Collisional damping of the surface ion-acoustic mode in semi-bounded plasmas

This paper discusses the effects of collision and temperature on the damping of surface ion-acoustic waves that propagate in a semi-bounded plasma. Research on the propagation and stability of surface waves in various types of plasma such as semi-bounded, slabs and columns has a long history [1–4]. After the pioneering work of Trivelpiece and Gould [1], a lot of information on the properties and propagation characteristics of surface waves has been accumulated [5]. The effects of finite temperature, density inhomogeneity [2], collisions between particles [6], dusty media with the presence of nonspherical grains [7], quantum plasma [8], etc, on the propagation of surface waves are a few examples and they still attract great interest since all plasmas are in some fashion bounded. The stability problem in a variety of confined configurations has also been extensively investigated [9].

We are motivated to conduct an analysis on the collisional damping of surface ion-acoustic waves that propagate in a fully ionized plasma. For low-frequency ion-acoustic waves, the ion collisions may provide a significant role in the damping of the waves if the ion mean free path is much smaller than the wavelength. Electron collisions may only be important for high-frequency electron plasma waves [10]. For weakly ionized plasma, the collision between ions and neutral particles is the most important component of wave dissipation [11, 12]. For completely ionized plasma, we consider the collisions between ions only because the Coulomb force between the charged particles falls very slowly with distance, the collisions with large impact parameters would play a major role in the collisional processes. Thus, the effect of collisions between ions and electrons on the ion-acoustic wave is neglected here. A large body of literature on surface waves has stimulated research activities on plasmas containing noble electrons and ions, so there is not enough information on the effect of collisional damping on the propagation and stability of surface waves. In addition, the influence of the ion-ion collision on the surface mode of ion-acoustic waves including Landau damping has not been investigated yet. Therefore, this work is motivated to present a general description of ion-acoustic waves propagating at the boundary surface of a plasma including the ion-ion collision, which seems to affect the dispersive property. Hence, the physical characteristics of electrostatic ion-acoustic surface waves are investigated to determine the wave frequency and the corresponding Landau damping rate including the ion collisional effect by using the transverse truncation method. Moreover, it is expected that the functional perturbation method with the transverse truncation approximation used in this work can provide insightful view points on the stable and unstable modes of surface plasma waves and also be useful to studies of the propagation of surface waves in the presence of the complex collective behaviors. The influence of compressible density of plasma species is not considered in this work since the ion collision effect on the propagation of the surface ion-acoustic wave including Landau damping based on the specular reflection condition is the main purpose of the current work. This situation occurs in nano-devices where the plasma is separated from the wall and spatially confined by the external field. As in the case of volume ion-acoustic wave in unbound plasma, surface ion-acoustic waves exist even if the ion thermal motion is completely ignored. Then, it is expected that the Landau dissipation on the wave would prevail. In this work, we take account of the ion temperature, although it is much less than the electron temperature. Thus, the effect of ion temperature as well as the effect of collision on the wave dissipation will be discussed. In order to obtain the analytic form for the damping rate as much as possible, we resort to the specular reflection boundary condition and the transverse truncation method which will be explained in the next section. We show that the total damping rate is the addition of a collisional damping rate arising from the ion thermal motion to a collisionless Landau damping rate arising from the wave-particle interaction.

We consider the case of longitudinal waves, in particular the low-frequency ion-acoustic mode existing in isotropic plasma with ${T_{\text{e}}}\,\, \gg \,\,{T_{\text{i}}}$ where ${T_{\text{e}}}$ and ${T_{\text{i}}}$ are the temperature of electrons and ions, respectively. In such a case (${T_{\text{e}}}\,\, \gg \,\,{T_{\text{i}}}$) with wave frequencies in the range $k{v_{Ti}}\,\, \ll \,\,\omega \,\, \ll \,\,k{v_{Te}}$, the collisional modification of the low-frequency ion-acoustic oscillations can be made on the longitudinal dielectric permittivity. Here, ${v_{Te}}$ and ${v_{Ti}}$ are the thermal speed of electrons (e) and ions (i), respectively, defined by ${v_{Tj}}{\mkern 1mu} {\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} \sqrt {2{T_j}/{m_j}}$ (j = e, i). To calculate the damping rate of the surface ion-acoustic mode which is caused by collisions between the charged particles, the effective collision frequency ${\nu _c}$ is considered in the longitudinal dielectric permittivity. When the particle collision is not too frequent, under the condition ${\nu _c}\,\, \ll \,\,\omega$ and $\left| {\omega \, + \,i{\nu _c}} \right|\,\, \ll \,\,k{v_{Te}}$, the longitudinal dielectric permittivity can be written as [13]

