Abstract
We present a theory that can compute the transverse coupled-bunch instability growth rates at any chromaticity and for any longitudinal potential provided only that the long-range wakefield varies slowly over the bunch. The theory is expressed in terms of the usual coupled-bunch eigenvalues at zero chromaticity, and when the longitudinal motion is simple harmonic our solution only requires numerical root-finding that is easy to implement and fast to solve; the more general case requires some additional calculations, but is still relatively fast. The theory predicts that the coupled-bunch growth rates can be significantly reduced when the chromatic betatron tune spread is larger than the coupled-bunch growth rate at zero chromaticity. Our theoretical results are compared favorably with tracking simulations for the long-range resistive wall instability, and we also indicate how damping and diffusion from sychrotron emission can further reduce or even stabilize the dynamics.
Highlights
Coupled-bunch transverse instabilities occur when longrange forces between particle bunches resonantly drive betatron oscillations
The resulting coupledbunch motion can lead to emittance growth and even particle loss, so that predicting the growth rates is an important part of storage ring design
We show that the theory predicts two distinct regimes depending upon whether the zero-chromaticity growth rate is smaller or larger than the chromatic tune spread over the bunch; in the former “weak” limit the instability is strongly suppressed by the chromaticity, while in the latter “strong” instability regime this no longer holds
Summary
Coupled-bunch transverse instabilities occur when longrange forces between particle bunches resonantly drive betatron oscillations. Laclare’s approach [9], in which the distribution function is only azimuthally expanded into synchrotron modes and the problem reduces to an eigenvalue equation when mode coupling is neglected, provides better results at large chromaticity [10] These methods have a few deficiencies: they can become complicated and somewhat opaque when the instability growth rate becomes comparable to the synchrotron frequency, they typically neglect the damping and diffusion due to synchrotron emission, and they do not directly apply when the longitudinal motion is not simple harmonic.
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