A cooled radiation emission device has an enclosure in which X-rays are produced. In the enclosure, there is a cathode, an anode situated facing the cathode and rotating on a shaft, and a fixed anode shaft support. The support includes a holding chamber, the shaft of the anode being held in the chamber. The cooling of the tube uses a gallium-indium-tin liquid alloy flow through the anode shaft. This alloy is a conductor of heat and electricity. At the same time as the lubrication of the bearings and the electrical powering of the anode, it provides for cooling of the anode.