Equilibrium of longitudinal bunch distributions in storage rings with arbitrary impedance sources and generic filling patterns

A new self-consistent semi-analytical method for calculating the stationary beam-induced voltage in the presence of arbitrary filling patterns and impedance sources in storage rings is presented. The theory was developed in space-domain with resonator wake-functions and in frequency-domain with arbitrary impedance functions. The SIRIUS storage ring parameters were used to benchmark the results, demonstrating good agreement between the two approaches and with macroparticle tracking simulations. Additionally, a different approach to simulate the beam-loading compensation of active rf cavities was investigated in frequency-domain, proving to be a more generic description than the methods generally used. The impact of broadband impedance on the longitudinal equilibrium was straightforwardly evaluated with the frequency-domain framework, without intermediate steps such as fitting broadband resonators or convolving short-range wakes with bunch distributions. Finally, a simple study of Touschek lifetime improvement with a passive higher harmonic cavity is presented.