A simple particle-core model for circular accelerators has been constructed assuming that dispersion effects are relatively weak. This model is applicable to a large class of high-intensity rings designed for modest density applications such as spallation neutron sources. Applying this model to isotropic beams in a smooth ring, halo formation processes in the presence of dispersion are investigated. In the analysis, it is found that dispersion matching is essential to suppress horizontal beam widening in injection if momentum spread is larger than a certain threshold. Even if the beam widening due to dispersion mismatch is suppressed, a halo can be formed in the same mechanism as in a linac, namely, parametric resonance between oscillating core and single particles. The width of halos formed by the particle-core resonance in modest-density rings is found to be a little narrower than that in a straight channel, and typically 1.6 to 1.8 times the maximum core width.
