Abstract: An apparatus is provided for generating high-intensity x-rays for medical, industrial, and scientific purposes. A thin radiator is placed inside a betatron as an internal target. The radiators are thin enough and the energy of the electron beam is high enough such that the electrons pass through the radiator and return a plurality of times. The average current through the thin radiator is increased by the average number of times the electrons pass through the radiator. Thus, both the average x-ray power and the wall-plug efficiency of the apparatus are increased. In addition, for the betatron the required electron-beam energy is much smaller than that require for recirculation in storage ring and microwave-power cavity acceleration booster rings. This is because the path length around betatron toroid is much shorter; thus, the recycling process is less affected by scattering and energy loss.