Diffusion model for the time evolution of particle loss rates in collimator scans: a method for measuring stochastic transverse beam dynamics in circular accelerators

A diffusion model of the time evolution of loss rates caused by a step in collimator position is presented. It builds upon the model of Seidel (1994) and its assumptions: (1) constant diffusion rate within the range of the step and (2) linear halo tails. These hypotheses allow one to obtain analytical expressions for the solutions of the diffusion equation and for the corresponding loss rates vs. time. The present model addresses some of the limitiations of the previous model and expands it in the following ways: (a) losses before, during, and after the step are predicted; (b) different steady-state rates before and after are explained; (c) determination of the model parameters (diffusion coefficient, tail population, detector calibration, and background rate) is more robust and precise. These calculations are the basis for the measurement of transverse beam diffusion rates as a function of particle amplitude with collimator scans. The results of these measurements in the Tevatron will be presented in a separate report.