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