Space charge effects in bunches for different rf wave forms

The interplay of nonlinear rf fields and space charge is studied using a particle simulation code together with analytic derivations. In the framework of the elliptic (“Hofmann-Pedersen”) distribution function the matched beam parameters are obtained. Using the simulation “Schottky” noise from matched bunches the coherent mode spectrum is analyzed and compared with analytic expression. The bunch response to a small rf phase modulation is studied over a large range of initial simulation parameters (modulation frequency, bunch intensity). These bunch response scans clearly show the location of the dipole mode frequency as well as the threshold for the loss of Landau damping due to space charge. In addition, bunch stability scans are performed in order to determine the stability boundaries for flattopped bunches in single and double rf wave forms. The results are related to previous work on beam transfer functions in single and double rf buckets and to experimental observations in the GSI synchrotron SIS.