When tungsten scanning tunneling microscopy (STM) tips are sharpened by self-sputtering with Ne+ ions, the Schiller-decapitation process can occur. We show that this process is accompanied by a very specific pattern in the sputter voltage as a function of time. This makes it possible to use the decapitation process for the manufacture of high quality STM tips. Our tips routinely produce atomic resolution in the [-110] and [001] directions on the {110} surfaces of GaAs and in. The occurrence of the decapitation process is recognized by the observed pattern in the sputter voltage alone, without the presence of a phosphor screen. Secondly, we present a model which enables us to calculate the radius of curvature of the tip apex based on Fowler-Nordheim plots. Verification with scanning electron microscopy shows that our model is correct within 5 nm. A slight deviation of the experimental data from the theory suggests that our polycrystalline tungsten wire possesses a preferential crystal orientation.