Effect of bunch length, chromaticity, and linear coupling on the transverse mode-coupling instability due to the electron cloud

The influence of bunch length, chromaticity, and linear coupling on the transverse mode-coupling intensity threshold is discussed for the case of a bunch interacting with a broad-band resonator impedance. Two regimes are possible according to whether the total bunch length is above or below a critical value, which is about the inverse of twice the resonance frequency. If the bunch length is greater than this value, the intensity threshold in the absence of linear coupling can be approximated by the coasting-beam value multiplied by the bunching factor. Hence, it is proportional to the bunch length, and increases linearly with the ratio between the chromatic and resonance frequencies. If the bunch length is smaller than the critical value, the intensity threshold is inversely proportional to the square of the bunch length, but it still increases slowly with chromaticity. In the presence of linear coupling, the intensity threshold can be increased up to a factor two when the second transverse plane has a negligible impedance. This formalism is then applied to the broad-band electron-cloud induced impedance, to evaluate the effect of bunch length, chromaticity and linear coupling on the intensity threshold of the CERN SPS beam for LHC.