Beam halo studies of double-periodic focusing structures at the low-energy end of superconducting linac

This paper presents a comprehensive study on the beam halo evolution in the low-energy end of high-current superconducting linear accelerators (SLAs) with double-period focusing structures. The structure consists of cryogenic cooling modules (macro-period structures) and focusing units composed of RF cavities and solenoids within each module (micro-period structures). We utilized a three-dimensional particle-core model based on an ellipsoidal bunch to study the mechanism of halo formation with greater realism and reference value. Numerical simulations indicate that, in line with envelope instability theory, maintaining the zero current phase advance ( σ 0s ) below 90° is critical for the formation of halo structures. Using 90° as the demarcation line, the resonance behavior between particles and the core shows significant differences. Fourier analysis of the envelope oscillation spectrum reveals that when the σ 0s of each micro-period structure is less than 90°, the resonance types differ between emittance-dominated beams and space-charge-dominated beams. 2:1 resonance occurs between particles and the 3rd harmonic of the envelope oscillation in space-charge-dominated beams; while 4:1 resonance occurs in emittance-dominated beams. As σ 0s increases, the influence of lower harmonics in the resonance becomes more significant. When σ 0s is slightly greater than 90°, 4:1 resonance form with higher harmonics of the envelope oscillation (specifically related to the number of micro-period structures in each module). When σ 0s reaches 110° or higher, 1:1 resonance form with the 2nd harmonic of the envelope oscillation. Unlike most previous studies based on two-dimensional particle-core models, we found that the longitudinal envelope significantly alters the charge density distribution of the core, leading to halo formation. This finding highlights the importance of three-dimensional effects in beam dynamics, especially under high current conditions. Additionally, we analyzed the impact of the number of micro-period structures within a module and their envelope modulation. Specifically, fewer micro-period structures within a module result in the envelope oscillation spectrum power to be concentrated at lower frequencies, making resonance with low harmonics more likely. Finally, we provide guidelines for the beam matching design at the low-energy end. By adjusting the focusing parameters of the matching structures, the likelihood of beam halo formation can be effectively suppressed. In summary, this study not only reveals the key mechanisms of halo formation in double-period focusing structures but also provides an effective approach to optimizing beam quality by adjusting structural parameters and matching methods. These findings offer clear guidance for beam design in the low-energy end of SLAs. • The 3D particle-core model ensures broad applicability and practicality. • The method for obtaining a matched beam in a double period structure is introduced. • Fourier analysis determines envelope oscillation frequency under non-smooth approximation. • Envelope modulation predicts particle-core resonance. • The impact of quasi-periodic structure quantity on beam quality is analyzed.