Abstract

A general formalism for treating simultaneously the transverse coupled-bunch and transverse coupled-mode instabilities is presented. In this approach, the equations of motion of a coupled multibunch beam are expanded to yield a system of equations involving correlation moments of the transverse and longitudinal motions. After a proper truncation, the system of equations is closed and can be solved. This approach allows us to formulate within one framework several known instability mechanisms including the single-bunch mode-coupling instability, the coupled-bunch instability, the mode-coupling instability, and the coupled-mode coupled-bunch instability as particular cases.

Highlights

  • EQUATION OF MOTIONA train of bunches is subject to various collective instabilities if the beam current is sufficiently high

  • The system of equations is closed and can be solved. This approach allows us to formulate within one framework several known instability mechanisms including the single-bunch mode-coupling instability, the coupled-bunch instability, the mode-coupling instability, and the coupled-mode coupled-bunch instability as particular cases

  • For a single-bunch beam, instabilities are traditionally analyzed by decomposing the collective motion of the particles in the bunch into collective modes; the instability results from the coupling among these modes

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Summary

EQUATION OF MOTION

A train of bunches is subject to various collective instabilities if the beam current is sufficiently high. CHAO is sufficient to consider only the lowest harmonics of the synchrotron frequency !s but taking into account correlation of the longitudinal and transverse motion in the same bunch. We consider transverse coherent effects assuming that the longitudinal motion is uncorrelated and can be described by Eq (2). In this way we neglect the longitudinal coherent effects the suggested formalism can be generalized to include this effect into account. We assume that there is no correlation of the longitudinal and transverse motion of particles belonging to different bunches, i.e. hziM zjN i 0, hziM AjN i 0 if N M, and neglect higher order correlation moments such as hziM z2jN i. If is small, we can neglect it in the argument of the impedance and obtain the approximate but explicit solution of the dispersion equation

TRANSVERSE DIPOLE COUPLED-BUNCH
HEAD-TAIL INSTABILITY
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