## Appendix A Derivation of Plasmon Frequencies for Finite Electrolyte System
### Surface Ionic Plasmons
For \(l\geq 1\) we get,  
\[\frac{\partial^{2}q_{lm}(t)}{\partial t^{2}}=-\frac{\omega_{p}^{2}l}{2l+1}q_{ lm}(t)+\sum\limits_{i}\omega_{p}^{2}\frac{l+1}{2l+1}A_{lm}n\int\limits_{0}^{a} \frac{r_{l}^{l+2}dr_{1}}{d^{l+2}}j_{l}(k_{li}r_{1})sin(\omega_{li}t), \tag{101}\]  
which induces the solution form as follows,  
\[ \begin{array}{l}q_{lm}(t)=B_{lm}/a^{2}sin(\omega_{p}\sqrt{\frac{l}{2l+1}}t) \\ +\sum\limits_{i}A_{lm}\frac{(l+1)\omega_{p}^{2}}{\omega_{p}^{2}-(2l+1)\omega_{ l}^{2}}\int\limits_{0}^{a}\frac{r_{l}^{l+2}dr_{1}}{d^{l+2}}j_{l}(k_{li}r_{1})sin( \omega_{li}t)\end{array}  \tag{102}\]  
and \(\delta\rho_{2}(\textbf{r},t)=\sum\limits_{l=1}^{\infty}\sum\limits_{m=-l}^{l }q_{lm}(t)Y_{lm}(\Omega)\delta(a-r)\). The first term in Equation (102) describes self-frequencies of surface plasmons, whereas the second one displays the surface plasmon oscillations induced by the volume plasmons. This induced part of surface oscillations is nonzero only when the volume modes are excited and their amplitudes, \(A_{lm}\), are nonzero. The frequencies of self-oscillations of the surface plasmons are equaled to \(\omega_{l0}=\omega_{p}\sqrt{\frac{l}{2l+1}}\), corresponding to various multipole modes (numbered with \(l\)). Note that these frequencies are lower that the bulk plasmon frequency (\(\omega_{p}=\sqrt{\frac{nq^{2}4\pi}{m}}\) in Gauss units or \(\sqrt{\frac{nq^{2}}{\epsilon_{0}m}}\) in SI), whereas the volume plasmon modes oscillate with frequencies higher than \(\omega_{p}\). Worth noting is also the absence of the temperature dependence of surface ionic plasmon resonances in contrary to volume plasmon self-frequencies. Concluding, we can state that in ionic finite systems we may observe plasmons similar as in the metallic nanoparticles. The structure of surface and volume plasmons for ions is repeated from the similar properties of electronic plasmons in metallic spherical systems, however, with the significant shift of resonant energy towards lower one correspondingly to by few orders larger mass of ions in comparison to electron mass and different concentration of ions in electrolyte. Thus corresponding to the resonant energy electro-magnetic wavelength is shifted to deep infra-red or even longer wavelengths depending on ion concentration. The typical for metal clusters cross-over in size dependence of plasmon damping between the scattering, leading to Ohmic-type energy dissipation, versus the irradiation losses also is observable in ionic spherical system with similar to metal size-dependence, though shifted toward the micrometer scale for ions instead of the nanometer scale for metals. Of particular interest is the high irradiation regime for dipole plasmons in ionic system with prospective application for signaling and energy transfer in ionic systems. The initial strong enhancement of efficiency of the Lorentz friction with the radius growth of electrolyte sphere is observed on the micrometer scale with typical \(a^{3}\) radius dependence above some threshold radius which value depends on electrolyte parameters. At certain value of the radius (variating in a wide range also depending on ion system parameters) this enhancement saturates and then slowly diminishes, which allows for definition of the most convenient sizes of electrolyte finite system for optimizing radiation mediated transport efficiency.