There is provided an exit window for an electron beam from a linear accelerator for use in producing radioisotopes. The exit window comprises a cylindrical channel operatively connectable at one end to a vacuum chamber configured for travel of the electron beam; and a domed dished head at the other end of the channel, the dished head comprising a convex portion having a protruding crown configured for pass-through of the electron beam wherein the geometry of the domed dished head is proportioned to resist pressure stress created by cooling medium circulating around the protruding crown and the vacuum in the cylindrical channel and to maintain the combined cooling medium pressure stress and pulsed electron beam thermal stress below the fatigue limit of the material forming the exit window.