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. Since the betatron is technically simple, inexpensive and robust, it is economically and technically superior to prior art synchrotron emitters and internal radiators inside storage and beam stretcher rings.