Catalyst component comprising Mg, Ti, and halogen atoms, and is characterized in that (a) the Ti atoms are present in an amount higher than 4% based on the total weight of the said catalyst component, (b) the amount of Mg and Ti atoms is such that the Mg/Ti molar ratio is higher than 2 and (c) by a X-ray diffraction spectrum, in which, in the range of 2θ diffraction angles between 47° and 52°, at least two diffraction lines are present at diffraction angles 2θ of 48.3±0.2°, and 50.0±0.2°, the most intense diffraction lines being the one at 2θ of 50.0±0.2°, the intensity of the other diffraction line being equal to or lower than the intensity of the most intense diffraction line.