$$\begin{align}{\varepsilon _L}\left( {\omega ,\,k} \right)\,\,& = \,\,1\,\, + \,\,\frac{{\omega _{pe}^2}}{{{k^2}v_{Te}^2}}\,\left( {1\,\, + \,\,i\sqrt {\frac{\pi }{2}} \frac{\omega }{{k{v_{Te}}}}} \right) - \,\,\frac{{\omega _{pi}^2}}{{{\omega ^2}}} \nonumber\\ & \quad \times \left( {1\,\, + \,\,3\frac{{{k^2}v_{Ti}^2}}{{{\omega ^2}}}} \right)\,\, + \,\,i\sqrt {\frac{\pi }{2}} \frac{{\omega _{pi}^2\omega }}{{{k^3}v_{Te}^3}}\,\exp \left( { - \,\frac{{{\omega ^2}}}{{2{k^2}v_{Ti}^2}}} \right)\nonumber\\ & \quad + \,\,i\frac{{\omega _{pi}^2{\nu _c}}}{{{\omega ^3}}}\,,\end{align} \tag{ 1 }$$

where ${\omega _{pj}}{\mkern 1mu} {\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} \sqrt {4\pi {n_j}{e_j}/{m_j}}$ is the plasma frequency of the particle species j (= e, i). In the frequency range of the ion-acoustic wave, equation (1) can be reduced to [9]

$$\begin{align}{\varepsilon _L}\left( {\omega ,\,k} \right)\,\,& = \,\,1\,\, - \,\,\frac{{\omega _{pi}^2}}{{{\omega ^2}}}\,\left( {1\,\, - \,\,i\frac{{{\nu _c}}}{\omega }} \right)\,\, + \,\,\frac{{\omega _{pe}^2}}{{{k^2}v_{Te}^2}}\nonumber\\ & \quad \times \left( {1\,\, + \,\,i\sqrt {\frac{\pi }{2}} \frac{\omega }{{k{v_{Te}}}}} \right).\end{align} \tag{ 2 }$$

Note that in the low-frequency range the electron collisions do not contribute to the dissipation of longitudinal oscillations. For a fully ionized plasma, the collisions between charged particles prevail. Under the condition ${\nu _c}\,\, \ll \,\,k{v_{Te}}$, the collisional effect due to electrons may be neglected since the wavelength is much smaller than the electron mean free path. Then, the dissipative collisional process is insignificant. However, under the condition, ${\nu _c}\,\, \gg \,\,k{v_{Ti}}$, ion collisions must be accounted for in the dissipative part of the dielectric permittivity. The collisional correction in the case of fully ionized plasmas is given by ${\nu _c}{\mkern 1mu} {\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} 8{k^2}v_{Ti}^2{\nu _{ii}}/5{\omega ^2}$ where ${\nu _{ii}}$ is the ion-ion collision frequency [9].

We consider an isotropic plasma occupying the half-space defined by the region $z\,\, \geqslant \,\,0$ and bounded by a vacuum half-space ($z\,\, < \,\,0$). This semi-bounded plasma is uniform and unmagnetized. We assume that the curl of electric field vanishes, so the analysis is limited to electrostatic. The perturbation about the equilibrium state is weak. If the characteristic length of the plasma is much greater than the scale length of the inhomogeneity, we can set the specular reflection boundary condition for the perturbed plasma distribution function ${f_1}$ at $z\,\, = \,\,0$. This condition must contain all the information on the interaction of plasma particles at the surface satisfying the condition [14]

$$\begin{equation}{\left. {{f_1}\left( {{v_x},\,{v_y},\,{v_z}} \right)} \right|_{x\, = \,0}}\,\, = \,\,{\left. {{f_1}\left( { - {v_x},\,{v_y},\,{v_z}} \right)} \right|_{x\, = \,0}}.\end{equation} \tag{ 3 }$$

It is well established that the dispersion equation for surface waves can be derived by applying the general electrodynamic boundary conditions and by equating the surface impedance of the semi-bounded plasma and that of the vacuum half-space. In [13], the surface impedance of the semi-bounded isotropic plasmas is calculated as

$$\begin{equation}{Z_s}\,\, = \,\, - \,\,\frac{{8i\omega }}{{{c^2}}}\int\limits_0^\infty \frac{{{\text{d}}{k_z}}}{{{k^2}}}\left[ {\frac{{k_x^2{c^2}}}{{{\omega ^2}{\varepsilon _L}\left( {\omega ,k} \right)}}\,\, - \,\,\frac{{k_z^2{c^2}}}{{{k^2}{c^2}\,\, - \,\,{\omega ^2}{\varepsilon _T}\left( {\omega ,k} \right)}}} \right]\end{equation} \tag{ 4 }$$

