Analysis of intense beam instability in a general quadrupole focusing channel with image charge effect

The stability properties of transverse envelopes of mismatched intense continuous charge particle beam propagating in a general quadrupole focusing channel have been investigated in the presence of image charge effect due to a cylindrical conducting pipe. Phase shifts and growth factors of the envelope oscillations in the case of instability are calculated by numerical evaluation of the eigenvalues of linearly perturbed envelope equations for small deviations from the matched beam conditions. A detailed study on the region of instability and its dependence on the system parameters like occupancy of the quadrupole focusing field, syncopation factor, zero current phase advance, beam intensity etc. have been carried out. It has been found that the strength and regions of envelope instability due to the lattice and confluent resonances in the parametric space are affected by the presence of image charge.