We investigate the origin of the abundance ratios and scatter of $alpha$- and neutron-capture elements of old, metal-poor stars, using cosmological, hydrodynamical simulations of galaxy formation. For this, we implement a novel treatment for the production and distribution of chemical products of Type II supernovae, which considers the effects of the rotation of massive stars on the chemical yields and the effects of the different life-times of stars that are progenitors of this type of supernovae. We focus on the stellar halo of a Milky Way-mass galaxy, studying the abundances and scatter of [O/Fe], [Mg/Fe], [Si/Fe], [Sr/Fe], [Eu/Fe] and [Ba/Fe]. Our model is able, for the first time in a cosmological simulation, to describe at the same time the low scatter in the abundances of $alpha$-elements and the higher scatter associated to neutron-capture elements in the halo stars, as suggested by observations of the Milky Way. We also reproduce the scatter observed in the [Sr/Ba] ratio, which results from the treatment of the fast-rotating stars and the dependence of the chemical yields on the metallicity, mass and rotational velocities. Our simulations show that such scatter patterns appear naturally if the different ejection times associated to stars of different mass are properly described, without the need to invoke for additional mixing mechanisms or a distinct treatment of the alpha- and neutron-capture elements. Simulations of this type will help characterizing and identifying the past accretion debris as well as the pristine in-situ populations in the Galaxy unveiled by Gaia and spectroscopic data.
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