# The Design of a High-Intensity Deuteron Radio Frequency Quadrupole Accelerator
## 5 Electromagnetic (EM) Analysis of RFQ
### Undercutting and Tuning  
The longitudinal distribution of the electric field within the RFQ is highly sensitive to the undercutting at both ends of the electrode. Hence, the field is best adjusted by  
Figure 16: The cross-section and parameters of the RFQ cavity.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(r_{0}\) & 4.23 & mm \\ \(r_{T}\) & 3.19 & mm \\ \(\rho\) & 0.75 & \\ \(L_{1}\) & 33.43 & mm \\ \(L_{2}\) & 11.15 & mm \\ \(\theta_{1}\) & 10 & degree \\ \(\theta_{2}\) & 10 & degree \\ \(R_{v}\) & 30 & mm \\ \(R_{w}\) & 30 & mm \\ \(L_{max}\) & 172.5 & mm \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters of the cavity cross-section.  
optimizing the undercut parameters. Upon introducing undercutting, the magnetic field between adjacent quadrants may form a closed loop, increasing the electric field at both ends. Figure 17 gives a schematic of the RFQ undercutting, and Table 6 lists the input and output undercutting parameters. By optimizing the undercutting angle \(\alpha\) (degrees) and undercutting depth D (mm), a smoother electric field was obtained between electrodes. For an RFQ operating in CW mode, a large \(\alpha\) value can lead to hot spots at the root of the undercutting and cause deformation at extremities due to poor water cooling.  
The optimized off-axis field is shown in Figure 18, where the voltage deviation between electrodes is less than 0.5%.  
Errors occurring during machining and assembly of the cavity can lead to mode-mixing, disrupting the field distribution of the operational quadrupole mode. The tuners can compensate for these errors by returning the field distribution and frequency to their designated values. Indeed, both the inductance-tuning and capacitance-tuning components are capable of altering the local frequency of the cavity. In our four-vane RFQ cavity, the electric field is concentrated in the small area of the electrode head, while the magnetic field is widely distributed in the cavity, so the tuner is inserted mainly to disturb the magnetic field, that is, inductance tuning. The local inductance decreases as the tuner is inserted deeper into the cavity, leading to a higher local frequency and smaller electric field. Our RFQ has a total of 64 tuners, with 16 in each quadrant. As shown in Figure 19, the tuners are evenly distributed about the horizontal axis, with a diameter of 50 mm each. Figure 20  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Input \(\alpha\) & 55 & degrees \\ Input D & 86.8 & mm \\ Output \(\alpha\) & 55 & degrees \\ Output D & 78.3 & mm \\ \hline \hline \end{tabular}
\end{table}
Table 6: Undercutting parameters.  
Figure 17: RFQ undercutting in both sides of vanes.  
Figure 18: Off-axis field distribution (**left**) and its unevenness (**right**).  
illustrates the linear relationship between the cavity frequency and tuner insertion depth. The tuner insertion depth ranges from 20 mm initially to 50 mm at maximum, spanning \(-0.82\) MHz to +1.65 MHz, with increments of 53 kHz/mm.