Halo formation in three-dimensional bunches

We have constructed, analytically and numerically, a class of self-consistent six-dimensional (6D) phase space stationary distributions. Stationary distributions allow us to study the halo development mechanism without it being obscured by beam redistribution and its effect on halo formation. The beam is then mismatched longitudinally and/or transversely, and we explore the formation of longitudinal and transverse halos in 3D axisymmetric beam bunches. We find that the longitudinal halo forms first for comparable longitudinal and transverse mismatches because the longitudinal tune depression is more severe than the transverse one for elongated bunches. Of particular importance is the result that, due to the coupling between longitudinal and transverse motion, a longitudinal or transverse halo is observed for a mismatch less than 10% if the mismatch in the other plane is large. copyright 1998 The American Physical Society