and the surface impedance of the vacuum half-space is given by

$$\begin{equation}{Z_v}\,\, = \,\,\frac{{4\pi i}}{c}\sqrt {\frac{{k_x^2{c^2}}}{{{\omega ^2}}}\,\, - \,\,1} \end{equation} \tag{ 5 }$$

where ${\varepsilon _T}$ is the transverse dielectric permittivity. Now we apply the boundary condition, i.e. the continuity of the tangential components of the electromagnetic fields for $z\,\, = \,\,0$ and equate ${Z_s}$ and ${Z_v}$ to yield the dispersion equation for surface waves such as

$$\begin{align}\sqrt {\frac{{k_x^2{c^2}}}{{{\omega ^2}}}\,\, - \,\,1} \,\, + \,\,\frac{{2\omega }}{{\pi c}}\int\limits_0^\infty \frac{{{\text{d}}{k_z}}}{{{k^2}}}&\Bigg[ \frac{{k_x^2{c^2}}}{{{\omega ^2}{\varepsilon _L}\left( {\omega ,k} \right)}}\,\nonumber\\ & \quad - \,\,\frac{{k_z^2{c^2}}}{{{k^2}{c^2}\,\, - \,\,{\omega ^2}{\varepsilon _T}\left( {\omega ,k} \right)}} \Bigg]\,\, = \,\,0.\end{align} \tag{ 6 }$$

In the limit of electrostatic perturbation $\left( {c\,\, \to \,\,\infty } \right)$, the dispersion equation for longitudinal surface waves in semi-bounded plasma can be obtained as

$$\begin{equation}D\left( {\omega ,\,{k_\parallel }} \right)\,\, \equiv \,\,1\,\, + \,\,\frac{{{k_\parallel }}}{\pi }\int\limits_{ - \,\infty }^\infty \frac{{{\text{d}}{k_ \bot }}}{{{k^2}\,\,{\varepsilon _L}\left( {\omega ,\,k} \right)}}\,\, = \,\,0\,,\end{equation} \tag{ 7 }$$

where ${k_\parallel }\,\left( { = \,{k_x}} \right)$ and ${k_ \bot }\,\left( { = \,{k_z}} \right)$ are the parallel and the perpendicular components of the wave vector $k\,\left( { = \,\sqrt {k_x^2\, + \,k_z^2} } \right)$ and ${\varepsilon _L}$ is given by equation (2) for the ion-acoustic wave in the fully ionized plasma. The component ${k_y}$ is set to zero since the y-coordinate has a translational invariance for the plasma bounded by the plane $z\,\, = \,\,0$ and thus can be ignored in our geometry. The finiteness of the fields would yield the continuity of the tangential components of the electric fields at $z\,\, = \,\,0$. This condition can be applied to yield equation (7) by equating the surface impedance of the semi-bounded isotropic plasma and the surface impedance of the vacuum half-space. Here, equation (7) can be used for our model in which the existence of a sufficiently sharp plasma surfaces is presumed. Hence, all plasma quantities with the dimension of physical length such as the Debye length usually exceed the size of the density variation near the interface. We are dealing with a linearized theory in which the perturbation is small. Then, the expansion of $D\left( {\omega ,\,{k_\parallel }} \right)$ is a useful technique to find the damping rate for the surface waves. Let $\omega \,\, = \,\,{\omega _r}\, + \,i\gamma$ where ${\omega _r}\,\, \equiv \,\,\operatorname{Re} \omega$ and $\gamma \,\, \equiv \,\,\operatorname{Im} \omega$. We also let ${\varepsilon _L}\,\, = \,\,{\varepsilon _r}\, + \,i{\varepsilon _i}$ where ${\varepsilon _r}\,\, \equiv \,\,\operatorname{Re} \varepsilon$ and ${\varepsilon _i}\,\, \equiv \,\,\operatorname{Im} \varepsilon$. If ${\omega _r}\, \gg \,\left| \gamma \right|$ and $\left| {{\varepsilon _r}} \right|\, \gg \,\,\left| {{\varepsilon _i}} \right|$, the expansion of $D\left( {\omega ,\,{k_\parallel }} \right)$ about $\omega \,\, = \,\,{\omega _r}$ gives $D\left( {\omega ,{\mkern 1mu} {k_\parallel }} \right){\mkern 1mu} {\mkern 1mu} \approx {\mkern 1mu} {\mkern 1mu} D\left( {{\omega _r},{\mkern 1mu} {k_\parallel }} \right){\mkern 1mu} {\mkern 1mu} + {\mkern 1mu} {\mkern 1mu} i{\mkern 1mu} \gamma {\mkern 1mu} \partial D\left( {{\omega _r},{\mkern 1mu} {k_\parallel }} \right)/\partial {\omega _r}$. The dielectric permittivity ${\varepsilon _L}$ in equation (7) allows us to write $D\left( {\omega ,\,{k_\parallel }} \right)$ in the form

