## 3 Results and Discussion
### Numerical Analysis  
To simulate the experimental results, we incorporated the NSD optical losses in the SiO\({}_{2}\)-Ag core-shell nanoparticles by introducing a thickness dependent collision frequency, \(\Gamma(\mathbf{R})\), in the Drude model introduced to describe the metal. The detailed permittivity model of a passive silver nano-sized shell (\(\leq\)100 nm), which incorporates the size effects and the interband transitions, was described by [PERSON] and [PERSON] [13] and by [PERSON] and [PERSON] [19]. It is expressed as  
\[\varepsilon(\mathbf{R},\mathbf{\omega})=\varepsilon_{\mathbf{Draude}}(\mathbf{R},\mathbf{ \omega})+\mathbf{\chi}_{\mathbf{Interband}}(\mathbf{\omega}) \tag{1}\]  
where \(\mathbf{R}\) is the thickness of the surrounding shell and the Drude permittivity has the form  
\[\varepsilon_{\mathbf{Draude}}(\mathbf{R},\mathbf{\omega})=1-\frac{\omega_{p}^{2}}{ \Gamma(\mathbf{R})^{2}+\ \mathbf{\omega}^{2}}+j\frac{\Gamma(\mathbf{R})\ \omega_{p}^{2}}{\mathbf{\omega}[\ \Gamma(\mathbf{R})^{2}+\ \mathbf{\omega}^{2}\ ]} \tag{2}\]  
where \(\mathbf{\omega_{p}}\) and \(\Gamma\) are the plasma and collision frequencies, respectively. The collision frequency becomes especially important in the nano-scale regime because the particle size and mean free path of the charge carriers become comparable and thus, a large resistance to their oscillations occurs. Consequently, both the optical NSD and electrical losses increase substantially. It is known [12; 18] that the collision frequency and particle size are inversely related as  
\[\Gamma(\mathbf{R})\ =\ \Gamma_{\infty}+\ \frac{\mathbf{A}\mathbf{V}_{\mathbf{F}}}{ \mathbf{R}} \tag{3}\]  
For silver at optical frequencies, the terms are \(\mathbf{A}\)- 1, a constant, and the other Drude parameters and Fermi velocity are m\({}^{\star}\)/m = 0.96, N = 5.85 \(\times\) 10\({}^{28}\) m\({}^{-3}\), \(V_{\mathbf{F}}\)= 1.39 \(\times\) 10\({}^{6}\) m/s, and \(\omega_{p}\)= 1.39269 \(\times\) 10\({}^{13}\) s\({}^{-1}\). All of the NSD losses described in the PL analysis section were included by incorporating this size-dependent collision frequency into the Drude model. Equation (3) indicates that a decrease in the collision frequency will occur with an increase in the size (R) of the nanoparticle. Consequently, the imaginary part of the Drude model (2) will become smaller and the observed blue shift with increasing particle size is demonstrated. The collision frequency \(\Gamma\) can be directly correlated with the NSD optical losses by adjusting its value according to the size of nanoparticle. A file containing the dielectric constant values of Ag, including the NSD losses specified by Equation (2) in the visible frequency range, was imported into the CST simulator. The core-shell nanoparticle was excited with a normally incident plane wave. All of the parameters used in the simulations and those used to fabricate the measured core-shell nanoparticles were the same. In particular, the thickness of the Ag shells for the three samples S1, S2, and S3, were 14, 18, and 22 nm, respectively, and the diameter of the silica core was 120 nm in all cases. The simulation results under these conditions are shown in Figure 3.  
Figure 3 gives the simulated absorption cross-section (ACS) for the three core-shell cases as functions of the wavelength of the incident plane wave. The results recover the higher order plasmonic resonances observed in the measured results at \(\lambda_{1}\): 470-483 nm and \(\lambda_{2}\): 553-565 nm.