Grazing function g and collimation angular acceptance

The grazing function g is introduced—a synchrobetatron optical quantity that is analogous (and closely connected) to the Twiss and dispersion functions β, α, η, and η′. It parametrizes the rate of change of total angle with respect to synchrotron amplitude for grazing particles, which just touch the surface of an aperture when their synchrotron and betatron oscillations are simultaneously (in time) at their extreme displacements. The grazing function can be important at collimators with limited acceptance angles. For example, it is important in both modes of crystal collimation operation—in channeling and in volume reflection. The grazing function is independent of the collimator type—crystal or amorphous—but can depend strongly on its azimuthal location. The rigorous synchrobetatron condition g=0 is solved, by invoking the close connection between the grazing function and the slope of the normalized dispersion. Propagation of the grazing function is described, through drifts, dipoles, and quadrupoles. Analytic expressions are developed for g in perfectly matched periodic FODO cells, and in the presence of β or η error waves. These analytic approximations are shown to be, in general, in good agreement with realistic numerical examples. The grazing function is shown to scale linearly with FODO cell bend angle, but to be independent of FODO cell length. The ideal value is g=0 at the collimator, but finite nonzero values are acceptable. Practically achievable grazing functions are described and evaluated, for both amorphous and crystal primary collimators, at RHIC, the SPS (UA9), the Tevatron (T-980), and the LHC.