$$\begin{equation}D\left( {\omega ,\,{k_\parallel }} \right)\,\, \approx \,\,1\,\, + \,\,\frac{{{k_\parallel }}}{\pi }\int\limits_{ - \,\infty }^\infty \frac{{{\text{d}}{k_ \bot }}}{{{k^2}\,\,{\varepsilon _r}\left( {{\omega _r},\,k} \right)}}\,\, - \,\,i\frac{{{k_\parallel }}}{\pi }\int\limits_{ - \,\infty }^\infty \frac{{{\varepsilon _i}{\text{d}}{k_ \bot }}}{{{k^2}\,\,\varepsilon _r^2\left( {{\omega _r},\,k} \right)}}.\end{equation} \tag{ 8 }$$

Therefore, the damping rate of the wave can be formulated succinctly as

$$\begin{equation}\gamma {\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} - {\mkern 1mu} {\mkern 1mu} \frac{{{D_i}\left( {{\omega _r},{\mkern 1mu} {k_\parallel }} \right)}}{{\partial {D_r}\left( {{\omega _r},{\mkern 1mu} {k_\parallel }} \right)/\partial {\omega _r}}}{\mkern 1mu} ,\end{equation} \tag{ 9 }$$

where ${D_r}\left( {{\omega _r},\,{k_\parallel }} \right)$ and ${D_i}\left( {{\omega _r},\,{k_\parallel }} \right)$ are the real and the imaginary part of the dispersion integral given by, respectively,

$$\begin{equation}{D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,1\,\, + \,\,\frac{{{k_\parallel }}}{\pi }\int\limits_{ - \,\infty }^\infty \frac{{{\text{d}}{k_ \bot }}}{{{k^2}\,\,{\varepsilon _r}\left( {{\omega _r},\,k} \right)}}\,,\end{equation} \tag{ 10 }$$

and

$$\begin{equation}{D_i}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,\,\, - \,\,\frac{{{k_\parallel }}}{\pi }\int\limits_{ - \,\infty }^\infty \frac{{{\varepsilon _i}{\text{d}}{k_ \bot }}}{{{k^2}\,\,\varepsilon _r^2\left( {{\omega _r},\,k} \right)}}.\end{equation} \tag{ 11 }$$

The dispersion relation of the surface ion-acoustic wave is obtained by evaluating ${D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,0$, i.e.

$$\begin{equation}1\,\, + \,\,\frac{{{k_\parallel }}}{{\left( {1\, - \,\frac{{\omega _{pi}^2}}{{\omega _r^2}}} \right)\sqrt {k_\parallel ^2\, + \,\frac{{\omega _{pe}^2}}{{v_{Te}^2}}{{\left( {1\, - \,\frac{{\omega _{pi}^2}}{{\omega _r^2}}} \right)}^{ - 1}}} }}\,\, = \,\,0.\end{equation} \tag{ 12 }$$

Therefore, the wave frequency is found as

$$\begin{equation}{\omega _r}\,\, = \,\,\frac{{{\omega _{pi}}{k_\parallel }{\lambda _{{\text{De}}}}}}{{\sqrt {k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\displaystyle\frac{1}{2}\,\, + \,\,\sqrt {k_\parallel ^4\lambda _{{\text{De}}}^4\,\, + \,\,\displaystyle\frac{1}{4}} } }}\,,\end{equation} \tag{ 13 }$$

where ${\lambda _{{\text{De}}}}\,\, = \,\,{v_{Te}}/{\omega _{pe}}$ is the electron Debye length. Surface ion-acoustic waves are possible only in the frequency range where ${\omega _r}{\mkern 1mu} {\mkern 1mu} < {\mkern 1mu} {\mkern 1mu} {\omega _{pi}}/\sqrt 2$ [14], and the above result satisfies this condition for the whole spectrum. In the region where ${k_\parallel }$ is small, we can see immediately that the frequency increases in proportion to ${k_\parallel }$ and converges to ${\omega _{pi}}/\sqrt 2$ as ${k_\parallel }$ increases.

