# Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization
## 2 Plasma Model of Hollow Cathode
### Plasma Model Equations
#### 2.3.3 Insert  
Four equations are written in order to obtain the unknown plasma number density (\(n_{e,e}\)), electron temperature (\(T_{e,e}\)), neutral density (\(n_{n,e}\)) and emitter sheath voltage drop (\(V_{p}\)) in the insert region. These are the balance for the ion flux, Equation (13), the current conservation, Equation (14), the power balance, Equation (15), and the continuity equations, Equation (16).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(\mathbf{p_{1}}\) & \(\mathbf{p_{2}}\) & \(\mathbf{p_{3}}\) & \(\mathbf{p_{4}}\) & \(\mathbf{p_{5}}\) & \(\mathbf{p_{6}}\) & \(\mathbf{q_{1}}\) & \(\mathbf{q_{2}}\) \\ \hline Values & 0.0350 & \(-\)0.3057 & \(-\)0.6698 & \(-\)18.5416 & \(-\)12.8821 & \(-\)9.9614 & 0.7093 & 0.5227 \\ \hline \end{tabular}
\end{table}
Table 1: Coefficients of xenon electron-neutral cross section numerical fit [13].  
\[\left\{\begin{aligned} & q\pi r_{e}^{2}L_{eff}n_{n,e}n_{e,e}K_{i}+j_{i,e} \pi r_{o}^{2}=\underbrace{j_{i,e}\Big{(}A_{eff}+\pi r_{e}^{2}-\pi r_{o}^{2} \Big{)}+j_{th,e}\pi r_{e}^{2}}_{h_{out,e}}\\ &\frac{I_{d}}{A_{eff}}=j_{i,e}+j_{em}-j_{\sigma}\\ & j_{i,o}\bigg{(}V_{ds}+2\frac{k_{B}T_{e}}{q}\bigg{)}\pi r_{o}^{2 }+RI_{d}^{2}+j_{em}\bigg{(}V_{p}+\frac{3}{2}\frac{k_{B}T_{s}}{q}\bigg{)}A_{eff} =\dot{n}_{out,e}\bigg{(}\langle\epsilon_{i}\rangle+2\frac{k_{B}T_{e}}{q}+j_{ \sigma}2\frac{k_{B}T_{e}}{q}A_{eff}+\frac{5}{2}\frac{k_{B}T_{e,e}}{q}I_{d}\\ &(n_{e,e}+n_{n,e})k_{B}T_{s}\bigg{(}1+\frac{\alpha T_{e,e}}{T_{s} }\bigg{)}=1.36\cdot 10^{-9}\frac{I_{d}\cdot 0^{.3}T_{n}\cdot 0^{.95}\mu^{. 57}L_{0}\cdot 0^{.15}}{M^{0.10}\langle\epsilon_{i}\rangle^{0.20}\mu^{. 05}{a_{e}}^{1.43}{d_{o}}^{1.71}}\end{aligned}\right. \tag{13}\]  
The plasma resistance in the insert region is computed as:  
\[R=\frac{3\eta}{4\pi L_{eff}} \tag{14}\]  
A voltage drop is supposed to be between the insert and orifice region; the intensity of this double sheath is computed according to the inteaterature [5; 6; 9] by using the following expression:  
\[V_{ds}=\bigg{(}\frac{9I_{d}k_{B}T_{e,o}}{7.5\pi n_{e,o}r_{o}^{2}q^{2}}\sqrt{ \frac{m_{e}}{2q}}\bigg{)}^{2/3} \tag{15}\]  
The xenon viscosity in \(Ns/m^{2}\) is computed by means of the formula proposed by [PERSON] and [PERSON] [1]:  
\[\mu=2.3\cdot 10^{-5}\cdot T_{r}^{0.71+\frac{0.29}{17}},T_{r}=T_{s}/289.7 \tag{16}\]  
which is true if \(T_{r}>1\). In the following, all the expressions for the current densities used in Equations (13)-(16) are recalled for completeness:  
\[j_{i}=0.61 qn_{e}\bigg{(}\frac{k_{B}T_{e}}{m_{i}}\bigg{)}^{1/2} \tag{17}\]  
\[j_{th}=0.25 qn_{e}\bigg{(}\frac{8k_{B}T_{e}}{\pi m_{i}}\bigg{)}^{1/2} \tag{18}\]  
\[j_{em}\text{= D }T_{s}^{2}\ exp\bigg{(}-\frac{q\phi_{eff}}{k_{B}T_{s}}\bigg{)} \tag{19}\]  
\[\phi_{eff}=\phi_{0}-\sqrt{\frac{qE_{c}}{4\pi e_{o}}} \tag{20}\]  
\[E_{c}=\sqrt{\frac{n_{e}k_{B}T_{e}}{\epsilon_{0}}}\Bigg{[}2\sqrt{1+2\frac{qV_{ p}}{k_{B}T_{e}}}-4\Bigg{]}^{1/2} \tag{21}\]  
\[j_{\sigma}\text{= q}\frac{1}{4}\text{n}_{e}exp\bigg{(}-\frac{qV_{p}}{k_{B}T_{s}} \bigg{)}\bigg{(}\frac{8k_{B}T_{e}}{\pi m_{e}}\bigg{)}^{1/2} \tag{22}\]