We present an analysis of Mg II $lambda2798$ and Fe II UV emission lines for archival Sloan Digital Sky Survey (SDSS) quasars to explore diagnostics of the magnesium-to-iron abundance ratio in a broad-line region cloud. Our sample consists of 17,432 quasars selected from the SDSS Data Release 7 with a redshift range of $0.72 < z < 1.63$. A strong anticorrelation between Mg II equivalent width (EW) and the Eddington ratio is found, while only a weak positive correlation is found between Fe II EW and the Eddington ratio. To investigate the origin of these differing behaviors of Mg II and Fe II emission lines, we have performed photoionization calculations using the Cloudy code, where constraints from recent reverberation mapping studies are considered. We find from calculations that (i) Mg II and Fe II emission lines are created at different regions in a photoionized cloud, and (ii) their EW correlations with the Eddington ratio can be explained by just changing the cloud gas density. These results indicate that the Mg II/Fe II flux ratio, which has been used as a first-order proxy for the Mg/Fe abundance ratio in chemical evolution studies with quasar emission lines, depends largely on the cloud gas density. By correcting this density dependence, we propose new diagnostics of the Mg/Fe abundance ratio for a broad line region cloud. Comparing the derived Mg/Fe abundance ratios with chemical evolution models, we suggest that $alpha$-enrichment by mass loss from metal-poor intermediate-mass stars occurred at $zsim2$ or earlier.
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