Orbital and spin moments in the ferromagnetic superconductor URhGe by x-ray magnetic circular dichroism

The ferromagnetic superconductor URhGe has been investigated by high field magnetic circular dichroism (XMCD) at the U ${M}_{4,5}$, Rh ${L}_{2,3}$, and Ge $K$ edges at 2.1 K and at applied fields up to 17 T. The XMCD performed at the ${M}_{4,5}$ absorption edges allows us to determine the spectroscopic branching ratio and the $5f$ electron contribution to the valence spin-orbit interaction. Combination with polarized neutron diffraction results allows us to derive the individual $\mathrm{U}$ orbital and spin moments and the magnetic-dipole contribution $\ensuremath{\langle}{T}_{z}\ensuremath{\rangle}$. There is no evidence for any change of the orbital-to-spin moment ratios across the spin reorientation transition at ${H}_{R}=12\phantom{\rule{0.16em}{0ex}}\mathrm{T}$, when the field is applied along the initial hard $b$ axis. We also confirm that the magnetism of URhGe is dominated by $\mathrm{U}$, with the contribution of Rh representing only about $10%$ of the macroscopic moment. The orbital and spin moments at the Rh site are found to be parallel to each other and parallel to the macroscopic magnetization, but an unexpectedly large orbital-to-spin moment ratio is observed. The XMCD at the Ge $K$ edge reveals the presence of a small induced Ge $4p$ orbital moment, parallel to the macroscopic magnetization. The results are discussed against predictions of the electronic band structure calculations by the density functional theory plus Coulomb $U$, including spin-orbit coupling $(\mathrm{DFT}+U+\mathrm{SOC})$.