Stability of anisotropic beams with space charge

Based on self-consistent Vlasov-Poisson equations, we derive coherent frequencies and stability properties of anisotropic or “nonequipartitioned” beams with different focusing constants and emittances in the two transverse directions. The thus obtained dispersion relations of transverse multipole oscillations with quadrupolar, sextupolar, and octupolar symmetries are solved numerically. We find that for sufficiently large energy anisotropy some of the eigenmodes become unstable in the space-charge-dominated regime. Applying our results to high-current linear accelerators, we find that “nonequipartitioned” beams may exist in relatively large regions of parameter space under stable conditions. It is only in beams with strongly space-charge-depressed betatron tunes that harmful instabilities leading to emittance exchange should be expected.