  We present a detailed first principles study on the magnetic structure of an
Fe monolayer on different surfaces of 5d transition metals. We use the
spin-cluster expansion technique to obtain parameters of a spin model, and
predict the possible magnetic ground state of the studied systems by employing
the mean field approach and in certain cases by spin dynamics calculations. We
point out that the number of shells considered for the isotropic exchange
interactions plays a crucial role in the determination of the magnetic ground
state. In the case of Ta substrate we demonstrate that the out-of-plane
relaxation of the Fe monolayer causes a transition from ferromagnetic to
antiferromagnetic ground state. We examine the relative magnitude of nearest
neighbour Dzyaloshinskii-Moriya (D) and isotropic (J) exchange interactions in
order to get insight into the nature of magnetic pattern formations. For the
Fe/Os(0001) system we calculate a very large D/J ratio, correspondingly, a spin
spiral ground state. We find that, mainly through the leading isotropic
exchange and Dzyaloshinskii-Moriya interactions, the inward layer relaxation
substantially influences the magnetic ordering of the Fe monolayer. For the
Fe/Re(0001) system characterized by large antiferromagnetic interactions we
also determine the chirality of the $120^{\circ}$ N\'eel-type ground state.
