  Betatron oscillation is a commonly known phenomenon in laser or beam driven
plasma wakefield accelerators. In the conventional model, the plasma wake
provides a linear focusing force to a relativistic electron, and the electron
oscillates in one transverse direction with the betatron frequency proportional
to $1/\sqrt{\gamma}$, where $\gamma$ is the Lorentz factor of the electron. In
this work, we extend this model to three-dimensional by considering the
oscillation in two transverse and one longitudinal directions. The long-term
equations, with motion in the betatron time scale averaged out, are obtained
and compared with the original equations by numerical methods. In addition to
the longitudinal and transverse damping due to radiation reaction which has
been found before, we show phenomena including the longitudinal phase drift,
betatron phase shift and betatron polarization change based on our long-term
equations. This work can be highly valuable for future plasma based high-energy
accelerators and colliders.
