Abstract
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.
Highlights
Longitudinal space charge effects play an important role in storage rings or synchrotrons for high current ion beams
The induced effects range from synchrotron tune shifts to coherent mode splitting [2] that can both be observed with high accuracy from the Schottky noise spectrum, as demonstrated in the GSI heavy ion cooler storage ring ESR [3]
Transition space charge reduces the effective rf voltage seen by the beam particles
Summary
Longitudinal space charge effects play an important role in storage rings or synchrotrons for high current ion beams. It is important to point out that the effect of rf phase or amplitude modulation on the single particle dynamics has been studied extensively in the literature [13] it was shown that for a modulated rf phase the response of the synchrotron motion in a single rf wave shows characteristics of chaotic particle dynamics in a parametric resonant system. The importance of these findings for the collective response of bunches affected by space charge is not clear. Afterward we briefly describe single and double rf waves and the resulting synchrotron motion including space charge (Sec. IV). The equation of motion in the z; v coordinates can be derived from the Hamiltonian
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