Crystal electric field excitations and an effective-spin-$1/2$ ground doublet in the hyperkagome magnet Yb$_3$Sc$_2$Ga$_3$O$_{12}$
We use inelastic neutron scattering (INS) to determine the crystal electric field (CEF) excitations of Yb$^{3+}$ in the rare-earth hyperkagome magnet Yb$_3$Sc$_2$Ga$_3$O$_{12}$. Three nearly dispersionless magnetic excitations are observed near 58, 68, and 74~meV, corresponding to transitions from the ground-state Kramers doublet to the three excited doublets of the $J=7/2$ multiplet. A Stevens-operator analysis constrained by the local $222$ ($D_2$) symmetry reproduces the excitation energies and spectral weights and yields an Ising-type ground-state $g$ tensor. The first excited doublet lies approximately 58~meV above the ground state, establishing a well-isolated effective $J_{\mathrm{eff}}=1/2$ degree of freedom over the low-temperature regime relevant to collective magnetism. Notably, the directly measured CEF spectrum substantially revises the level scheme previously inferred from bulk measurements, while preserving the essential low-energy pseudospin description. Two independent fitting protocols give consistent excitation energies, ground-doublet wave functions, and $g$ tensors, despite the nonuniqueness of the individual CEF parameters. The resulting single-ion model also reproduces the characteristic susceptibility, magnetization, and field evolution of the Schottky anomaly in specific heat. These results establish the microscopic single-ion basis needed to construct an effective exchange Hamiltonian and to interpret future measurements of low-energy collective excitations in Yb$_3$Sc$_2$Ga$_3$O$_{12}$.