## 2 Beam Dynamics Design  
### Design Considerations  
The design goal of the SC RFQ is to attain optimal beam acceleration and transmission efficiency. Acceleration efficiency depends on the accelerating gradient \(E_{acc}\), which is defined as the ratio of the maximum accelerating voltage \(V_{acc}\) experienced by the charged particle to the effective acceleration length \(d\) of the cavity:  
\[E_{acc}=\frac{V_{acc}}{d}=\frac{1}{d}\bigg{|}\int_{0}^{d}E_{z}(z)e^{i\omega_{ 0}z/\beta\epsilon}dz\bigg{|}, \tag{2}\]  
its unit is \(\mathrm{V/m}\). Here, \(\omega_{0}\) is the angular frequency of the cavity mode, \(\beta\) denotes the relativistic velocity of the ions, and \(c\) is the speed of light. In the RFQ accelerator, the axial electric field distribution is given by \[E_{z}=\frac{kAV}{2}I_{0}(kr)coskz, \tag{3}\]  
where \(A\) and \(V\) are the accelerating parameter and inter-vane voltage of the RFQ, respectively, and \(k=\frac{2\pi}{B\lambda}\) is the wave number. As described above, \(E_{acc}\) is proportional to the inter-vane voltage \(V\). However, the peak surface electric field \(E_{sp}\) and magnetic field \(B_{sp}\) of the cavity also increase with the inter-vane voltage. Excessive peak surface fields may lead to cavity quenching or multipacting effects. Consequently, the maximum accelerating gradient of an SC cavity is constrained by these fields. Therefore, in the design phase, it is imperative to minimize the peak surface electric and magnetic fields while preserving the highest achievable accelerating gradient. For SC cavities, the risk of quenching is pronounced when \(B_{sp}\) exceeds 80 mT under 4 K conditions. However, advancements in post-treatment technology enhance the cavity's ability to withstand larger surface electric fields. Consequently, the magnetic field, rather than the electric field, stands as the ultimate limiting factor determining the performance of SC cavities [24].  
Transmission efficiency is affected by various dynamic design parameters, which are crucial for controlling the beams. For SC accelerators, achieving a transmission efficiency as high as 100% is imperative to prevent quenching resulting from beam interactions with the cavity wall.  
Maintaining the superconducting state requires thorough consideration of cavity design to prevent quenching. Consequently, our design approach prioritizes simplicity by adopting a constant focusing strength \(B\), a dimensionless parameter defined as [25]:  
\[B=\frac{q\lambda^{2}V}{m_{0}c^{2}r_{0}{}^{2}}, \tag{4}\]  
where \(q\) and \(m_{0}\) are the charge and the rest mass of the accelerated ions, \(\lambda\) is the RF wavelength, and \(r_{0}\) is the average aperture of RFQ. When \(B\) remains unchanged, \(r_{0}\) also remains unchanged, which can simplify subsequent processing and facilitate cavity tuning.  
We employed the traditional four-step method [26] developed at Los Alamos National Laboratory (LANL) for the beam dynamics design. This method divides the RFQ into four distinct sections: the radial matching section, the shaper section, the gentle buncher section, and the accelerator section. The input beam undergoes a process of gradual bunching in the first three sections, then will be rapidly accelerated to the designed energy in the final section.  
The dynamics simulation was conducted using the PARMTEQM code [27]. PARMTEQM requires various input parameters, including operating frequency, ion species, beam current, input energy, inter-vane voltage, and synchronous phase, etc. In the simulation, the RFQ is divided into unit cells of length \(\beta\lambda/2\), and the input parameters are gradually changed in each cell. The optimal accelerator structure generated by PARMTEQM is then selected to achieve the desired particle distribution. The design goal is to attain adequate radial focusing, transverse emittance growth, overall length, and peak surface field of the accelerator.