# The Design of a High-Intensity Deuteron Radio Frequency Quadrupole Accelerator
## 3 Design Consideration and Strategy  
For the beam dynamics design, we adopted the conventional dynamics design strategy known as \"four-stage theory\" [20]. A constant electrode voltage and a steady average aperture were chosen to facilitate the machining of the RFQ structures. This approach has the potential to enable the utilization of a basic 2D cutter for machining vane modulations. Furthermore, during the EM analysis phase, the adjustment of voltage to align with the \begin{table}
\begin{tabular}{c c} \hline
**Ion** & **D\({}^{\star}\)** \\ \hline Frequency (MHz) & 162.5 \\ Duty factor & 100\% \\ Vane length (m) & 3.7 \\ Average aperture (mm) & 4.23 \\ Input/output energy (MeV/u) & 0.020/1.05 \\ Beam current (mA) & 10 \\ Intervane voltage (kV) & 60 \\ \(E_{k}\) (MV/m) & 18.98 (1.40 Ek) \\ Input trans. \(\epsilon_{\mathrm{norm,rms}}\) (mm \(\cdot\) mrad) & 0.200 \\ Output \(\varepsilon_{\mathrm{x,norm,rms}}\) (mm \(\cdot\) mrad) & 0.201 \\ Output \(\varepsilon_{\mathrm{y,norm,rms}}\) (mm \(\cdot\) mrad) & 0.202 \\ Output \(\varepsilon_{\mathrm{z,rms}}\) (mm \(\cdot\) mrad) & 0.180 \\ Transmission efficiency (E\({}_{\mathrm{limit}}\) = 0.08 MeV) & 98.63\% \\ \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the cavity. Figure 1: RFQ modeling diagram using SolidWorks. designed value is a more straightforward process compared with that of variable voltage and aperture design [21]. To increase mode separation, we ensured that the electrode length was twice the wavelength [22]. Consequently, the RFQ operates without the necessity of any \(\pi\)-mode stabilizers (PSL) or dipole-mode stabilizer rods (DSRs) to sustain cavity stability, while concurrently maintaining a mode separation of 3.3 MHz. Moreover, the following aspects were taken into consideration:  
1. Transmission efficiency: Deuterons lost in the acceleration process may collide with the electrode surface, causing instability and reduced efficiency over long-term operation. The loss of D\({}^{+}\) ions may also generate unintended neutrons through charged-particle activations with the apparatus. Especially for D\({}^{+}\) losses in higher energy sections, neutron radiation is more of a concern. 2. Intervane voltage: A higher intervane voltage offers the advantage of reducing the RFQ length; however, it also increases the power consumption per unit length and peak field strength, thereby raising the risk of RF sparking. A lower intervane voltage of 60 kV requires a smaller average aperture to generate enough transverse focusing factor (B), and a smaller average aperture needs higher requirements on the machining accuracy of the electrode. The \"973\" RFQ's successful facility commissioning substantiates the viability of employing an intervane voltage of 60 kV for CW operation [6; 23]. It also operates at a frequency of 162.5 MHz. 3. Input energy: The input ion energy was optimized at 40 keV. It affects the cavity length, space-charge force, and initial cell length. Lowering the input energy enlarges the space charge effects, negatively influencing beam transmission in the low-energy beam transport (LEBT) section. Additionally, the length of the initial segment shortens and impedes electrode processing. Conversely, an excessive input leads to an elongated bunching section, thereby increasing the cavity length and the associated construction and operational costs. 4. Large transverse acceptance: Due to the space-charge force, the size of the beam increases during transport, especially for low-energy beams. Consequently, adopting a large input aperture can enhance transverse acceptance. The main PARMTEQM parameters for the RFQ are shown in Figure 2. The synchronous phase \(\varphi_{\mathrm{s}}\) shifts gradually from \(-90^{\circ}\) to \(-29^{\circ}\), effectively preventing the loss of particles in the longitudinal direction. The modulation factor m increases smoothly from 1 to 1.9, while the average aperture R0 shrinks gradually to 4.2 mm. A large entry aperture method is adopted to enhance the transverse acceptance by approximately 4%.