  In particle accelerators, particle losses to surrounding components must be
minimized. These particles usually have been driven into an undesired "halo"
outside the desired distribution before being lost. The processes involved and
physical identification of such halo is an intense research topic especially
when the nonlinear space charge effect is taken into account. In order to
present a method, firmly based on the physics of the transient beam state, for
determining when a particle may be defined as being in a halo, the particular
example of an initially rms matched beam injected into a resonance of an
alternating-gradient (AG) focusing lattice is studied, in which the resonance
between the collective mode and single particle motion pushes some particles
into a halo defined by extending the halo description to the action-angle
frame. In this new frame of description, the halo particles can be physically
defined as those with larger actions affected by the outer quasi hyperbolic
tori, which contains the halo particles observed in the traditional approaches
on projected 2D phase space, as well as some particles still hidden in the
core.
