  A new class of self-consistent 6-D phase space stationary distributions is
constructed both analytically and numerically. The beam is then mismatched
longitudinally and/or transversely, and we explore the beam stability and halo
formation for the case of 3-D axisymmetric beam bunches using particle-in-cell
simulations. We concentrate on beams with bunch length-to-width ratios varying
from 1 to 5, which covers the typical range of the APT linac parameters. We
find that the longitudinal halo forms first for comparable longitudinal and
transverse mismatches. An interesting coupling phenomenon - a longitudinal or
transverse halo is observed even for very small mismatches if the mismatch in
the other plane is large - is discovered.
