Abstract
Transverse beam stability is strongly affected by the beam space charge. Usually it is analyzed with the rigid-beam model. However this model is only valid when a bare (not affected by the space charge) tune spread is small compared to the space charge tune shift. This condition specifies a relatively small area of parameters which, however, is the most interesting for practical applications. The Landau damping rate and the beam Schottky spectra are computed assuming that validity condition is satisfied. The results are applied to a round Gaussian beam. The stability thresholds are described by simple fits for the cases of chromatic and octupole tune spreads.
Highlights
Particle interaction via the walls of the vacuum chamber is conventionally described by the wake functions and impedances
If the phase space density of the resonant particles is sufficiently large, the instability is stabilized by this mechanism, called the Landau damping
The rigid-beam model is still a good approximation; but there is a small amount of the resonant particles in tails of the distribution, yielding small Landau damping
Summary
Particle interaction via the walls of the vacuum chamber is conventionally described by the wake functions and impedances. Perturbation of a particle motion depends on its amplitudes, this equation assumes that the beam oscillates as a rigid body when the coherent beam fields are computed. The beam coherent motion is completely described by the dipole offset x" This assumption is correct if all lattice frequencies iQi are identical. In this case, all particles respond identically to the coherent field, xi 1⁄4 x"; the beam oscillates as a rigid body, and the spread of the space charge tune shifts does not matter. Since individual responses are not identical, the beam shape is not preserved in the dipole oscillations, so the rigid-body model of Eq (1) is not self-consistent and generally cannot be justified. Were presented in several papers [4,5,6]; the range of their applicability was not clarified
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