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We present a high resolution transmission electron microscopy study of a Cr-coated W tip apex prepared for spin-polarized scanning tunneling microscopy (SP-STM). The characterization of the tip apex structure has been done with atomic resolution. We show that the Cr film is epitaxially grown on W and presents a monocrystalline phase. The surface analysis of the apex reveals roughness which gives rise to structures that can be considered as nanotips. In spite of the monocrystalline structure of these nanotips, we show that their spin arrangement and resulting magnetization direction cannot be controlled. SP-STM measurements on a Cr/MgO(001) sample confirm this conclusion.