# Beam Dynamics Design of a Multi-Ion RFQ for Medical Application
## 3 Beam Dynamics Design
### Design Strategy  
#### 3.2.1 General Design  
The beam dynamics design of the RFQ was performed using PARMTEQM [35], which includes collective effects in its simulation results, especially space charge effects. The initial design followed the general design strategy [36], which is one of the most widely used in RFQ design [37; 38; 39; 40]. This strategy also often serves as a starting point and reference for many studies involving beam dynamics design and optimization [18; 21; 41; 42; 43]. Therefore, it is suitable to serve as a reference for the first-stage design of the multi-ion RFQ. In the general design, the RFQ is divided into four sections: the radial matching section (RM), the shaper section (SH), the gentle buncher section (GB), and the accelerator section (AC). In the RM section, the beam is completely matched. In the SH section, a longitudinal phase-stable region is formed to capture the beam. In the GB section, a gentle and adiabatic bunching process is used to compress the beam and suppress space charge effects by maintaining a constant focusing strength factor and constant voltage. In the AC section, the beam is rapidly accelerated to its final output energy. The design parameters are illustrated in Table 2. In this design, the constant focusing strength factor 4.2, the voltage of 70 kV, and the synchronous phase range from \(-90^{\circ}\) to \(-30^{\circ}\) were adopted. The choice of these parameters was based on the operational experience of currently well-performing medical accelerators: the Heavy Ion Medical Machine (HIMM) [18], the Heavy Ion Medical Accelerator in Chiba (HIMAC) [31], and the Heidelberg Ion-Beam Therapy Center (HIT) [44]. The number of particles was set to 10,000 with a 4D water-bag distribution [45]. The beam dynamics simulation results are shown in Figures 2 and 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Particle & C\({}^{4+}\)/He\({}^{2+}\) \\ Frequency [MHz] & 200 \\ Beam current [mA] & 0.3 (C\({}^{4+}\))/1 (He\({}^{2+}\)) \\ Injection energy [keV/u] & 8 \\ Output energy [keV/u] & 750 \\ Input emittance [\(\pi\) mm-mrad] & 0.2 \\ Kilpatrick factor & \(\leq\)1.8 \\ Voltage [kV] & 60–80 \\ Length [m] & \(\leq\)2.8 \\ Transmission efficiency [\%] & \(\geq\)95 \\ Emittance growth [\%] & \(\leq\)5 \\ Extra beam loss [\%] & \(\leq\)10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design requirements of the multi-ion RFQ. In both figures, the top two plots show the transverse beam envelope changes along the RFQ, illustrating the transverse focusing process. The bottom two plots display the longitudinal beam envelope changes, indicating the phase and energy distributions generated by the design for beam acceleration. The transmission efficiency is 95.2% for 0.3 mA C\({}^{4+}\) and 89% for 1 mA He\({}^{2+}\) within the same RFQ. For the length and Kilpatrick factor meet the required specifications. However, several issues were identified in the general design, as listed below:  
1. For 1 mA He\({}^{2+}\), the transmission efficiency falls below the required minimum of 95%. 2. The beam was not well captured in the shaper section, as observed in Figure 4, resulting in increased longitudinal losses and longitudinal emittance. 3. Figure 5 shows the transmission efficiency of the RFQ versus beam current. As the beam current increases to 8 mA, the transmission efficiency decreases from 95.2% to 32%, representing a total reduction of 63.2%. This indicates that the RFQ exhibits  
Figure 3: Beam dynamics simulation results for He\({}^{2+}\) (general design). Plots from top to bottom show the beam envelope evolution along the RFQ in the x and y planes, and the phase and energy spectra, respectively (color online). Figure 2: Beam dynamics simulation results for C\({}^{4+}\) (general design). Plots from top to bottom show the beam envelope evolution along the RFQ in the x and y planes, and the phase and energy spectra, respectively (color online).