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The hyperfine structure of the ground state of erbium doped yttriumorthosilicate is analyzed with the use of electron paramagnetic resonanceexperiments in a tunable microwave resonator. This work was prompted by thedisagreement between the measurements made in zero magnetic field [Phys. Rev.B, 94, 075117, (2016)] and a previously published spin Hamiltonian, which isderived from standard EPR measurements at 9.5 GHz [Phys. Rev. B, 74, 214409,(2006)].  The ability to vary magnetic field strength, resonator frequency, and theorientation of our sample enabled us to monitor how the frequencies ofhyperfine transitions change as a function of a vector magnetic field. Comparedto [Phys. Rev. B, 74, 214409, (2006)], we arrived at a different set of spinHamiltonian parameters, which are also broadly consistent with their data. Wediscuss the reliability of our new spin Hamiltonian parameters to makepredictions outside the magnetic field and frequency regimes of our data. Wealso discuss why it proved to be difficult to determine spin Hamiltonianparameters for this material, and present data collection strategies thatimprove the model reliability.

Hyperfine interactions of $\text{Er}^{3+}$ ions in $\text{Y}_2\text{SiO}_{5}$: electron paramagnetic resonance in a tunable microwave cavity