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FORS2/VLT survey of Milky Way globular clusters II. Fe and Mg abundances of 51 Milky Way globular clusters on a homogeneous scale

(ABRIDGED) Globular clusters trace the formation and evolution of the Milky Way and surrounding galaxies, and outline their chemical enrichment history. To accomplish these tasks it is important to have large samples of clusters with homogeneous data and analysis to derive kinematics, chemical abundances, ages and locations. We obtain homogeneous metallicities and alpha-element enhancement for over 800 red giant stars in 51 Galactic bulge, disc, and halo globular clusters that are among the most distant and/or highly reddened in the Galaxy's globular cluster system. We observed R ~ 2000 spectra in the wavelength interval 456-586 nm and applied full spectrum fitting technique. We compared the mean abundances of all clusters with previous work and with field stars. We used the relation between mean metallicity and horizontal branch morphology defined by all clusters to select outliers for discussion. We find our metallicities are comparable to those derived from high-resolution data to within sigma = 0.08 dex over the interval -2.5 < [Fe/H] < 0.0. We also find that the distribution of [Mg/Fe] and [alpha/Fe] with [Fe/H] for the 51 clusters follows the general trend exhibited by field stars. It is the first time that the following clusters have been included in a large sample of homogeneous stellar spectroscopic observations and metallicity derivation: BH 176, Djorg 2, Pal 10, NGC 6426, Lynga 7, and Terzan 8. In particular, only photometric metallicities were available previously for the first three clusters, and the available metallicity for NGC 6426 was based on integrated spectroscopy and photometry. Two other clusters, HP 1 and NGC 6558, are confirmed as candidates for the oldest globular clusters in the Milky Way. The technique used here can also be applied to globular cluster systems in nearby galaxies with current instruments and to distant galaxies with the advent of ELTs.