0720012

We propose to observe Her~X-1 during extsl{INTEGRALs} AO-7 for a total observing time of extbf{~600ksec}: covering one ``Main-On with three INTEGRAL orbits the first near the early maximum and the next two during the decay of the ``Main-On. Our prime objectives are to systematically measure the centroid extbf{energy of the fundamental cyclotron line} $E_{cyc}$ as a function of extbf{time X-ray flux 35day phase and 1.24s phase}. Such a dedicated systematic study will yield information about extbf{fundamental physics} in this accreting binary pulsar with respect to: the structure of the extbf{magnetic field} at the polar caps the physics of extbf{sub-Eddington accretion} the generation of extbf{pulse profiles} and variation of $E_{cyc}$ with pulse phase. [truncated! Please see actual data for full text]