A free-ion-model crystal-field calculation has been performed for YAl${\mathrm{O}}_3$:${\mathrm{Er}}^{3+}$. A free-ion calculation was done by fitting a free-ion Hamiltonian, which included Coulomb, spin-orbit, configuration, spin-spin, and spin-other-orbit interactions. The coefficients which gave the best least-squares deviation of the calculated centers of gravity from the observed centers of gravity were $E^1=6174.1\mathrm{}$, $E^2=32.836\mathrm{}$, $E^3=651.69\mathrm{}$, $\xi =2350.5\mathrm{}$, $\alpha =27.825\mathrm{}$, $\beta =-722.32\mathrm{}$, $\gamma =3729.4\mathrm{}$, $M^0=3.525\mathrm{}$, $M^2=\left[1.974\right]\mathrm{}$, $M^4=\left[1.339\right]\mathrm{}$, $P^2=2.4779\mathrm{}$, $P^4=-0.0406\mathrm{}$, $P^6=0.1247\mathrm{}$, $T^2=157.04\mathrm{}$, $T^3=42.522\mathrm{}$, $T^4=62.23\mathrm{}$, $T^6=-259.93\mathrm{}$, $T^7=402.94\mathrm{}$, $T^8=66.377\mathrm{}$ with an rms deviation of 29.1; where all quantities have units of ${\mathrm{cm}}^{-1\mathrm{}}$. The intermediate-coupled free-ion vectors were then used to fit a $C_s$ ($C_{1h}$) Hamiltonian to the observed Stark splittings. The crystal-field parameters obtained from a previous calculation on YAl${\mathrm{O}}_3$:${\mathrm{Tm}}^{3+}$ were used as starting parameters. The parameters which gave the best fit to the observed YAl${\mathrm{O}}_3$:${\mathrm{Er}}^{3+}$ Stark spectra were $B_0^2=-183.2\mathrm{}$, $\mathrm{Re}{B}_2^2=385.6\mathrm{}$, $\mathrm{Im}{B}_2^2=185.2\mathrm{}$, $B_0^4=-816.2\mathrm{}$, $\mathrm{Re}{B}_2^4=444.0\mathrm{}$, $\mathrm{Im}{B}_2^4=39.7\mathrm{}$, $\mathrm{Re}{B}_4^4=637.0\mathrm{}$, $\mathrm{Im}{B}_4^4=-222.8\mathrm{}$, $B_0^6=-603.8\mathrm{}$, $\mathrm{Re}{B}_2^6=-42.2\mathrm{}$, $\mathrm{Im}{B}_2^6=305.4\mathrm{}$, $\mathrm{Re}{B}_4^6=229.7\mathrm{}$, $\mathrm{Im}{B}_4^6=523.8\mathrm{}$, $\mathrm{Re}{B}_6^6=317.0\mathrm{}$, $\mathrm{Im}{B}_6^6=87.0\mathrm{}$ with an rms deviation of 6.9; where all units are in ${\mathrm{cm}}^{-1\mathrm{}}$.

• Received 24 June 1976

DOI:https://doi.org/10.1103/PhysRevB.15.10