## 3 RF Structure Design  
The simulation software employed for this study is CST MWS 2023 [36], a comprehensive and highly integrated professional tool designed for 3D electromagnetic (EM), circuit, temperature, and structural stress simulations. CST MWS, a module within the CST software suite, stands out as the fastest and most accurate simulation tool in the field of high-frequency applications. For an SC cavity, on the one hand, simplicity in structure is essential to prevent the occurrence of large surface electric and magnetic fields, which could lead to quenching. On the other hand, the high \(Q\) value of the SC cavity necessitates mechanical stability to mitigate sensitivity to frequency fluctuations [37]. Considering these factors, we opted for the four-vane RFQ structure due to its superior mechanical and thermal stability, complemented by a circular shell design. In order to save the superconducting material niobium, the RFQ is constructed as a 3-millimeter-thick thin-wall structure, as shown in Figure 7.  
The four electrodes are installed in the four quadrants of the cavity, and their structural diagrams are shown in the figures below. Achieving the design resonant frequency of the cavity requires an iterative optimization of the electrode parameters, as outlined in Figure 8a. Simultaneously, to ensure a uniformly flat electric field distribution along the central axis of the cavity so as to achieve stable beam acceleration, it is necessary to optimize the undercut structure as illustrated in Figure 8b. It is essential to note that the values of _cutz_ differ at the front and end of the vanes. Figure 7: The cross-section view of SC RFQ. It is also crucial to minimize the peak surface electric field and magnetic field of the SC cavity to avoid sparking and quenching. To achieve this, we implemented chamfers on both the electrodes and vanes, as demonstrated in Figures 8 and 9. Ultimately, the structural parameters of the electrodes were fine-tuned to the values specified in Table 3. At this stage, the cavity could be excited by an EM field of a specific frequency, generating the expected quadrupole electric field. After modeling the cavity, we need to select the appropriate solver. For frequency and mode calculation of resonant cavities with high \(Q\) values, the Eigenmode Solver is typically employed. Then, the hexahedral mesh is adopted to obtain higher precision  
Figure 8: The structural diagram of the four electrodes installed in the cavity: (**a**) the front view of the electrodes; (**b**) the undercut shape of the electrodes. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline v1x (mm) & 17.00 \\ v1y (mm) & 45.00 \\ v2x (mm) & 31.50 \\ vaneh (mm) & 208.00 \\ \(\theta\) (deg) & 4.30 \\ cut1 (mm) & 62.00 \\ cut2 (mm) & 82.00 \\ cutz (mm) & 84.00 (front)/76.00 (end) \\ \(r_{1}\) (mm) & 20.00 \\ \(r_{2}\) (mm) & 10.00 \\ \(r_{3}\) (mm) & 30.00 \\ \(r_{4}\) (mm) & 1.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Structural parameters of the electrodes. Figure 9: Schematic diagram of electrodes and vanes chamfering. for complex model calculations. Given the finite integration (FIT) method used in the simulation, the mesh settings play an important role in both speed and accuracy. The global mesh significantly impacts frequency calculation results, while the local mesh in regions with large electric fields influences peak surface electric field outcomes, so a mesh convergence analysis was conducted. The simulation results indicate that with a mesh number around 3,000,000 and a local mesh step width of less than (3,3,3) in the central part of the cavity, both frequency and peak surface electric field converge. The EM field simulation results reveal that the operating frequency of the cavity is 162.53 MHz (refer to Figure 10), and it operates in TE210 mode. Figure 11 illustrates the EM field mode of the cavity.