We now proceed to the analysis of damping rate for the ion-acoustic surface waves in a semi-bounded plasma. We are interested in the case where analytical solutions can be possible. Unlike ${D_r}\left( {{\omega _r},\,{k_\parallel }} \right)$, it is very difficult to calculate ${D_i}\left( {{\omega _r},\,{k_\parallel }} \right)$ by directly performing the integration. Therefore, we use an approximate approach named the transverse truncation method to execute the integral in the imaginary part ${D_i}$. According to the condition of $\left| {{\varepsilon _i}/{\varepsilon _r}} \right|\,\, \ll \,\,1$ and weak contribution of the perpendicular component ${k_ \bot }$ of the wave vector to the function ${\varepsilon _i}/{\varepsilon _r}$ near the interface, the transverse truncation analysis, i.e. ${\left[ {{\varepsilon _i}\left( {{\omega _r},\,k} \right)/{\varepsilon _r}\left( {{\omega _r},\,k} \right)} \right]_{{k_ \bot }\, = \,0}}$ can be applied to extract the physical information on the imaginary part ${D_i}$. Hence, the constraint of the smallness of ${\varepsilon _i}/{\varepsilon _r}$ provides a very novel method and is useful for surface wave calculations, especially for surface waves propagating in semi-bounded plasmas. Near the sharp interface, the imaginary part ${D_i}$ is then approximated as follows,

$$\begin{align} {D_i}\left( {{\omega _r},\,{k_\parallel }} \right)\,\,& \approx \,\,\,\, - \,\,\frac{{{k_\parallel }}}{\pi }{{\left[ {\frac{{{\varepsilon _i}\left( {{\omega _r},\,k} \right)}}{{{\varepsilon _r}\left( {{\omega _r},\,k} \right)}}} \right]}_{{k_ \bot }\, = \,0}}\int\limits_{ - \,\infty }^\infty \frac{{{\text{d}}{k_ \bot }}}{{{k^2}\,\,{\varepsilon _r}\left( {{\omega _r},\,k} \right)}} \nonumber\\ & = \,\, - \,\,\frac{{{\varepsilon _i}\left( {{\omega _r},\,{k_\parallel }} \right)}}{{{\varepsilon _r}\left( {{\omega _r},\,{k_\parallel }} \right)}}\left[ {{D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, - \,\,1} \right].\end{align} \tag{ 14 }$$

After substituting the expressions for ${D_r}$, ${\varepsilon _r}$ and ${\varepsilon _i}$ into equation (14), we obtain ${D_i}$ in the form,

$$\begin{equation}{D_i}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,\,\, - \,\,\frac{{\sqrt {\frac{\pi }{2}} \frac{\omega }{{k_\parallel ^3\lambda _{{\text{De}}}^3{\omega _{pe}}}}\,\, + \,\,\displaystyle\frac{8}{5}k_\parallel ^2\lambda _{Di}^2\frac{{\omega _{pi}^4{\nu _{ii}}}}{{{\omega ^5}}}}}{{\left( {1\,\, - \,\,\frac{{\omega _{pi}^2}}{{{\omega ^2}}}} \right)\left( {1\,\, - \,\,\frac{{\omega _{pi}^2}}{{{\omega ^2}}}\,\, + \,\,\frac{1}{{k_\parallel ^2\lambda _{{\text{De}}}^2}}} \right)\sqrt {1\,\, + \,\,\frac{1}{{k_\parallel ^2\lambda _{{\text{De}}}^2}}\left( {\frac{{{\omega ^2}}}{{{\omega ^2}\,\, - \,\,\omega _{pi}^2}}} \right)} }}\,,\end{equation} \tag{ 15 }$$

where ${\lambda _{Di}}$ is the ion Debye length. Finally, the damping rate for the surface wave in the completely ionized plasma is calculated as

$$\begin{align} \gamma \,\, = \,\,\,\,{\gamma _{{\text{Ld}}}}\,\, + \,\,{\gamma _{{\text{Cd}}}}& = \,\,\,\, - \,\,\frac{{\left( {1\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} } \right){k_\parallel }{\lambda _{{\text{De}}}}}}{{\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} {{\left( {1\,\, + \,\,2k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} } \right)}^2}\frac{{{\omega _{pe}}}}{{{\omega _{pi}}}}}} \nonumber\\ & \quad \times \Bigg\{ \sqrt {2\pi } \,\, + \,\,\frac{{8{\nu _{ii}}{\omega _{pe}}}}{{5{k_\parallel }{\lambda _{{\text{De}}}}\omega _{pi}^2}}\left( {\frac{{{T_i}}}{{{T_e}}}} \right) \left[ {1\,\, + \,\,k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\left( {1\,\, + \,\,3k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} \right)\left( {2k_\parallel ^2\lambda _{{\text{De}}}^{\text{2}}\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} } \right)} \right] \Bigg\}\,, \end{align} \tag{ 16 }$$

