Abstract
Transverse instability of a bunch in a ring accelerator is considered with space charge and wakefield taken into account. It is assumed that space charge tune shift significantly exceeds the synchrotron tune. Bunch spectrum, instability growth rate, and effects of chromaticity are studied with different bunch and wake forms. Fast instability caused by coupling of transverse modes is studied in detail. It is shown that, for monotonic wakes, the transverse mode coupling instability is possible only with a certain sign of the wake. Its threshold and growth rate are calculated precisely over a wide range of parameters.
Highlights
Transverse instability of a bunched beam in a ring accelerator has been considered first by Pellegrini [1] and Sands [2] with synchrotron oscillations taken into account (‘‘head-tail instability’’)
The role of space charge has been studied first in Ref. [4] with the assumption that corresponding tune shift is small in comparison with synchrotron tune
A more detailed investigation has shown that the damping vanishes at larger tune shift [6]
Summary
Transverse instability of a bunched beam in a ring accelerator has been considered first by Pellegrini [1] and Sands [2] with synchrotron oscillations taken into account (‘‘head-tail instability’’). The space charge effects at low synchrotron frequency were thoroughly investigated in recent papers [6,7]. The conclusion was that, at zero chromaticity, small wake cannot violate bunch stability, and all its eigentunes remain to be real numbers. Wake strength should overcome a certain threshold to excite transverse mode coupling instability through approach, and coalescence of the eigentunes (vanished TMCI, by the author). It is shown that some small parts of the basic equation were lost at limiting transition to zero frequency in Ref. One of the base assumptions is that nonlinearity of the external field is negligible in comparison with space charge contribution. Another important assumption is that synchrotron oscillations are linear. The effect of chromaticity on the instability growth rate is studied and treated in the paper
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