# Research and Design of the RF Cavity for an 11 MeV Superconducting Cyclotron
## 4 Design and Simulation of the RF Cavity
### Optimized Design of the RF Cavity  
The RF cavity designed in this paper is partially placed in the magnet system, and this part of the RF cavity structure should not be changed significantly after the size is determined. The other part of the RF cavity (mainly the round coaxial resonator part in the back half of the cavity structure) is placed outside the magnet system so that the relevant structure parameters can be adjusted according to the design requirement.  
The main structural parameters that can be adjusted in the round coaxial resonant cavity part are the inner conductor radius \(r_{\rm ic}\), the outer conductor radius \(r_{\rm oc}\), and the length \(L\). The effect of these three structural factors on the resonant frequency of the cavity computed using 3D electromagnetic field numerical simulation software is shown in Figure 6. The RF cavity resonant frequency is smaller when \(r_{\rm ic}\) is larger, \(r_{\rm oc}\) is smaller, and \(L\) is shorter. When \(r_{\rm ic}\) is certain, \(r_{\rm oc}\) is roughly inversely proportional to \(L\) at the same resonant frequency. Therefore, it should be avoided that the length \(L\) of the round coaxial resonator is too short, which leads to a sharp increase in the radius \(r_{\rm oc}\) of the outer conductor of the round coaxial resonator.  
In order to further determine the selection of each structural parameter of the round coaxial resonator, the effect of these parameters on the unloaded quality factor \(Q_{0}\) of the cavity is calculated. Figure 7 shows the effect of \(r_{\rm oc}\) and \(L\) on the magnitude of the unloaded quality factor of the RF cavity for \(r_{\rm ic}\) = 20 mm. It can be determined that the larger \(r_{\rm oc}\) and  
Figure 4: Circular Stem diameter versus resonant frequency.  
Figure 5: Chamfer radius at the edge of Dee versus resonant frequency.  
are, the larger the \(Q_{0}\) value of the RF cavity is, and different \(Q_{0}\) values can be taken at the same resonant frequency.  
The equivalent circuit method is used to find the \(L\) corresponding to different \(r_{\text{oc}}\) at \(r_{\text{ic}}\) = 20 mm and the same resonant frequency, and the surface area size of the round coaxial cavity corresponding to each set of data and the unloaded quality factor are calculated by using 3D electromagnetic field simulation software. According to Figure 8, it is observed that at the same resonant frequency, the smaller \(L\) is, the larger the surface area of the cavity, and the larger the unloaded quality factor \(Q_{0}\). The surface area of the round coaxial resonator is related to the vacuum degree of the cavity; thus, the cavity design needs to follow the principle of minimizing the surface area of the cavity. This requirement works  
Figure 6: Influence of each parameter of a round coaxial resonator on the resonant frequency of the RF cavity.  
Figure 7: Influence of each parameter of the round coaxial resonator on the unloaded quality factor of the RF cavity.  
in the opposite way to the selection of the quality factor. Furthermore, it is established that \(L\) is inversely related to \(r_{\rm oc}\). In summary, choosing a smaller \(L\) will result in a larger \(Q_{0}\) but also lead to an overly large \(r_{\rm oc}\) and surface area of the cavity. Conversely, selecting a larger \(L\), although it minimizes the surface area of the cavity as much as possible and achieves a smaller \(r_{\rm oc}\), necessitates a reduction in the \(Q_{0}\) of the cavity. Consequently, the above influencing factors are considered in the actual design, and an appropriate value is chosen as a compromise based on demand.