We explore the emergence of chiral magnetism in one-dimensional monatomic Mn, Fe, and Co chains deposited at the Pt(664) step-edge carrying out an ab-initio study based on density functional theory (DFT). The results are analyzed employing several models: (i) a micromagnetic model, which takes into account the Dzyaloshinskii-Moriya interaction (DMI) besides the spin stiffness and the magnetic anisotropy energy, and (ii) the Fert-Levy model of the DMI for diluted magnetic impurities in metals. Due to the step-edge geometry, the direction of the Dzyaloshinskii vector (D-vector) is not predetermined by symmetry and points in an off-symmetry direction. For the Mn chain we predict a long-period cycloidal spin-spiral ground state of unique rotational sense on top of an otherwise atomic-scale antiferromagnetic phase. The spins rotate in a plane that is tilted relative to the Pt surface by $62^circ$ towards the upper step of the surface. The Fe and Co chains show a ferromagnetic ground state since the DMI is too weak to overcome their respective magnetic anisotropy barriers. Beyond the discussion of the monatomic chains we provide general expressions relating ab-initio results to realistic model parameters that occur in a spin-lattice or in a micromagnetic model. We prove that a planar homogeneous spiral of classical spins with a given wave vector rotating in a plane whose normal is parallel to the D-vector is an exact stationary state solution of a spin-lattice model for a periodic solid that includes Heisenberg exchange and DMI. The validity of the Fert-Levy model for the evaluation of micromagnetic DMI parameters and for the analysis of ab-initio calculations is explored for chains. The results suggest that some care has to be taken when applying the model to infinite periodic one-dimensional systems.
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