## 3 Model Definition
For ions of 3D shape of or of a linear shape, the inclusion of the rotational degrees of freedom result in factor of \(\frac{\delta k_{B}T}{2}\) or of \(\frac{\delta k_{B}T}{2}\), respectively, instead of \(\frac{3k_{B}T}{2}\) for the point-like ion model. Using the formula (15) and taking into account that \(\
abla\Theta(a-r)=-\frac{r}{r}\delta(a-r)\), one can rewrite Equation (13) in the following manner,  
\[ \begin{array}{ll}\frac{\partial^{2}\delta\hat{\rho}({\bf r},t)}{\partial t^{ 2}}=\left[\frac{k_{B}T}{m}\
abla^{2}\delta\hat{\rho}({\bf r},t)-\omega_{p}^{2 }\delta\hat{\rho}({\bf r},t)\right]\Theta(a-r)\\ -\frac{k_{B}T}{m}\
abla\left\{[n+\delta({\bf r},t)\right]\frac{r}{r}\delta( a-r)\right\}\\ -\left[\frac{kT}{m}\frac{kT}{r}\
abla\delta\hat{\rho}({\bf r},t)+\frac{\omega_ {p}^{2}}{4\pi}\frac{r}{r}\
abla\int d^{3}r_{1}\frac{1}{|{\bf r}-{\bf r}_{1}|} \delta\hat{\rho}({\bf r}_{1},t)\right]\delta(a-r).\end{array}  \tag{16}\]In the above formula \(\omega_{p}\) is the bulk ion-plasmon frequency, \(\omega_{p}^{2}=\frac{4\pi{m_{d}}^{2}}{m}\). The solution of Equation (16) can be decomposed into two parts related to the distinct domains--inside the sphere and on its surface,  
\[\delta\tilde{p}(\mathbf{r},t)=\left\{ \begin{array}{ll}\delta\tilde{\rho}_{1}( \mathbf{r},t),\ for\ r<a,\\ \delta\tilde{\rho}_{2}(\mathbf{r},t),\ for\ r\geq a,\ (r\to a+),\end{array} \right. \tag{17}\]  
corresponding to the volume and surface excitations, respectively. These two parts of ion local density fluctuations satisfy the equations (according to Equation (16)),  
\[\frac{\partial^{2}\delta\tilde{\rho}_{1}(\mathbf{r},t)}{\partial t^{2}}=\frac {k_{B}T}{m}\
abla^{2}\delta\tilde{\rho}_{1}(\mathbf{r},t)-\omega_{p}^{2}\delta \tilde{\rho}_{1}(\mathbf{r},t), \tag{18}\]  
and (here \(\epsilon=0+\))  
\[ \begin{array}{ll}\frac{\partial^{2}\delta\tilde{\rho}_{2}(\mathbf{r},t)}{ \partial t^{2}}=-\frac{k_{B}T}{m}\
abla\left\{\left[n+\delta\tilde{\rho}_{2}( \mathbf{r},t)\right]\frac{\epsilon}{r}\delta(a+\epsilon-r)\right\}\\ -\left[\frac{k_{B}T}{m}\frac{\mathbf{r}}{r}\
abla\delta\tilde{\rho}_{2}( \mathbf{r},t)+\frac{\omega_{p}^{2}}{4\pi}\frac{\mathbf{r}}{r}\
abla\int d^{3} r_{1}\frac{1}{|\mathbf{r}-\mathbf{r}_{1}|}\left(\delta\tilde{\rho}_{1}(\mathbf{r} _{1},t)\Theta(a-r_{1})\right.\right.\\ \left.\left.+\delta\tilde{\rho}_{2}(\mathbf{r}_{1},t)\Theta(r_{1}-a))\right] \delta(a+\epsilon-r).\end{array}  \tag{19}\]  
The Dirac delta in Equation (19) results due to the derivative of the Heaviside step function--the fictitious jellium distribution shape. In Equation (19) an infinitesimal shift, \(\epsilon=0+\), is introduced to fulfil requirements of the Dirac delta definition (its singular point has to be an inner point of an open subset of the domain). This shift is only of a formal character and does not reflect any asymmetry. Some kind of asymmetry is, however, caused by the last term in Equation (16), the gradient-type term, which describes the electric field induced by ion density fluctuations. The electric field due to surface charges is zero inside the sphere, and therefore cannot influence the volume excitations. Conversely, the volume charge fluctuation-induced-electric-field can excite the surface fluctuations. This asymmetry is visible by comparison of Equations (18) and (19), for the volume and surface of plasmons, respectively. The equation for the volume of plasmons is completely independent of the surface of plasmons, whereas the volume plasmons contribute the equation for the surface plasmons. The problem of separation between surface and volume plasmons has been thoroughly analyzed for metal clusters. For very small clusters the significance in this size-scale effect of the spill-out of electrons beyond the jellium edge makes the surface fuzzy resulting in coupling of volume and surface plasmon oscillations.