Selection Rules for an Electronic Raman Effect in a Situation where the Appropriate Point Group is a Double Group

The low-temperature Raman spectrum of the compound ytterbium gallium garnet is reported. The ${}_{}{}^2{}_{}{}^{}F_{\frac{7}{2}}^{}{}_{}{}^{}$ ground state of the ytterbium ion in this compound gives rise to four crystal-field components, all of them Kramers pairs. Raman scattering to two of these levels situated at 554 and 539 ${\mathrm{cm}}^{-1\mathrm{}}$ above the ground state has been found. These levels belong to the ${\Gamma }_5$ species of the double group $D_2$, and in order to find the Raman selection rules, the transformation properties of both the symmetric and antisymmetric parts of the scattering tensor for the double group have to be considered. This is in contrast to the first-order Raman spectrum of the garnet host lattice, where all phonon modes have to be labeled in the factor group $O_h\mathrm{}\left(10\right)\mathrm{}$ and where the scattering tensor is symmetric.