## 3 Results and Discussion
As the SEE coefficient is increased, collisionless pressure heating increases slightly in the sheath regions and does not change in the bulk plasma. This shows that the effect of the SEE coefficient is greater for electron ohmic heating, as shown in Figure 8a. The spatial distributions of cycle-averaged electron heating and electron power dissipation at different SEE coefficients are shown in Figure 9. As we know, electron heating is the sum of electron ohmic heating and collisionless pressure heating (\(P_{c}=e\mu_{e}E^{2}+eD_{e}E\cdot\
abla n_{e}\)). From Figure 8, we know that electron ohmic heating is dominant and is greater than collisionless pressure heating, which contributes to positive electron heating in most of the region and negative electron heating near the electrodes. The negative value can be considered to be the result of \"diffusion cooling\". When the SEE coefficient is increased, electron ohmic heating increases, as shown in Figure 8a, resulting in more frequent collisions between electrons and neutral particles, which leads to electron heating and also increases with increasing SEE coefficient, as shown in Figure 9a. Figure 9b shows the spatial distributions of electron power dissipation. Electron power dissipation can be written as \(P_{L}=\sum_{j}H_{j}\text{N}k_{j}\mu_{e}\). The electrons obtain energy from the electric field and lose energy by colliding with neutral particles. As the SEE coefficient is increased, electron density increases gradually across the entire discharge space, and electron energy loss also increases as a result. The effects of the SEE coefficient on the effective current density and effective power density were also considered based on the numerical calculations. The ion beam bombards the electrode, resulting in secondary electron generation. In Figure 10a, as the SEE increases, the maximum current density increases and the minimum value is negative growth. The positive and negative current density correspond to the electric field. The current density can be calculated using the above equations: \(j_{0}=\frac{\tau_{0}}{\tau_{0}}\frac{dV}{dt}+\frac{1}{L}\int_{0}^{L}j_{g}dx\), \(j_{g}=e(\sum_{j}l_{j}-\sum_{n}l_{e,n})\)  
Figure 8: Spatial distributions of cycle-averaged (**a**) electron ohmic heating and (**b**) collisionless pressure heating at SEE coefficients of 0.001, 0.005, and 0.01.  
and \(J_{e,i,n,m}=\pm\mu_{e,i,n}En_{e,i,n}-D_{e,i,m,n}\frac{\partial n_{e,i,m}}{\partial x}\). For electrons and ions, the diffusion term is smaller than the mobility term; thus it can be deduced that the positive and negative current density are related to the electric field. The effective current density was used instead of peak current density for analyzing the characteristic of discharge. The effective value is determined according to the formula \(X_{rms}=\sqrt{\frac{1}{n}\Sigma_{i=1,n}\,x_{i}^{2}}\)[38]. \(x\) is a certain quantity. Clearly, the form of the effective value is a root mean square (RMS) value. The power density \(P\) can be assumed to be \(P=f\int_{0}^{T}V(t)J_{0}(t)\), where \(T\) is a circle [39; 40]. The method for calculating effective power density is as the same as that used for effective current. The effective current density and effective power increase with increasing SEE coefficient. This is because the charged-particle density increased with the SEE coefficient. Obviously, the growth of the SEE coefficient can effectively regulate the effective current density and effective power density during discharge.