## 5 Damping of Plasmon Oscillations in Ionic Systems  
The semiclassical RPA treatment of plasmon excitations in finite ion systems as presented above, does not account for plasmon damping. The damping of plasmon oscillations can be, however, included in a phenomenological manner, by addition of an attenuation term to plasmon dynamic equations, i.e., the term, \(-\frac{2}{\tau_{0}}\frac{\Delta\delta\rho(\mathbf{r},t)}{\partial t}\), added to the r.h.s. of both Equations (18) and (19), taking advantage of their oscillatory form. The damping rate \(\frac{1}{\tau_{0}}\) accounts for ion scattering losses and can be approximated in analogy to metallic systems, by inclusion of energy dissipation caused by irreversible its transformation into heat via various microscopic channels similar to those for Ohmic resistivity [7],  
\[\frac{1}{\tau_{0}}\simeq\frac{v}{2\lambda_{b}}+\frac{Cv}{2a}, \tag{26}\]  
where \(a\) is the sphere radius, \(v\) is the mean velocity of ions, \(v=\sqrt{\frac{4k_{B}T}{3m}}\), \(\lambda_{b}\) is the ion mean free path in bulk electrolyte material the same as the sphere is made of (including scattering of ions on other ions, and on solvent particles and admixtures). The second term in Equation (26) accounts for the scattering of ions on the boundary of the finite ionic system, the sphere with the radius \(a\), and the constant \(C\) is of order of unity [7]. In order to explicitly express a forcing field which moves ions in the system, the inhomogeneous time dependent term should be added to homogeneous Equations (18) and (19). The forcing field may be the time dependent electric field. If one considers it as the electric component of the incident EM wave then the comparison of the resonant wavelength with the system size is of order. Similarlyas for metallic nanospheres also for finite ionic systems the surface plasmon resonant wavelength highly exceeds the system dimension and the forcing field is practically uniform along the whole system. Such a perturbation can excite only surface dipole plasmons, i.e., the mode with \(l=1\), which can be described by the function \(Q_{1m}(t)\) (\(l=1\) and \(m\) are angular momentum numbers related to the assumed spherical symmetry of the system). The corresponding dynamical equation for surface plasmons reduced to only mode \(Q_{1m}(t)\) has the following form,  
\[ \begin{array}{l}\frac{\partial^{2}Q_{1m}(t)}{\partial t^{2}}+\frac{2}{\tau_ {0}}\frac{\partial Q_{1m}(t)}{\partial t}+\omega_{1}^{2}Q_{1m}(t)\\ =\sqrt{\frac{4\pi}{3}}\frac{\partial m}{m}\left[E_{z}(t)\delta_{m,0}+\sqrt{2} \left(E_{x}(t)\delta_{m,1}+E_{y}(t)\delta_{m,-1}\right)\right],\end{array}  \tag{27}\]  
where \(\omega_{1}=\frac{\omega_{p}}{\sqrt{3}\varepsilon_{1}}\) (it is a dipole surface plasmon frequency, i.e., the Mie frequency [6], \(\varepsilon_{1}\) is the dielectric susceptibility of the system surroundings). Because only \(Q_{1m}\) contribute to the plasmon response to the homogeneous electric field, thus the effective ion density fluctuation has the form [1],  
\[\delta\rho(\mathbf{r},t)=\left\{ \begin{array}{ll}0,&r<a,\\ \sum\limits_{m=-1}^{1}Q_{1m}(t)Y_{1m}(\Omega)&r\geq a,&r\to a+,\end{array} \right. \tag{28}\]  
where \(Y_{1m}(\Omega)\) is the spherical function with \(l=1\). One can also explicitly calculate the dipole \(\mathbf{D}(t)\) corresponding to surface plasmon oscillations given by Equation (28),  
\[\left\{ \begin{array}{ll}D_{x}(t)&=q\int d^{3}rx\delta\rho(\mathbf{r},t)= \frac{\sqrt{2\pi}}{\sqrt{3}}eQ_{1,1}(t)a^{3},\\ D_{y}(t)&=q\int d^{3}ry\delta\rho(\mathbf{r},t)=\frac{\sqrt{2\pi}}{\sqrt{3}}eQ _{1,-1}(t)a^{3},\\ D_{z}(t)&=q\int d^{3}rz\delta\rho(\mathbf{r},t)=\frac{\sqrt{4\pi}}{\sqrt{3}}eQ _{1,0}(t)a^{3}.\end{array} \right.