## 3 Dust-Surface Potential (DSP) with \(Xe^{+}-F^{-}-Sf_{6}^{-}\) Plasma  
For xenon-fluorine-sulfur hexafluoride \(\left(Xe^{+}-F^{-}-SF_{6}^{-}\right)\) plasmas, two modes exist depending on the temperatures of the electrons and ions. We investigate a non-Maxwellian complex plasma that has electrons, positive ions (\(Xe^{+}\)), two negative ions (\(SF_{6}^{-}\) and \(F^{-}\)), and negatively charged dust particles as constituents. While in balance, the charge-neutrality precondition requires \(n_{e}-Z_{Xe^{+}}n_{Xe^{+}}+Z_{F^{-}}n_{F^{-}}+Z_{SF_{6}^{-}}n_{SF_{6}^{-}}-q_{d }n_{d}/e\), where \(n_{s}\) is the sth species of the unperturbed number density (\(s=e\) stands for electrons; \(Xe^{+}\) is the Xenonions; \(F^{-}\)and \(SF_{6}^{-}\) stand for fluorine and sulfur ions, respectively; and \(d\) stands for charged (negatively) dust particles). \(Z_{Xe^{+}}\), \(Z_{F^{-}}\), and \(Z_{SF_{6}^{-}}\) stand for the charging state of the argon, fluorine, and sulfur hexafluoride, respectively, and \(q_{d}\) stands for the charge on the dust grains. When \(q_{d}=\)constant, the electrons, the positive ion, and the two negative ions in the \((q,\alpha)\)-distribution currents may be represented, [38]  
\[I_{e}^{q\alpha}+I_{Xe^{+}}^{q\alpha}+I_{F^{-}}^{q\alpha}+I_{SF_{6}^{-}}^{q \alpha}=0, \tag{3}\]  
with  
\[\sigma_{j}^{d}=\pi r_{d}^{2}\left(1-\frac{2q_{j}\phi_{d}}{m_{j}V_{j}^{2}} \right), \tag{4}\]  
\[I_{j}^{q\alpha}=q_{j}\int V_{j}\sigma_{j}^{d}f_{CT}(\mathbf{V}_{j})d^{3} \mathbf{V}_{j}. \tag{5}\]  
\(\sigma_{j}^{d}\) stands for the cross-section for charging interactions among dust particles and plasmas constituents of charge \(q_{j}\), and \(\phi_{d}\) is the DSP of radius \(r_{d}\). To continue, we integrate over the \(f_{CT}(\mathbf{V}_{j})\) to simplify Equation (2). In spherical-coordinates, write the volume element as \(d^{3}\mathbf{V}_{j}=V_{j}^{3}dV_{j}d\mu d\phi\) to obtain the currents owing to electrons, positive ions, and the two negative ions due to negative charge dust particles [38], respectively. \[I_{e}^{q\alpha}=-\sqrt{2^{3}}\pi r_{d}^{2}\exp\left(\frac{\chi_{q\alpha}}{ \tau_{q\alpha}}\frac{e\phi_{d}}{m_{e}V_{Te}^{2}}\right)V_{Te}n_{e}\tau_{q \alpha}C, \tag{6}\]  
\[I_{Xe^{+}}^{q\alpha}=\sqrt{2^{3}}\pi r_{d}^{2}\left\{1-\frac{\chi_{q\alpha}} {\tau_{q\alpha}}\frac{e\phi_{d}}{m_{Xe^{+}}V_{Xe^{+}}^{2}}\right\}Z_{Xe^{+}}V_ {Xe^{+}}en_{X^{+}},\tau_{q\alpha}C, \tag{7}\]  
and  
\[I_{F^{-}}^{q\alpha}=-\sqrt{2^{3}}\pi r_{d}^{2}\left\{1+\frac{\chi_{q\alpha}} {\tau_{q\alpha}}\frac{Z_{F^{-}}e\phi_{d}}{m_{F^{-}}V_{F^{-}}^{2}}\right\}Z_{F^ {-}}V_{F^{-}}en_{F^{-}}\tau_{q\alpha}C, \tag{8}\]  
\[I_{SF_{6}^{-}}^{q\alpha}=-\sqrt{2^{3}}\pi r_{d}^{2}\left\{1+\frac{\chi_{q \alpha}}{\tau_{q\alpha}}\frac{Z_{SF_{6}^{-}}e\phi_{d}}{m_{SF_{6}^{-}}V_{SF_{6}^ {-}}^{2}}\right\}Z_{SF_{6}^{-}}V_{SF_{6}^{-}}en_{SF_{6}^{-}}\tau_{q\alpha}C, \tag{9}\]  
and  
\[C=\left(\frac{1}{1-q}\right)^{-\frac{3}{2}}\left[\frac{(1-q)^{2}\Gamma\left( \frac{1}{1-q}\right)}{\Gamma\left(\frac{1}{1-q}-\frac{5}{2}\right)\left[3 \alpha+\left(\frac{1}{1-q}-\frac{3}{2}\right)\left(\frac{1}{1-q}-\frac{5}{2} \right)\left(1-q\right)^{2}\right]}\right],\text{for }-1<q\leq 1,\]  
where \(\tau_{q\alpha}=\left[(6+24\alpha-17q+12q^{2})/\left(-6q+29q^{2}-46q^{3}+24q^{ 4}\right)\right]\) and \(\chi_{q\alpha}=1/q+8\alpha/\left(2q-7q^{2}+6q^{3}\right)\). The DSP of a negatively charged dust particle can be written as by substituting Equations (6)-(9) into Equation (3).