Correlation between superhyperfine constants and $4f$ crystal-field parameters of ${\mathrm{Tm}}^{2+}$ in Ca${\mathrm{F}}_2$, Sr${\mathrm{F}}_2$, and Ba${\mathrm{F}}_2$

The $4f$ crystal-field parameters of ${\mathrm{Tm}}^{2+}$ in Ca${\mathrm{F}}_2$, Sr${\mathrm{F}}_2$, and Ba${\mathrm{F}}_2$, after subtracting a calculated point-charge contribution, are shown to be linearly related to the scalar hyperfine constant of the first-shell fluorine ions. This provides direct experimental proof that admixture of the ligand wave functions with the metal-ion wave function plays a dominant role in determining the splitting of the $4f$ electronic levels in these highly ionic hosts. The electron-nuclear-double-resonance values for ${\mathrm{Tm}}^{2+}$ in Ba${\mathrm{F}}_2$ are reported here for the first time.