Effect of Head-Tail Mode Diffusion due to Synchrotron Radiation on Transverse Coupled-Bunch Instabilities with Harmonic-Cavity-Lengthened Bunches

Many present and future storage-ring-based light sources make use of cavities at a harmonic of the main RF to lengthen the electron bunches. This reduces Touschek and intrabeam scattering to improve the beam lifetime and transverse emittances respectively. These harmonic cavities are also beneficial in increasing the threshold currents of many collective instabilities in both longitudinal and transverse planes. The latter case has previously been explored in detail making extensive use of Laclare’s eigenvalue method. Such an approach does not take into account the diffusion of head-tail modes of nonzeroth order, which is a consequence of the quantum-mechanical nature of synchrotron radiation. In this paper, we show that Laclare’s eigenvalue method is a change of basis from another popular method of solving the Sacherer formalism, where Fokker-Planck diffusion terms can be readily included. For the case of the MAX IV 3 GeV ring, it is found that the effect is a large increase in the threshold currents of the resistive-wall coupled-bunch instability at positive chromaticity.