# Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization
## 4 Results  
This section reports the results of an application of the described methodology. The boundaries of the PSO exploration domain (Table 2) for each variable have been defined in agreement with the values estimated for the AlphCA Cathode, that is an OHC developed and tested by SITAEL [25; 26]. A keeper surface temperature of 800 K has been set for all the calculations according to the experimental measurements. AlphaCA has a Tantalum tube, 7.5 mm in diameter, which hosts a LaB6 insert that is 6 mmong and has an internal diameter of 3 mm. The tube is approximately thick 0.3 mm and 20.6 mmong and at its end there is an orifice that has a diameter of 0.4 mm and aength of 0.36 mm. The tube assembly is enclosed by means of a Molybdenum alloy keeper that has an orifice (diameter: 0.6 mm) at its end. Because theength of keeper orifice has not been found, a realistic value of 0.3 mm has been selected to perform the calculation. The propellant is pure xenon (contamination issues are not considered in the model). The cathode operates at a discharge current ranging from 1 to 3 A and at mass flow rates between 0.08 and 1 mg/s. The electrical characteristics in this operational envelope, measured in diode mode with keeper [26], are reported in Figure 3, as _test2_ and _test3_. The discharge voltage as a function of the mass flow rate decreases reaching a plateau after a certain value; this value states the transition from the so-called \"plume\" mode to the \"spot\" mode [1; 5]. The shaded zone in Figure 3 represents the confidence bound of the total discharge voltage, imited by the \(\hat{x}\pm\sigma\) curves, in which the PSO solutionsay. More in detail, the mean \(\hat{x}\), and the variance \(\sigma^{2}\), have been calculated for the best ten solutions obtained at each mass flow rate value. The mean and the variance are calculated with the standard equation, respectively, as follows:  
\[\hat{x}=\frac{1}{n}\sum_{z=1}^{n}x_{z} \tag{37}\]  
\[\hat{\sigma}^{2}=\frac{1}{n}\sum_{z=1}^{n}\left(x_{z}^{2}-\hat{x}^{2}\right) \tag{38}\]  
Hence, the standard deviation \(\sigma=\sqrt{\hat{\sigma}^{2}}\) is used to generate the confidence bound (\(\hat{x}\pm\sigma\)) around the mean values. Several considerations arise from the observation of Figure 3: \begin{table}
\begin{tabular}{c c c c} \hline
**Variable** & **Min** & **Max** & **Unit** \\ \hline \(n_{n,\ell}\) & \(1\times 10^{21}\) & \(1\times 10^{24}\) & m\({}^{-3}\) \\ \(n_{e,\ell}\) & \(1\times 10^{18}\) & \(1\times 10^{21}\) & m\({}^{-3}\) \\ \(V_{p}\) & 5 & 50 & V \\ \(T_{e,\ell}\) & 1 & 4 & eV \\ \(n_{n,\rho}\) & \(1\times 10^{21}\) & \(1\times 10^{23}\) & m\({}^{-3}\) \\ \(n_{e,\rho}\) & \(1\times 10^{18}\) & \(1\times 10^{22}\) & m\({}^{-3}\) \\ \(T_{e,\rho}\) & 1 & 4 & eV \\ \(T_{s}\) & 1700 & 1950 & K \\ \(L_{eff}\) & \(1\times 10^{-3}\) & \(6\times 10^{-3}\) & m \\ \(T_{u,\rho}\) & 1700 & 1950 & K \\ \hline \end{tabular}
\end{table} 
Table 2: Boundaries of the PSO exploration domain. Figure 3: Present calculation’s confidence bound of the total discharge voltage as function of the propellant mass flow rate computed at 1A (**a**), 2A (**b**), 3A (**c**) by means of PSO; experimental measurements from ref. [26] are reported for comparison as _test2_ and _test3_. The confidence bounds range between 2 V and 5 V for the current span examined;
* The discharge current initially decreases with the mass flow rate, as expected, but it increases at the highest mass flow rates; this could be attributed to the absence of a second double sheath at the cathode orifice exit that could be negative because the plasma density decreases passing from the orifice region to the keeper region (see Figures 4a and 5a);
* The shaded zone perfectly covers the experimental measurements obtained for a current of 2A and it is slightly higher at 3A; at 1A and atowest mass flow rates, the current is up to 40%ower, at its maximum, than the reference data, likely because the system of equations does not adequately describe the plasma physics in those conditions.