Multi-Bunch Energy Spread Induced by Beam Loading in a Standing Wave Structure

The interaction of a relativistic beam with the models of the TM010 pass-band of a multicell cavity does not cause any problem: although all the models are excited bythe RF generator, resulting in different cell excitations during the cavity filling and the beam pulse, the net accelerating field exhibits negligible fluctuations from bunch to bunch. However, when the beam is not fully relativistic, this is no more true.The phase slippage occurring in the first cells, between the non relativistic beam and the lower pass-band models, produces an effective enhancement of the shunt impedances, which is usually negligible for a relativistic beam in a well tuned cavity. Moreover, the voltage jumps (amplitude and phase) occurring at each bunch passage, as well as the beam detuning caused by the off-crest bunches, vary from cell to cell. These effects enhance dramatically the fluctuation of the accelerating voltage, with a dominant beating provided by the pass-band mode nearest to the pimode. The induced beam energy spread has been estimated by the help of two distinct codes, developed at Frascati and Saclay, with results in good agreement. While an interaction integral is computed at each bunch passage, the cavity refilling is calculated by solving coupled differential equations of the 'modes' of the pass-band, driven by a generator linked to one end-cell. It is shown also that the intermode coupling arises from the external Q of the drive end-cell, and not from the wall losses. For illustration, we applied the method to the beam-loading problem in the SC capture cavity of the low charge injector of TESLA Test Facility installed at Desy.