langle R_{\rm eq}\rangle)=0.61\). Applying the definition of the equivalent radius  
Figure 2.4  
the mean lateral dimension, _i.e._, \(\langle 2b\rangle\), can be easily calculated. For the given parameters (\(\langle R_{\rm eq}\rangle=5.5\) nm and \(x(\langle R_{\rm eq}\rangle)=0.61\)) it amounts to \(\langle 2b\rangle=(13\pm 2)\) nm, which is in excellent agreement with \(\langle 2b\rangle=(12\pm 2)\) nm extracted from the TEM images. All determined morphological parameters are summarized in Figure 2.5  
We emphasize that although several approximations have been made, the agreement between the TEM data and the nanoparticle morphology extracted by theoretical modeling combined with AFM is excellent. The shape and size of the nanoparticles within the ensemble is well reproduced by the rigorous modeling of the inhomogeneously broadened extinction spectrum in quasistatic approximation, using only the number density of nanoparticles as input parameter. \begin{table}
\begin{tabular}{c c c c} \hline  & \(\mathbf{n_{\rm p}}\) / \(\mathbf{10^{11}}\) cm\({}^{-2}\) & \(\langle\mathbf{R_{\rm eq}}\rangle\) / nm & \(\langle\mathbf{2b}\rangle\) / nm \\ \hline TEM & \(3.1\pm 0.3\) & & \(12\pm 2\) \\ AFM & \(2.9\pm 0.3\) & \(5.5\pm 0.3\) & \\ Opt. Spec. + Modeling & \(3.4\pm 0.3\) & \(5.5\pm 0.3\) & \(13\pm 2\) \\ \hline \end{tabular}
\end{table} 
Table 1: AFM and optical spectroscopy _versus_ TEM. Figure 6: Axial ratio as a function of the equivalent radius (solid line) and size distribution (dashed line). The dotted arrow indicates the axial ratio for the most probable nanoparticle size in the ensemble. Note, Equation (9) relates the optical spectrum to the morphology of the nanoparticles. Hence, the optical spectrum can be fitted also by the morphological parameters:  
\[S_{\mathrm{R}}(\omega)=\frac{n_{\mathrm{p}}}{c}\int_{0}^{\infty}f_{\mathrm{R}}(R _{\mathrm{eq}})\frac{4}{3}\pi R_{\mathrm{eq}}^{3}\omega\times\mathrm{Im}\left\{ \frac{\varepsilon(\omega)-1}{1+[\varepsilon(\omega)-1]L(x(R_{\mathrm{eq}}))} \right\}\mathrm{d}R_{\mathrm{eq}} \tag{11}\]  
Inserting \(x(R_{\mathrm{eq}})\) in Equation (11), the spectrum can be modeled again, now using the morphological parameters. Hence, Equation (11) together with Equation (7) allows to proof if the optical data are consistent with the morphological data. For this purpose, the equivalent radius and the nanoparticle density are used as free parameters for modeling the optical spectrum with Equation (11). The best fit has been obtained for \(\langle R_{\mathrm{eq}}\rangle\cong 5.5\) nm and \(n_{p}\cong 3.4\times 10^{11}\) cm\({}^{-2}\). The extracted particle density is in fair agreement with the TEM data and only slightly larger than the nanoparticle density obtained by AFM. The latter fact can be explained by the limited resolution of the AFM, whereby very small nanoparticles might be not observed. However, the derived equivalent radius is in perfect agreement with the AFM data (cf. Section 3.1). Hence, the optical measurements combined with an adequate modeling yield the same morphological parameters as the AFM and TEM analysis.