The galaxy formation process in the $Lambda$-Cold Dark Matter scenario can be constrained from the analysis of stars in the Milky Ways halo system. We examine the variation of chemical abundances in distant halo stars observed by the Apache Point Galactic Evolution Experiment (APOGEE), as a function of distance from the Galactic center ($r$) and iron abundance ([M/H]), in the range 5 $lesssim r lesssim$ 30 kpc and $-2.5 <$ [M/H] $<$ 0.0. We perform a statistical analysis of the abundance ratios derived by the APOGEE pipeline (ASPCAP) and distances calculated by several approaches. Our analysis reveals signatures of a different chemical enrichment between the inner and outer regions of the halo, with a transition at about 15 kpc. The derived metallicity distribution function exhibits two peaks, at [M/H] $sim -1.5$ and $sim -2.1$, consistent with previously reported halo metallicity distributions. We obtain a difference of $sim 0.1$ dex for $alpha$-element-to-iron ratios for stars at $r > 15$ kpc and [M/H] $> -1.1$ (larger in the case of O, Mg and S) with respect to the nearest halo stars. This result confirms previous claims for low-$alpha$ stars found at larger distances. Chemical differences in elements with other nucleosynthetic origins (Ni, K, Na, and Al) are also detected. C and N do not provide reliable information about the interstellar medium from which stars formed because our sample comprises RGB and AGB stars and can experience mixing of material to their surfaces.
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