In a process for the irradiation of strand-shaped irradiated material (19; 19a-e), in particular cable insulation or sheathing (31) capable of being cross-linked by irradiation, or tubes, hoses, or profile elements capable of being cross-linked by irradiation, with electron beams impinging transversely to the longitudinal axis of the irradiated material (19; 19a-e), which strike the irradiated material (19; 19a-e) from two fixed irradiation directions (R1, R2), located at an angle to one another which is other than zero and for preference is a right angle, a uniform irradiation is achieved in a simple manner in that a scanned electron beam (17) is created from an electron beam (11) in a scanner (12) by means of a scan device (13), which by means of a temporally-actuated back-and-forth slewing movement in a prescribed angle range creates a radiation fan transversely to the longitudinal direction of the irradiated material, and that the scanned electron beam (17) is deflected by a deflection magnet (16, 16′) arranged between the scan device (13) and the irradiated material (19; 19a-e) in such a way for each scan angle of the irradiation field that it impinges on the irradiation material (19; 19a-e) to be treated from one of the two fixed irradiation directions (R1, R2).