Electron-Nuclear Interactions as a Test of Crystal-Field Parameters for Low Symmetry Systems: Zeeman-Hyperfine Spectroscopy of Ho${}^{3+}$ Doped Y${}_2$SiO${}_5$

High-resolution Zeeman spectroscopy of electronic-nuclear hyperfine levels of ${}^5{\mathrm{I}}_8\to {}^5{\mathrm{I}}_7$ transitions in Ho${}^{3+}$:Y${}_2$SiO${}_5$ is reported. Crystal-field parameters determined for the two $C_1$ symmetry sites in Er${}^{3+}$:Y${}_2$SiO${}_5$ are successfully used to model the Zeeman-hyperfine data, including the prediction of avoided crossings between hyperfine levels under the influence of an external magnetic field. The two six- and seven-coordinate substitutional sites may be distinguished by comparing the spectra with crystal-field calculations.