  Rare-earth ion doped crystals are state-of-the-art platforms for processing
quantum information, particularly thanks to their excellent optical and spin
coherence properties at cryogenic temperatures. Experimental observations have
shown that the application of a static magnetic bias field significantly
improves the coherence times in the rare-earth ions ensemble, but only a few
studies have focused on its the dependency as a function of both magnetic field
direction and amplitude. This is especially true for magnetic field amplitudes
under the mT, and for low magnetic dipole moment ions. In this paper, we
investigate the relationship between the magnetic field parameters and the
decoherence caused by magnetic dipole-dipole coupling with the nearest
neighbors nuclear spins in the crystal. The primary non-Kramers rare-earth ions
investigated here are europium and praseodymium, but we also extend our study
to the ytterbium Kramers ion due to its low magnetic dipole in the mT range. We
perform theoretical investigations and simulations of the energy structure and
coherence time evolution and identify good correspondences between experimental
and simulated spin echo data. This work allows us to pinpoint the most relevant
decoherence mechanisms in the considered magnetic field regime, and to predict
favorable magnetic configurations.
