Simple Models for Beam Loss Near the Half Integer Resonance with Space Charge

The half integer resonance is often used to define the high intensity limit of medium or low energy hadron rings where transverse space charge is significant. However, the mechanism leading to particle loss as beam approaches this resonance, which thus defines the limit, is not clearly understood. In this paper we explore simple models, based on single particle resonance ideas, to see if they describe useful aspects of motion as observed in simulations and experiments of 2D coasting beams on the ISIS synchrotron. Single particle behaviour is compared to 2D self-consistent models to assess when coherent motion begins to affect the single particle motion, and understand the relevance of coherent and incoherent resonance. Whilst the general problem of 2D resonant loss, with non-stationary distributions and non-linear fields is potentially extremely complicated, here we suggest that for a well-designed machine, where higher order pathological loss effects are avoided, a relatively simple model may give valuable insights into beam behaviour and control.