Influence of spin-orbit coupling on the magnetic dipole term $T_{\alpha}$

The influence of the spin-orbit coupling (SOC) on the magnetic dipole term $T_{\alpha}$ is studied across a range of systems in order to check whether the $T_{\alpha}$ 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 $3\cos^{2}\theta-1$. It follows that there are circumstances when the influence of the SOC on $T_{\alpha}$ 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 $T_{\alpha}$ 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.