Calculation of Longitudinal Instability Threshold Currents for Single Bunches

Based on the publication by M. Venturini, et al. [1] a computer program has been developed that solves the Vlasov-Fokker-Planck equation numerically on a two dimensional grid. In this code different types of longitudinal interactions and their combinations are implemented. Calculations have been performed for the 1.7 GeV storage ring BESSY II and the 600 MeV ring MLS using the wake created by the shielded coherent synchrotron radiation (CSR). The results are compared with measurements on both rings which were based on the observation of the onset of time dependent fluctuations ("bursts") of CSR [2, 3]. Fair agreement is found between theory and experiment. The theoretical results complement calculations performed by Bane, et al. [4] for this type of interaction. The new results emphasize the resistive nature of the CSR-interaction, especially in regions where shielding effects are small or the resonance-like features of this wake are important. It is found that in these regions the instability is weak and thresholds depend on the damping time.