## 3 Optimization of Discharge Chamber Size and Magnetic Field
### Effects of Three Classical Axial Magnetic Field Distributions on Ionization Characteristics  
According to the research of ECR ion source workers, using an appropriate axial magnetic field distribution can help improve the extraction beam intensity of the ion source. The effects of three traditional axial magnetic field distributions often utilized in ECR ion sources on the ionization properties inside the discharge chamber are confirmed by modeling the plasma inside the discharge chamber [10; 17; 25; 26; 27]. In Figure 8a, the three types of field distribution are displayed. The size of the discharge chamber is \(\Phi\) = 30 mm \(\times\) L40 mm. Type 1 is a flat magnetic field with a uniform field of 875 G in the middle; type 2 is a Gaussian-distributed magnetic field with a peak value greater than 875 G, and type 3 is a saddle-shaped magnetic field with low sides and a high center. The distribution of the magnetic field and the contour line (860 G-5 G-890 G) are shown in parts (b)-(d) of Figure 8, where the red contour line is 875 G. To evaluate the impact of the magnetic field on the plasma from the perspective of the resonance plane, it is possible to estimate the size and form of the ECR resonance plane based on the distribution of the contour line. Figure 7: The number of particles, the average electron density, and the maximum electron density varies with the cavity: (**a**) is the number of particles, the average electron density, and the maximum electron density contrast with the radius of the discharge chamber. (**b**) is the number of particles, the average electron density, and the maximum electron density compared with the discharge chamber’s length. Figures 9 and 10 illustrate the ionization characteristics associated with the three field types. The variations in the average electron density and average electron temperature inside the discharge chamber, as seen in the time domain, are shown in Figure 9. It can be seen in Figure 9a that the electron density increases rapidly with time and tends to be stable at around 0.1 ms and no longer increases. This response time will not change with the change in the applied magnetic field. The saddle-shaped magnetic field can produce the highest average electron density compared to the flat magnetic field and the Gaussian-distributed magnetic field, which is nearly four times that of the former two magnetic field distributions. The electron density of the flat magnetic field is marginally higher than that of the Gaussian-distributed magnetic field. It further demonstrates that, under identical circumstances, the saddle-shaped magnetic field has the best effect on gas ionization and the most pronounced improvement for the ion source, which is consistent with the conclusion of the ion source extraction experiment [26]. Figure 9b illustrates this occurrence, which is compatible with the \"preglow\" phenomena seen in the experiment: the electron temperature would rise dramatically in a noticeably short period, reach its maximum value, decrease sharply, and then stabilize [21]. The ionization process within the discharge chamber is to be stable at about 0.1 ms since the period at which the electron temperature tends to stabilize is essentially the same as when the electron density tends to stabilize. This is further evidence that the reaction time of the plasma is not related to the distribution of the magnetic field. Figure 8: Three classic axial distributions and corresponding contour distributions: (**a**) is three classical axial distributions; (**b–d**) are contour distributions of the flat magnetic field, Gaussian magnetic field, and saddle magnetic field, respectively (860 G–5 G–890 G). Figure 10: Electron density under different magnetic field distributions: (**a**) is a flat magnetic field; (**b**) is a Gaussian-distributed magnetic field; (**c**) is a saddle-shaped magnetic field. Figure 9: The average electron density and average electron temperature change in the time domain: (**a**) is the electron density; (**b**) is the electron temperature.