Semi-analytical algorithms to study longitudinal beam instabilities in double rf systems

TL;DR

This work addresses the challenge of longitudinal beam instabilities in double RF systems essential for 4th-generation light sources by developing modern semi-analytical algorithms built on Haïssinski equilibrium and enhanced instability models. It replaces the traditional Bosch approach with two Haïssinski solvers (Venturini and Alves) and two instability formalisms (He criterion and Gaussian LMCI), enabling fast, scalable stability analysis and multimode coupling. The methods are validated against multibunch tracking (mbtrack2) and applied to the SOLEIL II storage ring, yielding accurate instability maps and revealing substantial Touschek lifetime gains through high-dimensional parameter optimization (ALBuMS). The open-source ALBuMS package facilitates rapid cavity-design iterations, stability analysis, and optimization campaigns, including partial treatment of short-range wakes, thereby offering a practical, computationally efficient tool for next-generation light sources. This approach significantly reduces the computational burden compared with tracking while preserving physical insight and broad applicability to double RF configurations.

Abstract

Double rf systems are critical for achieving the parameters of 4th-generation light sources. These systems, comprising both main and harmonic rf cavities, relax statistical collective effects but also introduce instabilities, such as Robinson and periodic transient beam loading (PTBL) instabilities. In this paper, we provide semi-analytical algorithms designed to predict and analyze these instabilities with improved accuracy and robustness. The algorithms leverage recent advancements in the field, offering a computationally efficient and accurate complement to multibunch tracking simulations. Using the SOLEIL II project as a case study, we demonstrate how these algorithms can optimize rf cavity parameters in high-dimensional parameter spaces, thereby maximizing the Touschek lifetime. An open-source Python package, ALBuMS (Algorithms for Longitudinal Multibunch Beam Stability), is provided as an accessible tool for double rf system stability analysis.

Figures (12)

• Figure 1: Bosch algorithm (left) and modified algorithm (right) for the study of longitudinal beam instabilities in double rf systems with passive harmonic cavities.
• Figure 2: Dipole and quadrupole mode frequencies as a function of the tuning angle $\psi_2$ in SOLEIL II storage ring at 200m A beam current. Passive parameters: $R_{s_2}/Q_{0_2}=90Ω$, $Q_{0_2} =36.0e3$.
• Figure 3: Instabilities predicted for the SOLEIL II storage ring, with the "Alves" solver (left) and with tracking (right). A is included with $f_\text{HOM} = 1.7G Hz$, $Q_{0_\text{HOM}}=670$ and $R_{s_\text{HOM}}=8.8k Ω$. Passive parameters: $R_{s_2}/Q_{0_2}=60Ω$, $Q_{0_2} = 31.0e3$.
• Figure 4: Tracking results (first four top plots) for the SOLEIL II lattice at 500m A beam current with and without . Dots and triangles show the mean value over the last 1000.0 turns while error bars show the standard deviation of the value over the last 1000.0 turns. Instability prediction using the "Alves" solver (bottom). Passive parameters: $R_{s_2}/Q_{0_2}=$ 60Ω, $Q_{0_2} =$ 31.0e3.
• Figure 5: Coupled bunch mode $\ell$ amplitude computed from tracking results for the SOLEIL II lattice at 500m A beam current with and without . Passive parameters: $R_{s_2}/Q_{0_2}=60Ω$, $Q_{0_2} =31.0e3$.