## 3 Electrostatic Scattering by a Sphere
Note that the resonant properties are here studied as a function of the permittivity; for the response of a sphere comprising a realistic material model, such as Au or Ag, see, for example, [6; 14; 67], and the references therein. The polarizability of a complementary structure, i.e., a spherical cavity of permittivity \(\varepsilon_{\rm host}\) surrounded by a material of \(\varepsilon_{1}\) permittivity, occurs by the mutual exchange between \(\varepsilon_{\rm host}\leftrightarrow\varepsilon_{1}\) on Equation (5). As a result, the resonant condition is \(\varepsilon_{1}=-\frac{\varepsilon_{\rm host}}{2}\), or  
\[\omega_{+}^{2}=\frac{2\omega_{p}^{2}}{2\varepsilon_{\infty}+\varepsilon_{\rm host}} \tag{10}\]  
known also as the antisymmetric (antibonding) resonance [1]. It is interesting to note that all plasmonic resonances occur at the negative permittivity range, i.e., a necessary condition for the existence of such resonances [68; 69]. The combination of a solid sphere and spherical cavity naturally delivers the case of a core-shell sphere, as depicted in Figure 5. Following a similar electrostatic analysis as before (see for example [70; 71]) the polarizability of a core-shell spherical inclusion reads  
\[B_{0}=\frac{C\varepsilon_{1}(\omega)-\varepsilon_{\rm host}}{C\varepsilon_{1}( \omega)+2\varepsilon_{\rm host}}r_{1}^{3}E_{0} \tag{11}\]  
where  
\[C=-2+3\frac{1}{1-\frac{\varepsilon_{2}-\varepsilon_{1}}{\varepsilon_{2}+2 \varepsilon_{1}}\eta^{3}} \tag{12}\]  
where \(\eta=\frac{r_{2}}{r_{1}}\) is the radius ratio. The coefficient \(C\) is a function of a the term \(\frac{\varepsilon_{2}-\varepsilon_{1}}{\varepsilon_{2}+2\varepsilon_{1}} \eta^{3}\), resembling the polarizability of a solid sphere of permittivity \(\varepsilon_{2}\) (core material) immersed into a host medium of permittivity \(\varepsilon_{1}\) (shell material). This formula suggests the appearance of one additional pole and the creation of a zero to the overall system response, directly influenced by the volume ratio between the two regions. In Figure 6, we can observe these effects for both thick (\(\eta=0.1\)) and thin (\(\eta=0.9\)) shells. The field distributions for both symmetric and antisymmetric resonances are presented, illustrating the main character of the resonances. Obviously, the case \(\varepsilon_{1}=\varepsilon_{2}\) leads to \(C=1\), collapsing to the homogeneous case of Equation (5). A complete discussion regarding the pole conditions with respect to the core and host medium can be found in [72]. \[A_{1}=-\frac{3\varepsilon_{\text{host}}}{\varepsilon_{1}+2\varepsilon_{\text {host}}}E_{0} \tag{6}\]0 Schematic representation of a core–shell sphere subject to a electrostatic field embedded in a host medium. Region 1 represents the shell region, while region 2 represents the core. \[A_{1}=-\frac{3\varepsilon_{\text{host}}}{\varepsilon_{1}+2\varepsilon_{\text {host}}}E_{0} \tag{6}\]1 Electric and magnetic field distribution (amplitude) of the symmetric (**top**) and antisymmetric (**bottom**) plasmonic resonances of a very small (\(x=0.01\)) core–shell sphere occurring for \(\eta=0.1\) (**left**) and \(\eta=0.9\) (**right**) radius ratio, respectively. The inset figures depict the scattering spectrum (extinction efficiency) for (**a**) \(\eta=0.1\) and (**b**) for \(\eta=0.9\). The center pictures illustrate the field distribution (electric and magnetic) of the plasmonic resonance for a solid sphere for comparison. Following the same, Drude-like philosophy as before (assuming \(\varepsilon_{2}=1\)), the case of a spherical hollow core-shell structure, after some calculations, gives the following two resonant conditions  
\[A_{1}=-\frac{3\varepsilon_{\text{host}}}{\varepsilon_{1}+2\varepsilon_{\text {host}}}E_{0} \tag{6}\]2  
corresponding to the first two, symmetric and antisymmetric dipolar-like resonances of a hollow core-shell sphere.