  We characterize the Europium (Eu$^{3+}$) hyperfine interaction of the excited
state ($^5$D$_0$) and determine its effective spin Hamiltonian parameters for
the Zeeman and quadrupole tensors. An optical free induction decay method is
used to measure all hyperfine splittings under weak external magnetic field (up
to 10 mT) for various field orientations. On the basis of the determined
Hamiltonian we discuss the possibility to predict optical transition
probabilities between hyperfine levels for the $^7$F$_{0} \longleftrightarrow
^5$D$_{0}$ transition. The obtained results provide necessary information to
realize an optical quantum memory scheme which utilizes long spin coherence
properties of $^{151}$Eu$^{3+}$:Y$_2$SiO$_5$ material under external magnetic
fields
