## 3 Results and Discussions
### Single Particle Characterization  
Yet, only the apparent mean axial ratio \(\langle\widetilde{a/b}\rangle=0.14\) of the nanoparticle ensemble has been obtained. However, the apparent mean axial ratio corresponds to the shape of particles that have a dominant impact  
Figure 3: Amplitude of the (1,1)-mode as a function of the polarization angle \(\varphi\) for gold particles with a \(\langle R_{eq}\rangle=(5.5\pm 0.3)\) nm. (**a**) Angle of incidence \(\vartheta=0^{\circ}\) and (**b**) angle of incidence \(\vartheta=45^{\circ}\). on the optical spectrum. Since the absorption cross section is proportional to \(R^{3}\)[15], \(\langle\widetilde{a/b}\rangle=0.14\) is not the most probable axial ratio in the nanoparticle distribution [23]. To obtain the most probable axial ratio, by taking the actual shape and size distribution into account, we use the following assumptions and simplifications:  
* The particles are small compared to the wavelength of light, _i.e._, the electromagnetic field within the particles is spatially nearly constant [13,14]. A comparison between calculations performed with the quasistatic approximations and the T-matrix method for gold nanoparticles can be found in reference [15], Figure 2.11. The figure demonstrates that the quasistatic approximation is fairly sufficient for the nanoparticles with \(R_{\mathrm{eq}}\leq 15\) nm. * The nanoparticles can be approximated by ellipsoidal nanoparticles with a Gaussian size distribution. This is usually fulfilled for particles grown on surfaces or deposited from gas phase [49, 50, 51]. * The optical properties of bulk materials \(\varepsilon(\omega)\) without modifications due to quantum size effects can be used. This simplification is justified, because previous observations demonstrate that major disagreements may be anticipated for particles smaller than 1 nm [14]. Furthermore, the resonance condition is only slightly influenced by variations of \(\varepsilon(\omega)\) due to the reduced dimensions for nanoparticles in our range of interest [52]. * Chemical interface damping (CID) is neglected. This assumption is justified, because for gold nanoparticles on sapphire, no CID has been observed [41]. On the other hand, for samples where CID is expected, it can be included in the theory by using an appropriate damping parameter in the size-dependent dielectric function [15]. * The influence of the substrate on the plasmon resonance can be described by the effective medium theory [14,15,53, 54, 55, 56, 57, 58, 59]. * A strong correlation between the nanoparticle size and shape exists, which is fulfilled for nanoparticles grown in Volmer-Weber growth. Hence, the nanoparticle ensemble can be characterized by a functional dependence of the shape and size. * Diploe-dipole interaction between the nanoparticles can be neglected. Note, particle-particle interaction might also cause a red shift of the plasmon resonance, but it has been demonstrated in a previous publication that a significant red-shift occurs only for very small distances between the particles [15]. The assumption is justified, because the broad shape distribution of the particles leads, in turn, to the broad distribution of plasmon frequencies, which reduces the mutual interaction between the particles. In other words, the adjacent particles are most probable of different shape and, hence, are not in resonance with each other. To demonstrate this, it is useful to treat the nanoparticles independently from their size. Thus, we define a dimensionless normalized distance \(d\) between the nanoparticles by \[d=\frac{\hat{d}}{\langle R_{\textit{eq}}\rangle}\] (4) where \(\hat{d}\) is the average distance between the nanoparticles that can be calculated from the particle number density measured by AFM. The functional dependence between \(d\) and the position of the plasmon resonance is displayed in Figure 4. The grey shaded area indicates the normalizeddistances of our nanoparticle ensembles for the coverages used in our experiments.