where ${T_{\text{i}}}/{T_{\text{e}}}\,\, = \,\,\lambda _{Di}^2/\lambda _{{\text{De}}}^2$. Here, ${\gamma _{{\text{Ld}}}}$ represents the collisionless Landau damping:

$$\begin{equation}{\gamma _{{\text{Ld}}}}\,\, = \,\, - \,\,\frac{{\sqrt {2\pi } \left( {1\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} } \right){k_\parallel }{\lambda _{{\text{De}}}}}}{{\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} {{\left( {1\,\, + \,\,2k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} } \right)}^2}\frac{{{\omega _{pe}}}}{{{\omega _{pi}}}}}},\end{equation} \tag{ 17 }$$

and ${\gamma _{{\text{Cd}}}}$ stands for the collisional dissipation:

$$\begin{align} {\gamma _{{\text{Cd}}}} & = - \frac{{\left( {1\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^{\text{4}}} } \right){k_\parallel }{\lambda _{{\text{De}}}}}}{{\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} {{\left( {1\,\, + \,\,2k_\parallel ^2\lambda _{{\text{De}}}^{\text{2}}\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4} } \right)}^2}\frac{{{\omega _{pe}}}}{{{\omega _{pi}}}}}} \nonumber\\ & \quad \times \,\,\Bigg\{ \frac{{8{\nu _{ii}}{\omega _{pe}}}}{{5{k_\parallel }{\lambda _{{\text{De}}}}\omega _{pi}^2}}\left( {\frac{{{T_i}}}{{{T_e}}}} \right)\Bigg[ 1\,\, + \,\,k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\Bigg( 1\,\, + \,\,3k_\parallel ^2\lambda _{{\text{De}}}^{\text{2}}\nonumber\\ & \qquad\quad + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^4 \Bigg)\left( {2k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\sqrt {1\,\, + \,\,4k_\parallel ^4\lambda _{{\text{De}}}^{\text{4}}} } \right) \Bigg] \Bigg\}.\end{align} \tag{ 18 }$$

The magnitude of damping rate for the surface ion-acoustic mode given by equation (16) is plotted in figures 1 and 2 for different values of collision frequency and ion temperature. In these figures, the damping rate, the wave number, the collision frequency and the electron plasma frequency are scaled to the dimensionless physical quantities $\left| {\bar \gamma } \right|{\mkern 1mu} {\mkern 1mu} \equiv {\mkern 1mu} {\mkern 1mu} \left| \gamma \right|/{\omega _{pi}}$, ${\bar k_\parallel }\,\, \equiv \,\,{k_\parallel }{\lambda _{{\text{De}}}}$, ${\bar \nu _{ii}}{\mkern 1mu} {\mkern 1mu} \equiv {\mkern 1mu} {\mkern 1mu} {\nu _{ii}}/{\omega _{pi}}$ and ${\bar \omega _{pe}}{\mkern 1mu} {\mkern 1mu} \equiv {\mkern 1mu} {\mkern 1mu} {\omega _{pe}}/{\omega _{pi}}$ (=40 is assumed for numerical results), respectively. In figure 1, the collisional damping is compared to the collisionless damping. As ${\bar k_\parallel }$ increases, the collisionless Landau damping rate $\bar \gamma$ increases initially and then decreases to zero after reaching the maximum value (solid line). It can be seen that the damping rate increases as the collision increases: the dashed line for ${\bar \nu _{ii}}\,\, = \,\,0.01$ and the dotted line for ${\bar \nu _{ii}}\,\, = \,\,0.02$. Therefore, it can be said that the damping rate is enhanced by the collision. However, we see from figure 1 that the collisional damping is significant for ${\bar k_\parallel }\, > \,1$, i.e. ion-ion collision is important when the wavelength is much smaller than the electron Debye length. This feature becomes more evident by taking the limits for the damping rate. For ${\bar k_\parallel }\, \to \,\infty$, i.e. in the short wavelength limit, the damping rate becomes

$$\begin{equation}\gamma \,\, \approx \,\,\,\, - \,\,\sqrt {\frac{\pi }{2}} \frac{{{\omega _{pi}}}}{{8k_\parallel ^3\lambda _{{\text{De}}}^3{\omega _{pe}}}}\,\, - \,\,\frac{8}{5}\left( {\frac{{{T_i}}}{{{T_e}}}} \right){\nu _{ii}}k_\parallel ^2\lambda _{{\text{De}}}^2\,,\end{equation} \tag{ 19 }$$

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which can be further reduced to $\gamma \,\, \approx \,\, -\left( {8/5} \right)\left( {{T_{\text{i}}}/{T_{\text{e}}}} \right){\nu _{ii}}k_\parallel ^2\lambda _{{\text{De}}}^2$. Therefore, for a large wave number, the collisionless Landau damping is ignorable, and the collision term is dominant.

