Using grazing incidence X-ray diffraction (GIRD), we have determined the atomic relaxations near the surface in the self-organised N/Cu(100) and O/Cu(110) systems. These periodic relaxations give rise to diffraction satellites whose intensity can be quantitatively measured. The comparison between the experiments and molecular dynamics simulations allows us to determine the difference between the surface stress of the chemisorbed phase and the surface stress of the bare surface, which is at the origin of self-organisation. In the case of O/Cu(110), we have analysed the kinetics of self-organisation by spot profile analysis low energy electron diffraction (SPA-LEED) and compared the experiments to Monte Carlo simulations taking into account the elastic interactions in the system.