# Beam Dynamics Design of a Multi-Ion RFQ for Medical Application
## 3 Beam Dynamics Design
### Design Strategy
#### 3.2.2 New Design  
Since the RFQ is designed for multi-ion and medical applications, it requires strong suppression capabilities to space charge effects. In addition to adiabatic bunching, the suppression of space charge effects can still be enhanced by varying the focusing strength factor B across the cells. The B is adjusted by Equations (2) and (6) to achieve a dynamic balance between focusing and defocusing forces, ensuring stable beam transmission. \[B(n+1)=B(n)-[\Delta(n+1)-\Delta(n)]\times\frac{4\pi}{B(n)} \tag{6}\]  
where \(n\) is the cell number along the RFQ, the term \(\frac{4\pi}{B(n)}\) is obtained by applying the smooth-approximation formalism [46]. To improve the RFQ's capture efficiency, a suitable stable area in the longitudinal phase space should be designed. The parameter \(S\)[47] represents the area of the separatrix, i.e., the stable area in longitudinal phase space. Adjustments to the stable area are achieved by modifying the modulation parameter and the synchronous phase. \[S=W_{s}\frac{\sigma_{L}}{2}\Psi\sqrt{1-\frac{\phi_{s}}{\tan(\phi_{s})}} \tag{7}\]  
where \(W_{s}\) is the synchronous energy, and \(\Psi\) is the stable phase zone. \[\Psi(\phi_{s})\approx-3\phi_{s}+0.27\phi_{s}^{3}-0.252347\phi_{s}^{5} \tag{8}\]  
\(\sigma_{L}\) and \(\sigma_{T}\) are the longitudinal and transverse phase advances at zero current. \(A_{10}\) is acceleration term, defined as \(A_{10}=AV/2\). Adjustments to the stable area must ensure that \(\sigma_{L}\
eq\sigma_{T}\) to avoid phase advance resonance and resultant extra beam loss. \[ \begin{split}\sigma_{L}^{2}&=-\frac{\pi^{2}}{2} \frac{qA_{10}V}{W_{s}}\sin(\phi_{s})\\ \sigma_{T}^{2}&=\frac{B^{2}}{8\pi^{2}}-\frac{1}{2} \sigma_{L}^{2}\end{split}  \tag{9}\]  
The optimization parameters are shown in Figure 6. Compared to the general design, the new design exhibits more stable and well-controlled variations in the defocusing parameter. The simulation results of the new design are presented in Figures 7 and 8. In the new design, the beam envelope is better controlled by transverse focusing compared to the general design, and there is no significant enlargement of the envelope. The transmission efficiency is increased from 95% to 98% for 0.3 mA C\({}^{4+}\), and for 1 mA He\({}^{2+}\) from 89% to 97%. As shown in Figure 10a, when the beam current reaches 8 mA, the transmission efficiency decreases from 98% to 84%, representing a total reduction of 14%. Compared to the general design, the reduction in transmission efficiency is reduced by 49%, and the ability to suppress space charge effects is significantly enhanced. In Figure 4b, the stable area in the shaper section is adjusted to enable well-controlled beam capture. Additionally, all sections are now well connected, with no abrupt transitions or disruptions, as shown in Figure 9a. This improvement contributes to a reduction in both the number of longitudinally lost particles and the longitudinal emittance, as illustrated in Figure 9b. Compared to the general design, the number of longitudinally lost particles (NL) decreases from 302 to 105 (a reduction of approximately 65%), and the longitudinal emittance decreases by 0.00579 \(\pi\) mm\(\cdot\)mrad. In addition, the maximum longitudinal emittance also decreases significantly, by 4.2 \(\pi\) mm\(\cdot\)mrad. The final parameters of the RFQ are summarized in Table 2. The key parameters meet the design requirements in Section 3.1. Furthermore, compared to general design, the RFQ achieves significant improvements in transmission efficiency while maintaining highoutput beam quality. This advancement is primarily attributed to enhancements in the design strategy. Figure 6: Comparison of parameters between the general design and the new design along the RFQ. (**a**) Focusing strength factor \(B\). (**b**) Defocusing factor \(\Delta\). (**c**) Modulation parameter \(m\). (**d**) Synchronous phase \(\phi_{\psi}\). Figure 7: New design simulation results for C\({}^{4+}\).