# Development of a Radio-Frequency Quadrupole Accelerator for the HL-2A/2M Tokamak Diagnostic System
## 2 Beam Dynamics Design  
The PARMTEQM [9] code is used to carry out the RFQ dynamics design. The choice of the input beam energy is a trade-off of the RFQ length and the space-charge effects. On the one hand, a low input energy beam makes the RFQ shorter. On the other hand, a higher energy beam can effectively reduce the effect of space charge. Accordingly, we choose 20 keV/u as the input energy for the RFQ design. A high inter-vane voltage results in a high accelerating gradient, which makes the RFQ shorter. However, it requires more RF power dissipation per unit length and increases the risk of RF breakdown. Based on the operational experience of RFQs, we select an inter-vane voltage of 70 kV to ensure the reliable operation of the RFQ. The Peak surface field is one of the most important parameters of the RFQ design. In our design, the peak surface field is 1.52 times the Kilpatrick limit of 13.6 MV/m [10]. Such a low surface electric field is beneficial to maintaining the stability of the RFQ in high-power operation. Another important issue is that the threshold energy of the deuteron breakup reaction is 100 keV [11]. Since this reaction can lead to the activation of the structure due to neutron generation, the beam losses at the high-energy section should be minimized.  
An approach is adopted to change the average aperture at the initial cells, which can reduce the difficulty of beam matching from the low energy beam transport (LEBT) into the RFQ [12,13]. In a typical RFQ design, the average aperture of the radial matching section (RMS) is reduced gradually, and the average aperture after the RMS remains constant, as shown in Figure 1. While in the new design, the average aperture after the RMS is larger than that of the typical design.  
The matched beam ellipse at the entrance of the RFQ under the two design methods is shown in Figure 2. It can be seen that the matched beam of the new design has a smaller incident angle and a larger envelope.  
The final RFQ design parameters as a function of the RFQ cell are given in Figure 3. \(\phi_{\mathrm{s}}\) is the synchronous phase. \(m\) is the modulation factor. \(a\) is the minimum aperture. \(W_{\mathrm{s}}\) is the kinetic energy of the synchronous particles. \(r_{0}\) is the average aperture. \(V\) is the inter-vane voltage. Figure 4 shows the transverse beam envelope and the longitudinal phase space evolution along the RFQ cell according to PARMTEQM. The beam distribution at the exit of the RFQ is shown in Figure 5. A summary of the RFQ parameters is listed in Table 1.  
Figure 1: Vane profile at the initial cells under two design methods.  
Figure 2: The matched beam ellipse at the entrance of the RFQ under two design methods. New design method (red): \(\alpha\) = 0.75, and \(\beta\) = 0.0425 mm/mrad. Typical design method (blue): \(\alpha\) = 0.99, and \(\beta\) = 0.0353 mm/mrad.  
Figure 4: Evolution of beam envelopes along the RFQ cell from PARMTEQM. Plots from top to bottom are the beam profiles in the \(x\) and \(y\) planes and phase and energy spectrums, respectively.  
Figure 5: Particles phase-space distribution at the exit of the RFQ.  
Figure 3: Evolution of the RFQ design parameters along the RFQ cell after optimization.