## 5 Damping of Plasmon Oscillations in Ionic Systems
#### Exact Inclusion of the Lorentz Damping to the Attenuation of Ionic Dipole Surface Plasmons  
Now we will consider the dynamic equation for surface plasmons in the ionic spherical system (33) with the Lorentz friction term, but without application of the perturbation method. We change to dimensionless variable \(t\to t^{\prime}=\omega_{1}t\). Then Equation (33) attains the form,  
\[\frac{\partial^{2}\mathbf{D}(t^{\prime})}{\partial{t^{\prime}}^{2}}+\frac{2}{ \tau_{0}\omega_{1}}\frac{\partial\mathbf{D}(t^{\prime})}{\partial t^{\prime}} +\mathbf{D}(t^{\prime})=\frac{2}{3}\left(\frac{\omega_{p}a}{\
u\sqrt{3}} \right)^{3}\frac{\partial^{3}\mathbf{D}(t^{\prime})}{\partial{t^{\prime}}^{3 }}. \tag{36}\]  
In the case of the solution of Equation (36) by the perturbation method we get the renormalized attenuation rate for the effective damping term, \(\frac{1}{\omega_{1}\tau_{0}}+\frac{1}{3}\left(\frac{\omega_{p}a}{\
u\sqrt{3}} \right)^{3}\). This term quickly achieves the value 1, for which the oscillator falls down into the over-damped regime. For system parameters as assumed for Figure 3, the achievement by the attenuation rate of the value equaled to 1 takes place at 25.5 um and 8 um for \(n=10^{-3}N_{0}\) and \(n=10^{-2}N_{0}\), respectively. At these values the frequency \(\omega_{1}^{\prime}=\omega_{1}^{\prime}\sqrt{1-\frac{1}{(\omega_{1}\tau)^{2}}}\) goes to zero, which indicates an apparent artifact of the perturbation method. To verify how behaves the exact damped frequency in the considered system one has to solve the dynamical equation without any approximations. As this equation is of the third order linear differential equation, one can find its solution in the form, \(\sim\)\(e^{i\Omega t^{\prime}}\), with the analytical expressions for three possible values of the exponent,  
\[ \begin{array}{rl}\Omega_{1}=&-\frac{i}{3\xi}-\frac{i2^{1/3}(1+6 gu)}{3g\left( 2+2\mathcal{F}g^{2}+18 gu+\sqrt{4(-1-6 gu)^{3}+(2+2\mathcal{F}g^{2}+18 gu)^{2}} \right)^{1/3}}\\ &-\frac{i\left(2+2\mathcal{F}g^{2}+18 gu+\sqrt{4(-1-6 gu)^{3}+(2+2\mathcal{F}g^ {2}+18 gu)^{2}}\right)^{1/3}}{3\times 2^{1/3}g}\in Im(=i\alpha),\\ \Omega_{2}=&-\frac{i}{3\xi}+\frac{i\left(1+i\sqrt{3}\right)\left(1+6 gu\right) }{3\times 2^{1/3}g\left(2+2\mathcal{F}g^{2}+18 gu+\sqrt{4(-1-6 gu)^{3}+(2+2\mathcal{F}g ^{2}+18 gu)^{2}}\right)^{1/3}}\\ &+\frac{i\left(1-i\sqrt{3}\right)\left(2+2\mathcal{F}g^{2}+18 gu+\sqrt{4(-1-6 gu )^{3}+(2+2\mathcal{F}g^{2}+18 gu)^{2}}\right)^{1/3}}{6\times 2^{1/3}g}=\omega+i\frac{1}{ \tau},\\ \Omega_{3}=&-\frac{i}{3\xi}+\frac{i\left(1-i\sqrt{3}\right)\left(1+6 gu\right) }{3\times 2^{1/3}g\left(2+2\mathcal{F}g^{2}+18 gu+\sqrt{4(-1-6 gu)^{3}+(2+2\mathcal{F}g ^{2}+18 gu)^{2}}\right)^{1/3}}\\ &+\frac{i\left(1+i\sqrt{3}\right)\left(2+2\mathcal{F}g^{2}+18 gu+\sqrt{4(-1-6 gu )^{3}+(2+2\mathcal{F}g^{2}+18 gu)^{2}}\right)^{1/3}}{6\times 2^{1/3}g}=-\omega+i\frac{1}{ \tau},\end{array}  \tag{37}\]  
where \(u=\frac{1}{\tau_{0}\omega_{1}}\) and \(g=2/3\left(\frac{\omega_{p}}{c\sqrt{3\xi_{1}}}\right)^{3}\). In Figure 4 we have plotted the damping rate (\(Im\Omega\)) and the self-frequency (\(Re\Omega\)) (also translated for resonance wavelength--right panels) with respect to the system radius \(a\). For comparison, the approximate perturbative solutions are plotted--the blue line, whereas the exact solution of Equation (36) is plotted in the red line. The blue line finishes at \(a_{limit}\), when the attenuation rate within the perturbation approach reaches the critical value 1 (then \(\lambda\rightarrow\infty\)). For the accurate solution of Equation (36) this singular behavior completely disappears and the oscillating solution, \(e^{i\Omega t}\), exists for larger \(a\) as well. We notice that the red-shift of the plasmon resonance is strongly overestimated in the framework of the perturbative approach to the Lorentz friction unless \(a<\bar{a}\), where \(\bar{a}\) is sensitive to ionic system parameters and especially to ion concentration (as is demonstrated in Figure 4).