The space charge effects on the intra-bunch motion under large chromaticity at the Main Ring in the Japan Proton Accelerator Research Complex

TL;DR

This work addresses intrabunch motion under large chromaticity in the J-PARC MR by developing a compact 2D tracking code that incorporates direct and indirect space-charge forces and the resistive-wall impedance. It combines ABS/ABH theoretical models with Gaussian-beam extensions to interpret head-tail modes and validates the approach through detailed beam measurements, showing that the maximum intrabunch frequency $f_{max}$ scales with chromaticity as $f_{max}=\left|2 f_0 \frac{\xi_x}{\eta}\right|$ and that indirect space charge can suppress instabilities while not altering this fundamental relation. The study reveals that recoherence times saturate at high space-charge strength and that radial modes become less relevant at high chromaticity unless instabilities are excited. Overall, the validated simulations provide a practical tool for optimizing chromaticity and IBFB parameters to sustain stable, high-intensity proton operation in the J-PARC MR.

Abstract

In general, optimizing chromaticity and transverse feedback parameters is important in suppressing beam instabilities for stable operation in high-intensity proton machines. Meanwhile, space charge effects impact the intrabunch motion of the proton bunch within a large chromaticity region in the machines, such as the main ring (MR) in the Japan Proton Accelerator Research Complex (J-PARC). To address this issue, the decoherence and recoherence of the transverse motion of the particles comprising the proton bunch are investigated. The analysis reveals that the space charge effects have a significant influence on the recoherence period through the chromaticity. Nevertheless, the relationship between the maximum frequency in the bunch and the chromaticity is not affected by the space charge effects. These findings are demonstrated with particle tracking simulations including the direct and indirect space charge effects, and the impedance source of the J-PARC MR. Furthermore, we illustrate the influence of the indirect space charge effect on particle motion by examining the excitation patterns of radial and head-tail modes.

Figures (26)

• Figure 1: Sum of multiturn wakes for different $N_\lambda$(a) and the deviation from the analytical estimate with infinite $N_\lambda$(b). Here, $V$ represents the accumulated wake function values, assuming they are stored only for the past $N_\lambda$ turns. (a) $V(N_\lambda)$ and (b) $1-V(N_\lambda)/V(\infty)$
• Figure 4: Head-tail tune spectra in the harmonic potential model with indirect space charge effect for various values of chromaticity: (a) $\xi_x=-1.5$, (b) $\xi_x=-6.5$, and (c) $\xi_x=-12.5$. The dots indicate only the results for $|a_m|>0.1$. The blue, green and red dashed lines and black, red, green, blue, orange, magenta, cyan, dark green, purple, gray, dark red, dark cyan and yellow lines represent the frequency shifts calculated by Eq. \ref{['eq:84']} for each head-tail mode in the range of $m = -3, -2, \cdots 12$, respectively.
• Figure 8: Head-tail tune spectra in the harmonic potential model with weak space charge force. The dots indicate only the results for $|a_m|>0.1$. The orange, blue, green and red dashed lines and black, red, green, blue and orange lines represent the frequency shifts calculated by Eq. \ref{['eq:84']} for each head-tail mode in the range of $m = -4, -3, \cdots 4$, respectively.
• Figure 9: Head-tail tune spectra in the harmonic potential model with indirect space charge effect for various values of intensity: (a) $N_B=4.2\times10^{12}$ ppb and (b) $N_B=8.6\times10^{12}$ ppb. The dots indicate only the results for $|a_m|>0.1$. The black, red, green, blue, orange, magenta, cyan, dark green, purple, gray, dark red, dark cyan and yellow lines represent the frequency shifts calculated by Eq. \ref{['eq:84']} for each head-tail mode in the range of $m = 0, 1, 2, \cdots 12$, respectively, but due to the resolution they are drawn as if the low modes were degenerate.
• Figure 12: Relative amplitudes $|a_m|$ tracking simulation of the harmonic potential model with indirect space charge effect. The black, red, green, blue, orange, magenta, cyan, dark green, purple, and gray curves denote the cases of $m = 0, 1, 2, \cdots 9$, respectively. (a) $N_B=4.2\times10^{12}$ ppb and (b) $N_B=8.6\times10^{12}$ ppb.