## 3 Results and Discussions
### Single Particle Characterization
Hence, the red-shift of the plasmon resonance due to a dipole-dipole interaction between the nanoparticles can be neglected for nanoparticles with \(\langle R_{eq}\rangle=(5.5\pm 0.3)\) nm and a particle density of about \(n_{\rm p}=3\times 10^{11}\) particles/cm\({}^{2}\), as investigated here. To obtain the most probable axial ratio, a _probability distribution_ function of the axial ratios \(f_{a/b}(x)\) has to be introduced. For simplicity, we have defined the axial ratio as \(a/b=x\). \(f_{a/b}(x)\) has to satisfy the normalization condition [23]:  
\[\int_{0}^{1}f_{a/b}(x)dx=1 \tag{5}\]  
The integration limits correspond to spherical particles (upper limit) and infinite flat particles (lower limit). Since the optical spectrum is a linear superposition of the extinction spectra of single particles, the extinction spectrum of the ensemble can be expressed as:  
\[S_{a/b}=n_{p}\int_{0}^{1}f_{a/b}(x)\cdot V(x)\cdot s(x)dx \tag{6}\]  
where \(n_{p}\) is the density of particles. \(V(x)\cdot s(x)\) is the extinction spectrum of a single particle, with axial ratio \(x\) and volume \(V(x)\)[23]. Note that the volume of the particle is not an independent variable but rather a function of the axial ratio \(V(x)\). It is useful to define the function \(\hat{f}_{a/b}(x)\) as \(\hat{f}_{a/b}(x)=f_{a/b}(x)V(x)\), which describes the axial ratio distribution and the impact of the particle volume on the spectrum, thus:  
\[S_{a/b}=n_{p}\int_{0}^{1}\hat{f}_{a/b}(x)\cdot s(x)dx \tag{7}\]  
From reference [23] on can obtain \(s(x)\) to be:  
\[s(x)=\frac{\omega}{c}\mathrm{Im}\left\{\frac{\varepsilon(\omega)-1}{1+[ \varepsilon(\omega)-1]L(x)}\right\} \tag{8}\]  
Figure 4: Photon energy of the plasmon resonance versus normalized nanoparticle distance. The shaded area indicates the coverages used in this study. where \(L(x)\) is the depolarization factor known for any axial ratio \(x\), \(c\) is the speed of light, and \(\varepsilon(\omega)=\varepsilon_{1}(\omega)/\varepsilon_{\rm m}\). \(\varepsilon_{1}(\omega)\) and \(\varepsilon_{m}\) being the dielectric permittivities of the particle material and the surrounding medium, respectively. To find \(\hat{f}_{a/b}(x)\), Equation (7) is fitted to the experimental data by a spline with about ten to twenty sampling points, and the mean error squares are minimized [60]. This procedure holds, since the spectral dependence of the plasmon resonance on the axial ratio follows a steady and monotonic function [13]. Figure 5(a) demonstrates that the modeled spectrum \(S_{a/b}\) ideally reproduces the experimental one. The function \(\hat{f}_{a/b}(x)\), which has been extracted from the modeling, is depicted in Figure 5(b). It should be stressed here that the value of \(x=0.14\) at which \(\hat{f}_{a/b}(x)\) peaks is the axial ratio previously denoted as _apparent mean axial ratio_\(\langle\widetilde{a/b}\rangle\). It is neither the most probable axial ratio nor the mean axial ratio of the particle ensemble. Due to the convolution with the volume, the maximum of \(\hat{f}_{a/b}(x)\) peaks at lower values compared to \(f_{a/b}(x)\)[23]. Nevertheless, the agreement to the simple modeling (cf. Section 3.1) is perfect. To extract the axial ratio probability distribution \(f_{a/b}(x)\), the function \(V(x)\) is needed. For this purpose, the following Equation  
\[\int_{0}^{R_{\rm eq}}f_{\rm R}(R^{\prime}_{\rm eq})\frac{4}{3}\pi(R^{\prime} _{\rm eq})^{3}{\rm d}R^{\prime}_{\rm eq}=\int_{x}^{1}f_{\rm ab}(x^{\prime})V( x^{\prime}){\rm d}x^{\prime} \tag{9}\]  
has to be numerically integrated, as demonstrated in reference [23]. \(R_{\rm eq}\) is the equivalent radius of a nanoparticle and \(f_{R}(R_{\rm eq})\) the size distribution, obtained from AFM measurements. The numerical integration of Equation (9) readily provides both functions \(x(R_{\rm eq})\) and \(R_{\rm eq}(x)\), _i.e._, the functional dependence between nanoparticle size and shape. These functions are very powerful, because they permit to determine the morphology of certain particles within the nanoparticle ensemble.