Hyperfine Structure and Exchange Coupling of Vacancy-Induced Ce$^{3+}$ Spin Centers in Nuclear-Spin-Dilute CeO$_2$

Oxygen vacancies in ceria CeO$_2$ donate electrons that localize as Ce$^{3+}$ ($4f^1$, $S=1/2$) small polarons, creating rare-earth spin centers through native defect chemistry rather than implantation or extrinsic doping. We investigate the magnetic environment of these centers using first-principles PBE$+U$ calculations with a linear-response Hubbard parameter ($U=5.8382$ eV), hyperfine tensors from the all-electron reconstruction of the projector-augmented-wave method, and Korringa-Kohn-Rostoker exchange calculations within the coherent-potential approximation. Four vacancy configurations spanning concentrations from $3.125\%$ to $12.5\%$ are considered. A distinctive feature of the host follows from cerium isotopics: all naturally occurring cerium isotopes possess nuclear spin $I=0$, eliminating on-site hyperfine interactions at the Ce$^{3+}$ center and leaving the nuclear-spin bath entirely on the oxygen sublattice, whose sole magnetic isotope, $^{17}O$ ($I=5/2$), occurs at $0.038\%$ natural abundance. The resulting $^{17}O$ hyperfine landscape consists of a small number of strongly coupled, nearly axial first-shell nuclei with contact couplings reaching $6$~MHz, surrounded by a weakly coupled and strongly anisotropic outer shell. These tensors define experimentally accessible signatures for $^{17}O$ ESEEM and HYSCORE measurements and provide the microscopic hyperfine parameters required for cluster-correlation-expansion calculations of spin coherence. Exchange interactions between neighboring polarons are weak and oxygen-mediated, leaving the vacancy-generated spins largely independent over the concentration range considered. Together, these results establish oxygen-deficient CeO$_2$ as a chemically generated and intrinsically nuclear-spin-dilute host for rare-earth spin centers, and provide the first-principles magnetic parameters needed to assess their coherence properties.