# Design of a CW BISOL RFQ for Three Kinds of High-Charge-State Ions Simultaneous Acceleration
## 2 Beam Dynamics Design  
### Design Requirements and Beam Dynamics Scheme  
The design parameters of the post RFQ are summarized in Table 1. Sn\({}^{22+}\) is taken as the reference particle, the input transverse normalized rms emittance is 0.25 mm-mrad, and the peak beam current is 2.2 emA. According to the Kilpatrick criterion Equation (1) [8]:  
\[f=1.643E_{k}^{2}e^{-\frac{8.5}{k_{k}}} \tag{1}\]  
where \(f\) is the cavity frequency in MHz and \(E_{k}\) is the maximum surface electric field value in MV/m. For CW cavities, it is usually required that the maximum surface electric field in the cavity does not exceed \(1.8E_{k}\)[9] to reduce the risk of cavity sparking. The surface electric field in the cavity should be as low as possible for safety; commissioning experiences from the current running CW RFQ accelerators show that it is safe to keep the peak surface electric field around \(1.5E_{k}\). For example, the SPIRAL2 RFQ is \(1.65E_{k}\)[10], the SSC-LINAC RFQ is \(1.523E_{k}\)[11] and the C-ADS RFQ is \(1.3E_{k}\)[12]. For our 81.25 MHz RFQ cavity, the maximum surface electric field in the cavity is \(16.7\) MV/m, almost \(1.58E_{k}\), which is a pretty conservative design.  
Figure 1: Layout diagram of BISOL [1].  
RFQ's dynamic design adopted PARMTEQ software [13; 14], in which the linac is generated cell-by-cell in an iterative procedure. In order to simplify the RF design and tuning process for RFQ, the inter-vane voltage and average aperture remain constant. The \({}^{132}\)Sn\({}^{22+}\) beam is accelerated from 3 keV/u to 500 keV/u with a vane length of 5.604 m. The simulated transmission efficiency reached 95.6%. The output longitudinal normalized rms emittance is only 0.20 keV/u-ns, indicating that a good beam quality is realized. The Kilpatrick coefficient is 1.58. The beam transmission envelope along the RFQ is shown in Figure 2. Beam losses in the transverse direction are mainly at the end of the gentle buncher section, and in the longitudinal direction are at the beginning of the accelerating section.  
To verify the design results, TraceWin [15] and Impact-T [16] were also used to simulate the particle transport separately. The comparison of the simulation results of the three software is shown in Table 2. There are some differences in the beam emittance and transmission efficiency, mainly because of the differences in the criteria for particle loss and the methods of calculating space charge force.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Reference particle & \({}^{132}\)Sn\({}^{22+}\) \\ Frequency (MHz) & 81.25 \\ Input beam energy (keV/u) & 3 \\ Output beam energy (keV/u) & 504 \\ Vane length (mm) & 5604 \\ Peak beam current (emA) & 2.2 \\ Duty factor (\%) & 100 \\ Inter-vane voltage (kV) & 70 \\ Kilpatrick coefficient & 1.58 \\ Average aperture (mm) & 5.73 \\ Synchronous phase (deg) & \(-90\)\(\sim\)\(-25\) \\ Maximum modulation factor & 2.07 \\ Output transverse normalized rms emittance (\(\tau\)mm-mrad) & 0.19 \\ Output longitudinal normalized rms emittance (keV/u-ns) & 0.20 \\ Transmission efficiency (\%) & 95.6 (\(>\)95) \\ \hline \end{tabular}
\end{table}
Table 1: Main dynamic design parameters.  
Figure 2: Beam transmission envelope of PARMTEQ.