# Beam Dynamics Design of a Multi-Ion RFQ for Medical Application
## 1 Introduction  
Particle radiotherapy is an advanced radiotherapy technique that utilizes high-energy particles, such as protons and carbon ions, to destroy tumour cells [1]. By leveraging characteristics of different charged particles, such as the Bragg peak and relative biological effectiveness (RBE), personalized treatment plans can be tailored [2]. However, most of the existing medical accelerator systems are designed for single-ion acceleration [3, 4, 5, 6], which makes it difficult to cover all clinical treatment scenarios, restricting treatment flexibility and increasing treatment costs. Therefore, there is a need to design a medical accelerator capable of accelerating a variety of ions, which can flexibly adjust the treatment plan according to different clinical needs, so as to achieve personalized treatment [7, 8, 9].  
The RFQ is a linear accelerator that provides both transverse focusing and longitudinal bunching, offering significant advantages in accelerating beams with low velocities [10]. Many medical accelerators have used RFQ as injectors for the initial acceleration and bunching of particle beams, ensuring high-quality beam injection into downstream accelerators [11, 12, 13, 14]. Thus, one of the key steps in developing a medical accelerator for multi-ion radiotherapy is designing a multi-ion-compatible RFQ [15, 16, 17].  
In the study of RFQ beam dynamics, several innovative strategies have been developed to improve beam quality, transmission efficiency, and structural compactness. For example, fast-bunching designs [18] shorten the bunching section to reduce overall RFQ length and boost transmission efficiency; truncated-vane modulation introduces regions of zero longitudinal electric field to lower longitudinal emittance [19]; high-frequency compact [20,21] significantly reduce both RFQ length and RF power consumption; and optimized radial-matching sections adjust potential-function boundary conditions to enhance transmission stability [22]. However, these methods are typically designed for a specific ion species or charge-to-mass ratio and do not consider performance when accelerating additional or different ions. Consequently, in multi-ion radiotherapy or other applications requiring the acceleration of multiple ion species, a single RFQ often cannot simultaneously account for the space charge effects and bunching characteristics of different ions. As a result, a separate RFQ accelerator is typically required for each species [23,24,25], leading to higher costs or greater design complexity.  
To address this issue, a new beam dynamics design strategy focusing on multi-ion acceleration within the same RFQ is proposed in this study. Its main contributions are as follows:  
1. **Achieving Multi-Ion Compatibility via Dynamic Focus-Defocus Control**: By adjusting the focusing strength to achieve a dynamic balance with the defocusing strength, the efficient transmission of multiple ion species within the same RFQ is improved.
2. **Quantification and Optimization of Stable Longitudinal Phase Space Area**: A parameter is introduced to characterize the stable longitudinal phase space area and optimized to maximize beam capture efficiency.
3. **Error Tolerance Analysis**: Using comprehensive error studies, acceptable tolerances for input-beam errors and mechanical fabrication errors are determined, and practical guidelines for RFQ construction and operation are provided.  
The structure of this paper is as follows: Section 2 introduces the basic principles of the RFQ. Section 3 describes the beam dynamics design strategy along with simulation results that validate the effectiveness of the proposed method. Section 4 analyzes the operational and manufacturing error tolerances of the RFQ. Finally, Section 5 presents the conclusions.