A scenario for the formation of the bi-modality in the chemical space [$alpha$/Fe] vs [Fe/H] of the Milky Way was recently proposed in which $alpha$-enhanced stars are produced early and quickly in clumps. Besides accelerating the enrichment of the medium with $alpha$-elements, these clumps scatter the old stars, converting in-plane to vertical motion, forming a geometric thick disc. In this paper, by means of a detailed analysis of the data from smooth particle hydrodynamical simulations, we investigate the geometric properties (in particular of the chemical thick disc) produced in this scenario. For mono-age populations we show that the surface radial density profiles of high-[$alpha$/Fe] stars are well described by single exponentials, while that of low-[$alpha$/Fe] stars require broken exponentials. This break is sharp for young populations and broadens for older ones. The position of the break does not depend significantly on age. The vertical density profiles of mono-age populations are characterized by single exponentials, which flare significantly for low-[$alpha$/Fe] stars but only weakly (or not at all) for high-[$alpha$/Fe] stars. For low-[$alpha$/Fe] stars, the flaring level decreases with age, while for high-[$alpha$/Fe] stars it weakly increases with age (although with large uncertainties). All these properties are in agreement with observational results recently reported for the Milky Way, making this a plausible scenario for the formation of the Galactic thick disc.
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