  Rare-earth ion doped crystals are promising systems for quantum communication
and quantum information processing. In particular, paramagnetic rare-earth
centres can be utilized to realize quantum coherent interfaces simultaneously
for optical and microwave photons. In this article, we study hyperfine and
magnetic properties of a Y$_2$SiO$_5$ crystal doped with $^{171}$Yb$^{3+}$
ions. This isotope is particularly interesting since it is the only rare--earth
ion having electronic spin $S=\frac{1}{2}$ and nuclear spin $I=\frac{1}{2}$,
which results in the simplest possible hyperfine level structure. In this work
we determine the hyperfine tensors for the ground and excited states on the
optical $^2$F$_{7/2}(0) \longleftrightarrow ^2$F$_{5/2}$(0) transition by
combining spectral holeburning and optically detected magnetic resonance
techniques. The resulting spin Hamiltonians correctly predict the
magnetic-field dependence of all observed optical-hyperfine transitions, from
zero applied field to the high-field regime where the Zeeman interaction is
dominating. Using the optical absorption spectrum we can also determine the
order of the hyperfine levels in both states. These results pave the way for
realizing solid-state optical and microwave quantum memories based on a
$^{171}$Yb$^{3+}$:Y$_2$SiO$_5$ crystal.
