Fractional filling induced Landau damping of longitudinal instabilities at the ESRF

Longitudinal coupled bunch instabilities are a major obstacle for the increase of beam current in modern electron storage rings. At the ESRF, threshold limits for multi-bunch operation have been considerably increased by using fractional fillings, from about 60 mA for a homogeneous filling to well beyond the nominal intensity of 200 mA for a filling of one third of the circumference. The gap in the bunch train induces a modulation of the cavity voltage and a subsequent spread in synchrotron frequencies. This results in additional Landau damping. An appropriate set of coupled equations, which completely models the problem, has been derived. With slight simplifications one obtains analytical formulae which still accurately describe the observed effect. The theoretical results have been soundly confirmed by experiments carried out at the ESRF