  A classical model of radiation reaction for the betatron oscillation of an
electron in a plasma wakefield accelerator is presented. The maximum energy of
the electron due to the longitudinal radiation reaction is found, and the
betatron oscillation damping due to both the longitudinal and transverse
radiation reaction effects is analyzed. Both theoretical and numerical
solutions are shown with good agreements. The regime that the quantum radiation
takes effect is also discussed. This model is important for designing future
plasma based super accelerators or colliders.
