## 2 Beam Dynamics Design
### Choice of Beam Dynamics Parameters
#### 2.2.5 Final Design Parameters  
Figure 2 provides an overview of the main parameters in the SC RFQ design. The cavity is divided into 112 cells. To get an optimal performance, we set the focusing strength \(B\) to be 13, ensuring effective beam focusing for high-intensity beams and an improved transmission efficiency. Additionally, we carefully control the maximum valueof the modulation parameter \(m\) to approximately 2.07. This control enhances our ability to manage the beam envelope and facilitates the efficient transmission of high-intensity beams. The design strategically achieves a smooth variation of various parameters along the SC RFQ, mitigating abrupt changes in the accelerator that could lead to beam instability.  
The final dynamics design parameters are summarized in Table 1, where we compared the SC and NC situations. Both the average aperture \(r_{0}\) and the minimum aperture \(a\) of the SC cavity surpass those of the NC cavity, significantly reducing beam loss. Despite the nearly threefold increase in inter-vane voltage under the SC condition, resulting in a larger \(E_{sp}\), it remains within established safety limits. Moreover, the transmission efficiency of the SC cavity reaches 100.0%, higher than the value under the NC condition, while the length of the SC cavity is only half of the NC length, which can significantly enhance acceleration efficiency and save construction costs. As for the power loss of the cavity, the SC case demonstrates a power consumption about five orders of magnitude lower than that in the NC case, as detailed in Section 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value (SC)** & **Value (NC)** \\ \hline Frequency (MHz) & & 162.5 & \\ Particles & & proton & \\ Beam current (mA) & & 10 & \\ Input energy (MeV/u) & & 0.030 & \\ Output energy (MeV/u) & 2.55 & & 2.54 \\ Inter-vane voltage (kV) & 240 & & 70 \\ Focusing parameter \(B\) & 13.0 & & 10.0 \\ Average aperture \(r_{0}\) (mm) & 8.18 & & 4.86 \\ Minimum aperture \(a\) (mm) & 5.10 & & 2.12 \\ Synchronous phase \(\varphi_{\mathrm{s}}\) (\({}^{\circ}\)) & & \(-90\)\(--30\) & \\ Peak surface electric field (MV/m) & 41.8 & & 20.8 \\ Transmission efficiency (\%) & 100.0 & & 99.1 \\ Cell number & 112 & & 190 \\ Cavity length (m) & 2.03 & & 4.20 \\ \(Q\) & \(2.52\times 10^{9}\) & & 9949 \\ Power consumption (W) & 2.48 & & \(1.49\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dynamics design parameters of the SC RFQ.  
Figure 2: Evolution of main design parameters along SC RFQ. (\(B\) is the focusing strength, \(m\) is the modulation factor, \(a\) is the minimum aperture, \(\varphi_{\mathrm{s}}\) is the synchronous phase, and \(W_{\mathrm{s}}\) is the synchronous energy).  
In addition, we conducted a comparative analysis of key parameters in this SC RFQ design with two counterparts, SRFQ1 and SRFQ2, proposed by INFN, as presented in Table 2. The INFN-SRFQs are built to accelerate heavy ions with a charge-to-mass ratio (Z/A) of 8.5, and its cavity operating frequency is lower than our design, at 80 MHz. In RFQ cavities, a lower frequency corresponds to a smaller tolerable \(E_{sp}\), and thus, INFN SC RFQs are configured with \(E_{sp}\) values around 25.0 MV/m. According to Equation (3), \(r_{0}\) increases with \(V\) if the wavelength (i.e., the frequency of the cavity) is held constant, and it increases with the wavelength (i.e., decreasing cavity frequency) if \(V\) remains constant. The \(r_{0}\) values for these three cavities listed in the table adhere to this principle. The \(B_{sp}\) values for all three SC RFQs are consistent and all far below the upper limit of the magnetic field.