Polarization analysis by means of individual soft x-ray absorption spectra of rare earths

X-ray magnetic circular dichroism (XMCD), which by virtue of the sum rules provides element-specific spin and orbital moments, is obtained from the difference between two polarized spectra by reversing the direction of either the light helicity or the applied magnetic field. Usually, it is tacitly assumed that these two spectra are obtained using the same absolute degree of light and magnetic polarization. This is, however, not always possible and depends on circumstances that can be beyond control. First, we recapitulate the conventional XMCD sum rule method to obtain the values of the moments and emphasize some of the complications in the case of the rare-earth ${M}_{4,5}$ edges, such as the presence of strong core-hole $jj$ overlap, linear dichroism, and magnetic dipole term $\ensuremath{\langle}{T}_{z}\ensuremath{\rangle}$. Instead, we propose an alternative method. Using the individual polarized x-ray absorption spectra obtained at the Ho and Dy ${M}_{5}$ edges, where each of the $\mathrm{\ensuremath{\Delta}}J=\ensuremath{-}1,0$, and $+1$ transitions are separated by $\ensuremath{\sim}2$ eV in photon energy, we are able to determine independently the degree of circular dichroism in a single spectrum. Since light is a transverse wave, we need to include, apart from the circular dichroism, also a linear dichroism contribution in order to fit the circularly polarized spectra. In the measurements on paramagnetic rare-earth dopants it was found that reversing the field produces the same degree of circular dichroism, while reversing the helicity yields a $\ensuremath{\sim}$20% difference in the degree of circular dichroism.