This paper presents a study of the chemical compositions in cool gas around a sample of 27 intermediate-redshift galaxies. The sample comprises 13 massive quiescent galaxies at z=0.40-0.73 probed by QSO sightlines at projected distances d=3-400 kpc, and 14 star-forming galaxies at z=0.10-1.24 probed by QSO sightlines at d=8-163 kpc. The main goal of this study is to examine the radial profiles of the gas-phase Fe/{alpha} ratio in galaxy halos based on the observed Fe II to Mg II column density ratios. Because Mg+ and Fe+ share similar ionization potentials, the relative ionization correction is small in moderately ionized gas and the observed ionic abundance ratio N(Fe II)/N(Mg II) places a lower limit to the underlying (Fe/Mg) elemental abundance ratio. For quiescent galaxies, a median and dispersion of log <N(Fe II)/N(Mg II)> =-0.06+/-0.15 is found at d<~60 kpc, which declines to log <N(Fe II)/N(Mg II)> <-0.3 at d>~100 kpc. On the other hand, star-forming galaxies exhibit log <N(Fe II)/N(Mg II)> =-0.25+/-0.21 at d<~60 kpc and log <N(Fe II)/N(Mg II)> =-0.9+/-0.4 at larger distances. Including possible differential dust depletion or ionization correction would only increase the inferred (Fe/Mg) ratio. The observed N(FeII)/N(Mg II) implies super-solar Fe/{alpha} ratios in the inner halo of quiescent galaxies. An enhanced Fe abundance indicates a substantial contribution by Type Ia supernovae in the chemical enrichment, which is at least comparable to what is observed in the solar neighborhood or in intracluster media but differs from young star-forming regions. In the outer halos of quiescent galaxies and in halos around star-forming galaxies, however, the observed N(Fe II)/N(Mg II) is consistent with an {alpha}-element enhanced enrichment pattern, suggesting a core-collapse supernovae dominated enrichment history.
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