Calculation of space-charge tune shift in a cylindrical chamber for bunched beams employing Green's function formalism

Shobuda, Yoshihiro   

When computing the space charge tune shift for a relativistic bunched beam within a cylindrical chamber, mirror currents for a coasting beam, initially introduced to replace the chamber wall, are employed. Subsequently, the obtained result is extended to encompass the bunched beam, taking into account the bunching factor which quantifies the distribution of bunches around the accelerator ring. In the process of derivation, the terms that characterize the bunch length are intuitively integrated into the formula. As a result, the validity of this approach has never been established. This study provides the derivation of the space charge tune shift formula for both relativistic and non-relativistic bunched beams right from the outset, employing the Green function formalism. Subsequently, it is compared with the earlier formula derived using mirror currents.