As well as the dynamic role of collisions on the dissipation of surface ion-acoustic waves, the ion temperature is also important in the fully ionized plasma. If the ions are cold, i.e. ${T_i}\, \to \,0$, then the dissipation becomes purely collisionless Landau damping. In fact, the effect of ion-ion collision on the damping is important only for ${T_i}\,\, \ne \,\,0$, regardless of the range of ${\bar k_\parallel }$. Figure 2 is provided to compare with figure 1 to see the effect of ion temperature. Figures 1 and 2 are drawn for ${T_{\text{i}}}/{T_{\text{e}}}\,\, = \,\,0.01$ and 0.02, respectively, with the same variation of collision rate. We easily observe that the ion temperature enhances the collisional damping rate as indicated in equation (19). Finally, the variation of the wave amplitude is drawn in figures 3–5 to show visually how the perturbed surface ion-acoustic wave would dissipate over time at an arbitrary point on the boundary. The normalized form of the perturbed wave can be written as $\bar E\,\, = \,\,{e^{\bar \gamma \bar t}}\,\operatorname{Re} \left[ {{e^{i\left( {{{\bar k}_\parallel }\bar x\, - \,{{\bar \omega }_r}\bar t} \right)}}} \right]$ where $\bar t\,\, \equiv \,\,t{\omega _{pi}}$ is the scaled time and $\bar x\,\, \equiv \,\,x/{\lambda _{{\text{De}}}}$ is the scaled position of the wave. Figure 3 shows the change of $\bar E$ in time for ${\bar \nu _{ii}}\,\, = \,\,0.01$ and ${T_{\text{i}}}/{T_{\text{e}}}\,\, = \,\,0.01$. It can be seen that the perturbation of the wave rapidly decreases around ${\bar k_\parallel }\,\, \approx \,\,0.5$ at which the damping rate peaks. If the collision rate and the ion temperature increase, the attenuation of the perturbation will be noticeable as ${\bar k_\parallel }$ increases past the peak. This property is seen in the large ${\bar k_\parallel }$ area in figure 4 where ${\bar \nu _{ii}}\,\, = \,\,0.02$ and ${T_{\text{i}}}/{T_{\text{e}}}\,\, = \,\,0.02$ are used. As equation (19) implies, the strong dissipation can be seen more clearly in figure 5 where the range of ${\bar k_\parallel }$ is further expanded. In weakly ionized plasmas, it is shown that the collisional moment [15] should include both Coulomb and change-exchange scattering. Hence, the investigation of the dissipation of ion-acoustic surface waves in a weakly ionized and semi-bounded plasma will also be treated elsewhere since the fractional force in weakly coupled plasmas can also be caused by the charge-exchange with neutral species. For surface wave experiments, the radially resolved probe measurements of the electron energy distribution have been obtained in an argon discharge [16] and the stationary microwave discharge in helium gas was investigated by applying field radial decay measurements with optical spectroscopy methods [17]. The excitation of a surface plasma wave was investigated by a relativistic electron beam over a plasma cylinder [18]. In addition, the experiment for the excitation of surface plasma waves was carried out by using an intense short laser pulse [19]. However, the experimental investigations on the dissipation of ion-acoustic surface waves has not been carried out in a fully ionized and semi-bounded plasma. So, in the future, we may observe and resolve the decay of ion-acoustic surface waves in a semi-bounded plasma including the influence of ion collisions.

