# Beam Dynamics Design of a Multi-Ion RFQ for Medical Application
## 4 Error Analysis  
### Input Errors Analysis  
The RFQ is capable of directly receiving the input beam from the ion source and the LEBT. However, due to the instabilities of the ion source [50] and the LEBT [51; 52], the practical values of the RFQ input beam may deviate from the design parameters. Thus, it is important to investigate the RFQ's tolerance to a range of input errors. In this study, the beam emittance in the multi-ion RFQ is well controlled through optimization using the new strategy. Given this effective control, input errors have a more pronounced effect on transmission efficiency than on emittance. Therefore, the analysis primarily focuses on the impact of various input beam errors on the transmission efficiency. The input errors include the input beam current, vane-voltage multiplier, emittance, energy spread, spatial displacements, and divergence-coordinate displacements. The scan range of the beam parameters at the entrance of the RFQ is shown in Table 5. During the simulation, each parameter is varied individually while keeping the others constant using PARMTEQM. Figure 10 shows the transmission efficiency of the RFQ versus the input beam parameters. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Program**} & **Transmission Efficiency** & \(\varepsilon_{n,x}\)/\(\varepsilon_{n,y}\) & \(\varepsilon_{n,l}\) \\  & **[\%]** & **[\(\pi\) mm\(\cdot\)mrad]** & **[\(\pi\) mm\(\cdot\)mrad]** \\ \hline PARMTEQM & 97.00 & 0.1972/0.1980 & 0.08158 \\ TRACEWIN & 96.50 & 0.1968/0.1973 & 0.08162 \\ TOUTATIS & 96.90 & 0.1973/0.1975 & 0.08177 \\ \hline \end{tabular}
\end{table} 
Table 4: Multi-particle simulation results using different programs (He\({}^{2+}\)). \begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Item**} & **Scan Range** & **Unit** \\ \hline Beam current & 0–8 & mA \\ vane-voltage multiplier & 0.9–1.1 & - \\ Input emittance & 0.05–0.35 & \(\pi\) mm\(\cdot\)mrad \\ Energy spread & 0–10 & \% \\ Spatial displacement for centroid & \(-\)0.1–0.1 & cm \\ Divergence-coordinate displacement for centroid & \(-\)80–80 & mrad \\ \hline \end{tabular}
\end{table} 
Table 5: The scan range of the beam parameters at the entrance of the RFQ. Since the density of the ion source plasma fluctuates with changes in the source chamber temperature, voltage jitter or gas pressure, the beam current injected into the RFQ will fluctuate. As the beam current increases, space charge effects become more pronounced, leading to transverse and longitudinal phase shifts that destabilize the beam and reduce transmission efficiency [53]. For this RFQ, the required operational conditions are a beam current of 1 mA, and the transmission efficiency should be no less than 95% to ensure a safety margin of compatibility. The impact of varying input beam current on transmission efficiency is shown in Figure 9(a). At 3.2 mA, the transmission efficiency remains above 95%. The analysis of space charge effects under the designed beam dynamics conditions indicates that the overall design possesses a certain level of beam current tolerance. The vane-voltage multiplier is a scaling factor for the vane voltage [54]. As the vane voltage increases, both the longitudinal and transverse acceptances of the particles increase. Consequently, a higher inter-vane voltage leads to higher transmission efficiency. As shown in Figure 9(b), when the vane-voltage multiplier exceeds 0.927 (corresponding to an inter-electrode voltage greater than 60.255 kV), a transmission efficiency of over 95% is achieved. At the design voltage of 65 kV, the transmission efficiency reaches 98.1%. However, further increasing the voltage not only requires a more powerful power source but also heightens the risk of cavity breakdown, while the improvement in transmission efficiency becomes negligible. Figure 9(c) illustrates the effect of the normalized transverse emittance growth at the entrance of the RFQ. To maintain a transmission efficiency of 95%, the normalized transverse emittance should not exceed 0.242 \(\pi\) mm\(\cdot\)mrad.