 Solution

Hund's rules maximizes total spin within the partially filled shell: parallel spins require an antisymmetric spatial wavefunction, reducing Coulomb overlap through exchange. Among terms with that spin, the second rule selects the largest total orbital angular momentum. The third rule, from spin-orbit coupling within the selected term, gives $J=|L-S|$ below half filling and $J=L+S$ above half filling, with the half-filled case treated directly.

For seven $4f$ electrons, put one parallel-spin electron in each $m_l=-3,-2,-1,0,1,2,3$ orbital. Their orbital projections sum to zero and form the half-filled $L=0$ shell; all seven spins add. ThusThere is no ambiguity from applying the below- or above-half-filled formula because $L=0$. The absence of a first-order orbital moment is important for the weak magnetic anisotropy discussed in c. These are isolated-ion term rules; a crystal field or strong covalency can alter other ions' orbital behavior, but the half-filled Eu${}^{2+}$ shell is unusually robust.