## 3 Results and Discussions  
### Determination of the Mean Particle Size and the Apparent Mean Shape  
Figure 1(a) depicts an AFM image of the gold nanoparticle ensemble recorded after deposition of \(\Theta=(1.5\pm 0.2)\times 10^{16}\) atoms/cm\({}^{2}\) on a sapphire support, demonstrating the three-dimensional island growth. Figure 1: **(a)** AFM image of gold particles on a quartz support for a coverage of \(\Theta=(1.5\pm 0.2)\times 10^{16}\) atoms/cm\({}^{2}\) and **(c)** extracted size distribution; **(b)** TEM image of a similar sample and **(d)** extracted size distribution. The particle density has been extracted from the AFM image and amounts to \(n_{\mathrm{p}}^{\mathrm{AFM}}=(2.9\pm 0.3)\times 10^{11}\) particles\(/\)cm\({}^{2}\). With this particle density, we derive a mean equivalent nanoparticle radius of \(\langle R_{\mathrm{eq}}\rangle=(5.5\pm 0.3)\) nm. In addition, a HRTEM image of a sample prepared under the same experimental conditions and with the same optical properties as the sample shown in Figure 1(a) is depicted in Figure 1(b). From a set of HRTEM images we have extracted a particle density of \(n_{\mathrm{p}}^{\mathrm{STM}}=(3.1\pm 0.3)\times 10^{11}\) particles\(/\)cm\({}^{2}\), in good agreement with the data obtained by AFM. In addition, we have extracted the mean lateral diameter of the ensemble from the HRTEM images, which amounts to \(\langle 2b\rangle=(12\pm 2)\) nm. This value will be later compared to the mean lateral diameter determined with our new modeling technique. However, neither AFM nor TEM is capable to extract the precise size or shape of the nanoparticles. While in the AFM images, the apparent nanoparticle size and shape is convoluted with the AFM tip, the TEM images reveal only the lateral dimensions, but no information on the nanoparticle height. Therefore, a direct comparison of the nanoparticle morphology is not possible. However, the width of the relative size distribution \(R_{\mathrm{eq}}/\langle R_{\mathrm{eq}}\rangle\), obtained with AFM (Figure 1(c)) and the width of the lateral particle diameter distribution, obtained by HRTEM (Figure 1(d)) can be compared. In both cases, the width of the size distribution is in the same range of 30% to 35%. To extract the apparent mean axial ratio, _i.e._, the axial ratio that corresponds to nanoparticles which exhibit the same plasmon resonance frequency as the nanoparticle ensemble, optical spectroscopy in combination with simple quasistatic modeling of the optical properties of single nanoparticles is applied. Figure 2(a) depicts extinction spectra of gold nanoparticles for a coverage ranging from \(\Theta=0.1\times 10^{16}\) atoms/cm\({}^{2}\) to \(\Theta=1.5\times 10^{16}\) atoms/cm\({}^{2}\). It can be clearly observed that for increasing coverage the (1,1)-mode of the plasmon resonance increases in amplitude and shifts to lower photon energies. The latter effect is due to the flattening of the nanoparticles during growth [15, 17, 42]. Although atoms from the vapor phase can be directly deposited on top of the already grown nanoparticles, this process is rather unlikely. In practice, atoms are deposited on the substrate and attach from the side to the nanoparticles. A subsequent upstepping, _i.e._, an upward movement of the atoms, is necessary to obtain the equilibrium shape. However, upstepping is unfavorable compared to diffusion along the substrate surface. As a consequence, the mass transport across the surface is faster than in the direction normal to the surface and the nanoparticles grow in a non-thermodynamical equilibrium [14, 15], _i.e._, with oblate shape. Since surface diffusion takes place also after deposition of atoms, a shape change towards more spherical shape occurs after preparation. To allow the nanoparticle ensembles to complete their morphological changes, the samples are rested after preparation in the UHV chamber for 24 h. As a consequence of the shape changes towards more spherical, the center position of the plasmon resonance is slightly back-shifted to higher photon energies.