Energy Levels of ${\mathrm{Ho}}^{3+}$ Ions in Yttrium Gallium Garnet and Yttrium Iron Garnet, and Exchange between ${\mathrm{Ho}}^{3+}$ and ${\mathrm{Fe}}^{3+}$ Ions in Yttrium Iron Garnet

Energy levels of a ${\mathrm{Ho}}^{3+}$ ion in YGaG are firstly calculated within the framework of the crystal-field theory. Observed structures of the ground ${}_{}{}^5{}_{}{}^{}I_8^{}{}_{}{}^{}$ and the excited ${}_{}{}^5{}_{}{}^{}I_7^{}{}_{}{}^{}$ manifolds are fairly well reproduced by the three-parameter fit. Then the energy levels of a ${\mathrm{Ho}}^{3+}$ ion in YIG are calculated as functions of the direction of the net magnetization, taking into account the orbitally anisotropic exchange as well as the isotropic one. Comparison between the calculated and observed results on the variation of the energy levels reveals that the orbitally anisotropic exchange plays an important role in the Ho-Fe exchange interaction, and further that the Ho-Fe exchange interaction takes place primarily through the next-nearest-neighbor oxygen ions.