Throbbing Beam Instabilities in Particle Accelerators and Storage Rings

Finite conductivity of the vacuum chamber wall can cause unstable coherent transverse oscillations of the center of charge and also oscillations of the transverse cross section of a beam of charged particles. The former case, which can be characterized by a dipole oscillation, has been studied extensively by others. 1, 2 In this work, a study has been conducted of the cross sectional oscillations of a nearly circular beam centered in a circular pipe, and a self-consistent solution has been obtained for both monopole and quadrupole oscillations. Dispersion relations for the oscillation frequencies have been found, and conditions for stability have been deduced. For the quadrupole instability the growth rate and threshold are very close to those obtained for the dipole instability, 1, 2 differing only by a geometrical factor of order unity; whereas, for the monopole instability, the growth rates are so small that the oscillations in present electron accelerators and storage rings will be suppressed by radiation damping.