# Design of a CW BISOL RFQ for Three Kinds of High-Charge-State Ions Simultaneous Acceleration
## 2 Beam Dynamics Design
### Three Kinds of High-Charge-State Ions Simultaneous Acceleration  
The FRIB project has already realized the simultaneous transport of ions in multiple charge states [18], as a similar scientific facility, BISOL is expected to transport three kinds of mixed heavy ions simultaneously. Before simulating, the possibility of simultaneous transport is analyzed. According to Equation (3), the energy gain per nucleon of the synchronous particle in an RFQ accelerating cell is [19]:  
\[\Delta W_{\mathrm{s}}=\frac{q}{A}eE_{0}TL\cos\varphi_{\mathrm{s}} \tag{3}\]  
In order to accelerate particles with different charge-mass ratios, Equation (4) must be satisfied:  
\[\big{(}\frac{q}{A}\big{)}_{i}\cos\varphi_{i}=\big{(}\frac{q}{A}\big{)}_{j}\cos \varphi_{j} \tag{4}\]  
\(i\) and \(j\) represent particles with different charge states. When the mass number of the two particles is equal, then:  
\[\varphi_{i}=-\arccos\big{(}\frac{q_{j}}{q_{i}}\cos\varphi_{j}\big{)} \tag{5}\]  
When \(\frac{q_{j}}{q_{i}}\) approaches 1, \(\varphi_{i}\) will approach \(\varphi_{j}\), indicating that the higher the charge state is, the closer the synchronous phase of the nearby charge state particles is, the easier it is to be simultaneously accelerated in an RFQ accelerating cell. Impact-T is used to simulate the simultaneous acceleration of the three high-charge-state ions. Some research has shown that the high-charge-state mixed ions and the single-high-charge-state ion have similar beam parameters [20,21]. Therefore, we assume that the beams with different charge states have the same input parameters. The output phase distributions are shown in Figure 9; the main parameters are listed in Table 3.  
Figure 8: Effect of dipole field components on transmission efficiency. For the same inter-vane voltage, the energy gain of the low-charge state particles will be lower than that of the higher-charged state particles in each acceleration cell, which will lead to a trailing in the longitudinal phase space. This scheme requires Sn\({}^{22+}\) as the reference particle; the transmission efficiency will be lower than the single transmission efficiency of Sn\({}^{22+}\) when the mixed ions contain Sn\({}^{21+}\) and the output longitudinal emittance increases when high-charge state ions containing Sn\({}^{21+}\) are accelerated simultaneously. The RFQ buncher sections usually dominate the length of the accelerator. Usually, significantly less than half the length of an RFQ is devoted to accelerating the beam at a full longitudinal gradient. In addition, the introduction of an unbunched beam to the RFQ entrance requires an adiabatic transverse matching section that transforms the time-independent envelope into a time-varying envelope [22]. It has been proven that the longitudinal output emittance of an RFQ can be improved by replacing the adiabatic buncher with external discrete bunchers. The transmission of the RFQ is almost equal to that of a conventional RFQ with adiabatic bunching, and the RFQ length is reduced [23]. Figure 10 shows the phase distributions of the beam arriving at the RFQ entrance in the LEBT section without an MHB (internal bunching scheme) and with an MHB (external bunching scheme), respectively. In an external bunching scheme, there is a three-harmonic buncher with a fundamental frequency of 20.3125 MHz or 40.625 MHz. The design work for voltage and power was completed in 2020, and the details are given in Ref. [24]. The detailed simulation parameters given by the two schemes are listed in Table 4. Compared with the internal bunching scheme, the external bunching scheme can shorten the RFQ length by about 1 m at the same output beam energy, are reduce the cavity power consumption while reducing the beam output emittance, which will simplify the subsequent MEBT matching and DTL design. When the mixed ions are accelerated simultaneously, the beam quality obtained by the new scheme is better than the original one.