Nonthermal Synchrotron Radiation from Gamma Ray Burst External Shocks and the X-ray Flares Observed with Swift

An analysis of the interaction between a spherical relativistic blast-wave shell and a stationary cloud with a spherical cap geometry is performed assuming that the cloud width 0, where \Delta_0 + \eta x/\Gamma_0^2 is the blast-wave shell width), and perhaps in the thin-shell approximation (\eta ~ 1/\Gamma_0), but not when \eta ~ 1. If this approximation is valid, then external shock processes could make the dominant prompt and afterglow emissions in GRB light curves, consistent with short delay two-step collapse models for GRBs