  We characterize the magnetic properties for thulium ion energy levels in the
Y$_3$Ga$_5$O$_{12}$ (Tm:YGG) lattice with the goal to improve decoherence and
reduce line-width broadening caused by local host spins and crystal
imperfections. More precisely, we measure hyperfine tensors for the lowest
level of the, $^3$H$_6$, and excited, $^3$H$_4$, states using a combination of
spectral hole burning, absorption spectroscopy, and optically detected nuclear
magnetic resonance. By rotating the sample through a series of angles with an
applied external magnetic field, we measure and analyze the orientation
dependence of the Tm$^{3+}$ ion's spin-Hamiltonian. Using this
spin-Hamiltonian, we propose a set of orientations to improve material
properties that are important for light-matter interaction and quantum
information applications. Our results yield several important external field
directions: some to extend optical coherence times, another to improve spin
inhomogeneous broadening, and yet another that maximizes mixing of the spin
states for specific sets of ions, which allows improving optical pumping and
creation of lambda systems in this material.
