# Numerical Analysis of the Breakdown Process of CF\({}_{3}\)I at Low Pressure
## 4 Discussions
### Power Balance  
The energy conservation of the discharge process is important to consider. Two primary sources of energy for particles are the electric field between the plates, known as the heating effect, and the secondary electron emission effect induced by the impact of ions on the plate. Two energy loss terms must also be taken into account: particles absorbed by the plate at the boundary and particles dissipating energy through collision with one another. In the DC discharge process, the change value of the total kinetic energy \((dE_{k}/dt)\) of all particles is typically too small to be compared to other power terms. Consequently, the energy conservation principle can be employed to derive the power balance equation of the discharge device as follows:  
\[P_{\text{heating}}\,+P_{SEE}=P_{\text{collision}}\,+P_{\text{boundary}} \tag{3}\]  
Among them, \(P_{\text{heating}}\) is the heating power, which is the main production term of the particle energy:  
\[P_{\text{heating}}\,=\int E\cdot(j_{e}+j_{i}) \tag{4}\]Among them, \(E\) is the intensity of the electric field, \(j_{e}\) is the electron current density, and \(j_{i}\) is the ion current density. This algorithm obtains the heating power of each position in the discharge space by diagnosing the electric field intensity, electron current density, and ion current density of each position in the discharge space and then integrates the heating power of each position in the discharge space to obtain the heating power. \(P_{SEE}\) is the secondary electron emission (\(SEE\)) power, another term for small energy production:  
\[P_{SEE}=\frac{E_{SEE}}{dt} \tag{5}\]  
Among them, \(E_{SEE}\) is the total energy of secondary electrons emitted by the electrode in unit time, which can be obtained by counting the number and energy of secondary electrons. \(P_{\text{collision}}\) is collision loss power, which is lost through particle collisions:  
\[P_{\text{collision}}=\frac{E_{\text{beforeMCC}}-E_{\text{afterMCC}}}{dt} \tag{6}\]  
Among them, \(E_{\text{beforeMCC}}\) and \(E_{\text{afterMCC}}\) are the total particle energies before and after the Monte Carlo collision (MCC), respectively. \(P_{\text{boundary}}\) is the boundary loss power, which is the power absorbed by the boundary when the particles move to the boundary:  
\[P_{\text{boundary}}=\frac{E_{\text{boundary}}}{dt} \tag{7}\]  
Among them, \(E_{\text{boundary}}\) is the total energy of all particles bombarded with the plate in unit time, and the boundary loss power is obtained by dividing the total energy of all particles bombarded with the plate in unit time by unit time. Because the heating power of electrons is of special research significance, this paper divides the heating power into the electronic heating power \(P_{\text{eheating}}\) and the ion heating power \(P_{\text{heating}}\). The corresponding equations are as follows:  
\[P_{\text{heating}}=P_{\text{eheating}}+P_{\text{iheating}} \tag{8}\]  
\[P_{\text{eheating}}=P_{\text{collision}}+P_{\text{eboundary}}-P_{SEE} \tag{9}\]  
\[P_{\text{iheating}}=P_{\text{collision}}+P_{\text{boundary}} \tag{10}\]  
where \(P_{\text{collision}}\) is the electron collision loss power, \(P_{\text{eboundary}}\) is electron boundary loss power, \(P_{\text{collision}}\) is ion collision loss power, and \(P_{\text{boundary}}\) is ion boundary loss power. The power balance in a discharge process can be studied by diagnostically calculating the total, electron, negative ion, and positive ion powers according to the appropriate equations. Figure 15 shows these powers, with (a) representing the total power balance, (b) the electron power balance, (c) the negative ion power balance, and (d) the positive ion power balance. The analysis and verification of the power balance in the discharge process can be carried out using these data. In order to reduce the noise of the simulated data, the adjacent average method was employed to smooth the curves, using 200 window points.