Single-beam collective phenomena: transverse I Coasting beams

If a charged particle beam is locally displaced or kicked (e.g. by noise), it will start to oscillate around an equilibrium orbit ('betatron oscillations') due to the external focusing fields of an accelerator or storage ring. These oscillations will excite electromagnetic fields which are modified by the presence of the vacuum chamber walls or other material boundaries in the neighbourhood of the beam. The electromagnetic fields react back on the oscillating beam. If there is an out of phase component of the forces, such as caused by the finite resistivity of the walls, the original oscillations may be reinforced. Their amplitude will then grow exponentially, i.e. the beam is unstable. Linear theory is only concerned with the conditions for the onset of the instability, i.e. the thresholds and initial growth rates, or with the means to stabilize the beam, e.g. by Landau damping or feedback.