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The effect of collisional damping of the surface ion-acoustic wave is investigated for a fully ionized plasma separated sharply by a vacuum. Hence, the physical properties of an ion-acoustic surface wave are investigated to determine the wave frequency and the Landau damping rate including the influence of the ion collisional effect. It has been shown that the ion temperature, as well as the ion-ion collision rate, can play an important role in the dissipation of the surface ion-acoustic wave. It has been found that the collisionless Landau damping is significant for a wavelength larger than the electron Debye length, but the collision rate and the ion temperature dominate over the Landau damping in the range where the wavelength is smaller than the electron Debye length. Surface waves possess a lot of advantages over bulk waves since they can produce plasmas with more spatially uniform density profiles, resulting in a wide range of applications such as gas discharge for surface treatment of solids, laser physics, gas detoxification, medical industry, etc. Since the collisional frequency is not often small under experimental conditions for plasma wave research, one should take into account the collision frequency for the dielectric permittivity of the plasma if the deviation of the experimental results from the theoretical dispersion diagram is evident. The mirror reflection model employed in this work would be especially useful to describe the confinement of plasma particles in metals and semiconductor plasmas. Moreover, the specular reflection condition and the transverse truncation method will be quite useful for the analysis of plasma confined within a finite layer such as a thin slab in a variety of applicational devices. It is shown that the functional perturbation method with the transverse truncation method used in this work provide insightful view points on the stable and unstable modes of surface waves and would also be useful to explore the propagation mode of surface waves in the presence of complex collective behaviors which provide complex character of the plasma susceptibility. In this work, it is shown that the collisionless damping is significant in the region where the wavelength is larger than the electron Debye length. In addition, it is shown that the collision dynamics dominates the wave damping when the wavelength is smaller than the electron Debye length. In this case, the relation between the thickness of the layer and the wavelength of the wave is expected to be an important factor for the propagation and damping of the wave. It should be pointed out that the methods used here would be helpful to understand the physics of ion-acoustic surface waves propagating in various geometric shapes.

The work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (NRF-2019R1A2C1003363).

The data that support the findings of this study are available upon reasonable request from the authors.

Appendix. Derivation of the wave frequency

The frequency of ion-acoustic wave can be derived from ${D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,0$ where ${D_r}$ is defined by equation (10) as

$$\begin{equation}{D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,1\,\, + \,\,\frac{{{k_\parallel }}}{\pi }\int\limits_{ - \,\infty }^\infty \frac{{{\text{d}}{k_ \bot }}}{{{k^2}\,\,{\varepsilon _r}\left( {{\omega _r},\,k} \right)}}.\end{equation} \tag{ A1 }$$

The real part of the permittivity in equation (A1), ${\varepsilon _r}\left( {{\omega _r},\,k} \right)$, can be found as

$$\begin{equation}{\varepsilon _r}\left( {{\omega _r},\,k} \right)\,\, = \,\,1\,\, - \,\,\frac{{\omega _{pi}^2}}{{{\omega ^2}}}\,\, + \,\,\frac{{\omega _{pe}^2}}{{{k^2}v_{Te}^2}}.\end{equation} \tag{ A2 }$$

If we insert equation (A2) into equation (A1), we get

$$\begin{equation}{D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,1\,\, + \,\,\frac{{{k_\parallel }}}{\pi }\frac{1}{{\left( {1\,\, - \,\,\frac{{\omega _{pi}^2}}{{\omega _r^2}}} \right)}}\int\limits_{ - \,\infty }^\infty \frac{{{\text{d}}{k_ \bot }}}{{k_ \bot ^2\,\, + \,\,{\beta ^2}}}\end{equation} \tag{ A3 }$$

where

$$\begin{equation}{\beta ^2}\,\, = \,\,k_\parallel ^2\,\, + \,\,\frac{{\omega _{pe}^2}}{{v_{Te}^2}}{\left( {1\,\, - \,\,\frac{{\omega _{pi}^2}}{{\omega _r^2}}} \right)^{ - 1}}.\end{equation} \tag{ A4 }$$

The integral in equation (A3) can be evaluated in the complex ${k_ \bot }$-plane. Picking up the residue at the pole ${k_ \bot }\,\, = \,\,\beta$, we obtain the following result:

$$\begin{equation}{D_r}\left( {{\omega _r},\,{k_\parallel }} \right)\,\, = \,\,1\,\, + \,\,\frac{{{k_\parallel }}}{{\left( {1\,\, - \,\,\frac{{\omega _{pi}^2}}{{\omega _r^2}}} \right)\beta }}\,\, = \,\,0\end{equation} \tag{ A5 }$$

which is equivalent to equation (12). If we solve equation (A5) for ${\omega _r}$ with the condition ${\omega _r}\, < \,{\omega _{pi}}$, one finally obtains the wave frequency in the form:

$$\begin{equation}{\omega _r}\,\, = \,\,\frac{{{\omega _{pi}}{k_\parallel }{\lambda _{{\text{De}}}}}}{{\sqrt {k_\parallel ^2\lambda _{{\text{De}}}^2\,\, + \,\,\displaystyle\frac{1}{2}\,\, + \,\,\sqrt {k_\parallel ^4\lambda _{{\text{De}}}^4\,\, + \,\,\displaystyle\frac{1}{4}} } }}.\end{equation} \tag{ A6 }$$