# Numerical Analysis of the Breakdown Process of CF\({}_{3}\)I at Low Pressure
## 4 Discussions
### Paschen Curve  
The Paschen curve, which is derived from a model based on an analytical fit of the first ionization Townsend coefficient, describes the breakdown voltage of a gas \(U_{b}\) as a function of pressure \(P\) and gap distance between electrodes \(d\):  
Figure 15: Power balance of (**a**) sum, (**b**) electron, (**c**) negative ions, (**d**) positive ions (Smoothed). \[U_{b}=f(Pd) \tag{11}\]  
By systematically scanning a wide range of parameters for CF\({}_{3}\)I, we calculated and compared the Paschen curves of CF\({}_{3}\)I and SF\({}_{6}\), as shown in Figure 16. This comparative analysis facilitates a comprehensive evaluation of the breakdown characteristics of CF\({}_{3}\)I and SF\({}_{6}\) at low pressures [21]. The red and gray dots represent successful and unsuccessful breakdown points, respectively, under these conditions. The blue curve represents the theoretical Paschen curve of CF\({}_{3}\)I with a secondary electron emission coefficient of 0.2. Meanwhile, the green and purple curves represent the theoretical Paschen curves of SF\({}_{6}\) with secondary electron emission coefficients of 0.2 and 0.1, respectively. The black curve corresponds to the experimentally obtained Paschen curve of SF\({}_{6}\). The horizontal axis of the Paschen curve is the product of pressure and the gap distance Pd, and the unit is generally \(\times\)133.3 Pa-cm. This conversion is because the pressure unit used in the simulation program is Torr and 1 Torr = 133.3 Pa, so after multiplying the distance between the two, the unit becomes Torr \(\cdot\) cm. The breakdown voltage of the gas gap on the longitudinal axis is Ub and the unit is V.  
In a uniform electric field, the Paschen curve of the gas is generally \"U\" shaped, that is, there is a point at which the breakdown voltage reaches a minimum. It is evident that the simulated Paschen curves align perfectly with the characteristic \"U\" shape. For the CF\({}_{3}\)I parameter scan, a fixed secondary electron emission coefficient of 0.2 was used. In contrast, during the SF\({}_{6}\) Paschen curve investigation, two cases were considered, which incorporated secondary electron emission coefficients of 0.1 and 0.2, respectively. These cases were juxtaposed with experimental measurements of the SF\({}_{6}\) Paschen curves, revealing a significant influence of the secondary electron emission coefficient on the Paschen curves. This suggests that the experimental measurements for the SF\({}_{6}\) Paschen  
Figure 16: Paschen curve of trifluoroiodomethane and sulfur hexafluoride [21]. curves were conducted using a material with a secondary electron emission coefficient of approximately 0.1. Unfortunately, the parameters for the Paschen curve of CF\({}_{3}\)I at low pressures could not be obtained. However, through a comparative analysis with the Paschen curve of SF\({}_{6}\) at low pressures, we assert the reliability of our results. As illustrated in Figure 16, under low-pressure conditions, CF\({}_{3}\)I demonstrates a lower breakdown voltage compared to SF\({}_{6}\). Under high pressure conditions, the breakdown voltage of pure CF\({}_{3}\)I has been shown to exceed that of SF\({}_{6}\)[44; 45]. However, experiments also indicate that under low-pressure conditions, the breakdown voltage of CF\({}_{3}\)I may be lower than that of SF\({}_{6}\), attributable to CF\({}_{3}\)I's heightened sensitivity to changes in the electric field [46]. Therefore, the results of this article are consistent with the experiments qualitatively. In addition, according to the formula for calculating the Paschen curve (12), when \(A\), \(B\), \(\gamma_{se}\) are determined, the fitted Paschen curve can be obtained, as follows:  
\[U_{b}=f(Pd)=\frac{Bpd}{\ln Apd-\ln[\ln(1+1/\gamma_{\rm se})]} \tag{12}\]  
Among them, \(A\) and \(B\) are determined by experiments that depend on the constants of gas properties. \(\gamma_{se}\) is the average value of the secondary electron emission. According to the previous text, this paper sets it to 0.2.