Orientation of ground-state orbital in CeCoIn$_5$ and CeRhIn$_5$

We present core level non-resonant inelastic x-ray scattering (NIXS) data of the heavy fermion compounds CeCoIn$_5$ and CeRhIn$_5$ measured at the Ce $N_{4,5}$-edges. The higher than dipole transitions in NIXS allow determining the orientation of the $\Gamma_7$ crystal-field ground-state orbital within the unit cell. The crystal-field parameters of the Ce$M$In$_5$ compounds and related substitution phase diagrams have been investigated in great detail in the past; however, whether the ground-state wavefunction is the $\Gamma_7^+$ ($x^2\,-\,y^2$) or $\Gamma_7^-$ ($xy$ orientation) remained undetermined. We show that the $\Gamma_7^-$ doublet with lobes along the (110) direction forms the ground state in CeCoIn$_5$ and CeRhIn$_5$. For CeCoIn$_5$, however, we find also some contribution of the first excited state crystal-field state in the ground state due to the stronger hybridization of 4$f$ and conduction electrons, suggesting a smaller $\alpha^2$ value than originally anticipated from x-ray absorption. A comparison is made to the results of existing density functional theory plus dynamical mean-field theory calculations.