## 5 Damping of Plasmon Oscillations in Ionic Systems
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Material** & **Ionic System** & **Sample 1** & **Sample 2** \\ \hline ion concentration & \(n\) (\(N_{0}\) is one molar electr.) & \(10^{-2}\)\(N_{0}\) & \(10^{-3}\)\(N_{0}\) \\ effective ion mass & \(m\) (\(m_{e}\) electron mass) & \(10^{4}m_{e}\) & \(10^{4}m_{e}\) \\ charge of effective ion & \(q/\sqrt{\varepsilon}\) & \(3\)\(e\) & \(3\)\(e\) \\ temperature & \(T\) & \(300\) K & \(300\) K \\ mean velocity of ions & \(v=\sqrt{\frac{3 kT}{m}}\) & \(1168\) m/s & \(1168\) m/s \\ bulk plasmon frequency & \(\omega_{p}\) & \(9.3\times 10^{13}\) 1/s & \(2.93\times 10^{12}\) 1/s \\ dielectric constant of souroundings & \(\varepsilon_{1}\) & \(2\) & \(2\) \\ Mie frequency & \(\omega_{1}=\omega_{p}/\sqrt{3\varepsilon_{1}}\) & \(3.8\times 10^{13}\) 1/s & \(1.2\times 10^{12}\) 1/s \\ constant in Equation (26) & \(C\) & \(2\) & \(2\) \\ bulk mean free path (room temp.) & \(\lambda_{b}\) & \(0.5\)\(\mu\)m & \(0.1\)\(\mu\)m \\ radius for minimal damping & \(a^{*}=\left(\frac{3^{2}C^{2}c^{3}}{2\omega_{1}\omega_{p}^{2}}\right)^{1/4}\) & \(2.7\times 10^{-7}\) m & \(8.6\times 10^{-7}\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ion-system parameters assumed for calculation of damping rates and self-frequency for dipole surface plasmons. Figure 3: The cross-over in ionic system-size dependence of damping rate for surface plasmons, for \(T=300\) K, \(m=10^{4}m_{e}\), \(q=3e\), \(n=10^{-2}N_{0}\) (\(N_{0}\) is the concentration of one molar electrolyte) (**left**) and for \(n=10^{-3}N_{0}\) (**right**); ion scattering losses, \(\sim\)\(\frac{1}{a}\), are of lowering importance with growth of the electrolyte sphere size (\(a\) is the sphere radius), whereas the irradiation losses dominate at larger size, as it initially grows as \(a^{3}\); in the size-region close to the cross-over the perturbative treatment for Lorentz friction perfectly coincides with the exact approach. where  
\[\frac{1}{\tau}=\frac{1}{\tau_{0}}+\frac{\omega_{1}}{3}\left(\frac{\omega_{p}a}{c \sqrt{3}}\right)^{3}. \tag{35}\]  
Within the first step of perturbation, the Lorentz friction can be included into the total attenuation rate \(\frac{1}{\tau}\). Nevertheless, this approximation is justified only for sufficiently small perturbations, i.e., when the second term in Equation (35), proportional to \(a^{3}\), is small enough to fulfill the perturbation restrictions. The related limiting value, \(\bar{a}\), of the ionic system size depends on the ion concentration, charge, mass, dielectric susceptibility, as is exemplified below in the following subsection. The solution of Equation (34) is of the form \(\mathbf{D}(t)=\mathbf{A}e^{-t/\tau}cos(\omega_{1}^{\prime}t+\phi)\), where \(\omega_{1}^{\prime}=\omega_{1}^{\prime}\sqrt{1-\frac{1}{(\omega_{1}\tau)^{2}}}\), which gives the red shift of the plasmon resonance due to strong, \(\sim a^{3}\), growth of the attenuation caused by the irradiation. The Lorentz friction term in Equation (35) dominates plasmon damping for \(a\leq\bar{a}\) due to this \(a^{3}\) dependence--cf. Figure 3. The plasmon damping grows rapidly with \(a\) and this results in the pronounced red-shift of the resonance frequency.