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It has been known since decades that the alignment of a beam of high-energy electrons with particular crystal directions involves a significant increase of bremsstrahlung radiation emission. This enhancement lies at the conceptual foundation of innovative positron source schemes for future lepton colliders. In particular, the so-called hybrid scheme makes use of a heavy-metal radiator in crystalline form, which is then followed by an amorphous metallic converter for positron generation from electrons by means of a two-step electromagnetic process. This work presents the most recent simulation results obtained on the development of a hybrid positron source for the FCC-$ee$ from the standpoint of the features of both the crystalline radiator and the amorphous converter.