$3d$ spin-orbit photoemission spectrum of nonferromagnetic materials: The test cases of CoO and Cu

The x-ray photoemission spectrum of the valence states of $3d$ transition-metal systems is spin polarized when using circularly polarized photons. The integral of the spin-orbit spectrum is proportional to the expectation value of the angular part of the $3d$ spin-orbit operator in the initial state. We show that this quantity can be used to get an estimate of the atomic orbital moment. While the measurement is sensitive to the magnetization axis, it does not require a net macroscopic magnetization nor the presence of a long-range magnetic order, and is therefore suitable for any transition-metal systems being antiferromagnetic or paramagnetic or magnetically disordered. In the case of full $3d$ shell the integral of the spin-orbit spectrum is zero, but the spectral shape can give a direct estimate of the $3d$ spin-orbit energy splitting $\Delta E_{\mathrm{SO}}.$ We have used Cu and CoO to experimentally test this technique. As expected Cu provides a vanishing result for $〈L_z〉,$ whereas for ${\mathrm{Co}}^{2+}$ in CoO we find $〈L_z〉=1.36\mathrm{}ħ$ at 0 K. On the other hand we find $\Delta E_{\mathrm{SO}}\simeq 280\hspace{0.5em}\mathrm{meV}$ for Cu.