  Synchrotron radiation of relativistic electrons in storage rings naturally
leads to the process of damping of betatron oscillations. Damping time and
transverse beam emittance can be reduced by wigglers or undulators while the
beam parameters are still well defined by the common radiation integrals, based
on the properties of synchrotron radiation. However, the quantum excitation of
betatron oscillations in principle can be considerably reduced if an electron
radiation occurs due to the Thomson scattering in the periodic electromagnetic
field. After a brief introduction we compare radiation properties for different
cases and suggest the modification of the radiation integrals.
