= Solution
An $f$ shell has orbital quantum number $l=3$ and seven orbital states with $m_l=-3,-2,\ldots,3$. The first of <Hund's rules> maximizes total <spin> subject to the <Pauli exclusion principle>. Seven <electrons> occupy the seven orbitals singly with parallel spins, giving
$$
\boxed{S=7\times\frac12=\frac72.}
$$
Their orbital projections sum to zero, so the half-filled shell also has total orbital <angular momentum> $L=0$, total $J=7/2$, and <Landé g-factor> $g_J=2$. These further values identify the magnetic moment used below.
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