# Beam Dynamics Design of a Multi-Ion RFQ for Medical Application
## 3 Beam Dynamics Design  
### Design Considerations  
This RFQ is designed to accelerate ions with charge-to-mass ratios ranging from 1/3 to 1/2, primarily C\({}^{4+}\) and He\({}^{2+}\), focusing on the ion species commonly used in current and next-generation medical radiotherapy [27; 28]. The design parameters of the RFQ should fully take into account the current technological capabilities, as well as the operational requirements of both upstream components (ion source and Low Energy Beam Transport, LEBT) and downstream components (Drift Tube Linac, DTL). * **Beam current**: The selection of RFQ beam current is closely related to the dose rate required for clinical particle therapy. Considering the clinical dose rate require  
Figure 1: Schematic Diagram of the Modulated Electrodes in an RFQ. ments [29; 30] and referencing the beam current settings of the existing medical RFQ accelerator [31; 32], the beam current is set to 0.3 mA for C\({}^{4+}\) and 1 mA for He\({}^{2+}\). * **Operating frequency**: In order to maintain sufficient focusing strength when increasing the RFQ operating frequency, the average aperture must be reduced accordingly. If the average aperture is too small, it is not conducive to beam transmission. On the other hand, if the aperture is not reduced, the voltage needs to be increased, which increases the risk of RF breakdown. Considering the size of the cavity and the frequency of commonly available power sources, the operating frequency of the RFQ is set to 200 MHz. * **Injection energy**: The input energy should be consistent with the extraction voltage of the ion source. In addition, it affects the design of the LEBT. If the input energy is too low, space charge effects will be intensified, placing stricter requirements on the vacuum level and magnetic focusing system of the LEBT. Conversely, an excessively high injection energy may lead to an increase in the length of the LEBT. Considering prior experience with stable RFQ operation and established design practices [11; 33], the injection energy is ultimately set to 8 keV/u. * **Output energy**: The output energy of the RFQ not only affects its own length but is also directly related to the design of the subsequent DTL. If the output energy is too high, the length of the RFQ will increase significantly, leading to higher manufacturing costs and a larger equipment footprint. In contrast, if the injection energy into the DTL is too low, it may result in an uneven radial electric field distribution within the drift tube, leading to degraded beam quality. Considering both the length of the RFQ and the complexity of the subsequent DTL design, the RFQ output energy is ultimately set to 750 keV/u. * **Input emittance**: An excessively large input emittance increases beam loss, while an overly small emittance leads to higher beam density and enhanced space charge effects. Therefore, an appropriate input emittance is essential not only for reducing the probability of beam loss but also for ensuring stable beam transport. Ultimately, the input emittance is set to 0.2 \(\pi\) mm\(\cdot\)mrad. * **Voltage and Kilpatrick factor**: To maintain sufficient focusing strength and accelerating gradient, the inter-vane voltage is selected within the range of 60-80 kV. Simultaneously, the Kilpatrick factor is limited to below 1.8 in order to minimize the risk of RF breakdown within the cavity. This choice aligns with the design practices of many RFQs currently in operation at frequencies close to 200 MHz [31; 33; 34], where similar voltage ranges and Kilpatrick factors are commonly adopted to ensure reliable and stable operation. In addition to considering the design parameters, there are several specific operational and performance requirements. * **Multi-ion compatibility**: Ions with different charge-to-mass ratios exhibit varying sensitivities to space charge effects. Therefore, when designing an RFQ capable of accelerating multi-ion, one of the primary considerations is the influence of space charge effects.