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Influence of spin-orbit coupling on the magnetic dipole term $T_α$

The influence of the spin-orbit coupling (SOC) on the magnetic dipole term $T_α$ is studied across a range of systems in order to check whether the $T_α$ term can be eliminated from analysis of x-ray magnetic circular dichroism spectra done via the spin moment sum rule. Fully relativistic Korringa-Kohn-Rostoker (KKR) Green function calculations for Co monolayers and adatoms on Cu, Pd, Ag, Pt, and Au (111) surfaces were performed to verify whether the sum over magnetic dipole terms $T_{x}+T_{y}+T_{z}$ is zero and whether the angular dependence of the \ta\ term goes as $3\cos^{2}θ-1$. It follows that there are circumstances when the influence of the SOC on $T_α$ cannot be neglected even for 3$d$ atoms, where the SOC is nominally small. The crucial factor appears to be the dimensionality of the system: for 3$d$ adatoms, the influence of SOC on $T_α$ can be significant while for monolayers it is always practically negligible. Apart from the dimensionality, hybridization between adatom and substrate states is also important: small hybridization enhances the importance of the SOC and vice versa.