A semi-classical treatment of channeling radiation reaction

Radiation reaction including damping and quantum excitation has been studied extensively in synchrotrons and storage rings, where the effect is mainly due to the bending field. Recent development in advanced acceleration concepts requires very strong transverse focusing to maintain beam stability, and novel ideas such as channeling acceleration utilize the superstrong microscopic focusing field existing in a crystal channel. Here, a semi-classical formalism is used to calculate the radiation reaction of a relativistic particle in a straight, continuous focusing system. Due to the absence of quantum excitation in such a focusing system, the radiation damping rate of the transverse action obtained using this formalism agrees exactly with the result from the classical Lorentz-Dirac radiation reaction equation. In the limit where the pitch angle of the particle is much smaller than the radiation opening angle, the transverse action damps exponentially with an energy-independent rate that is much faster than the energy decay rate. In the opposite limit, both the transverse action and the energy damp with power laws and their relative rates are comparable. The general time-dependence of the transverse action damping and the energy decay are obtained analytically from these rate equations.