Beam halo definitions based upon moments of the particle distribution

Two different parameters for the quantitative description of beam halo are discussed. Both are based on moments of the particle distribution and represent a convenient and model-independent method for quantifying the magnitude of beam halo observed in either spatial or phase-space projections. One parameter is a measure of spatial profile of the beam and has been defined by Wangler and Crandall previously. The current authors defined a new parameter using kinematic invariants to quantify halo formation in 2D phase space. Here we expand the development and present detailed numerical results. Although the spatial-profile parameter and the phase-space halo parameter both reduce to the same value when the distribution has the elliptical symmetry, in general these parameters are not equal. Halo in the 1D spatial profiles is relatively easily measured, but is variable as the beam distribution evolves and can hide as it rotates in phase space. The 2D phase-space halo is more difficult to measure, but it varies more smoothly as the halo evolves. It provides a more reliable characterization of the halo as an intrinsic property of the beam.