= Single-particle nuclear magnetic moment
{title2=$\mu=g_jj\mu_N$}
In the simplest <nuclear shell model> of an odd-<mass number> <nucleus>, paired <nucleons> contribute zero total <angular momentum> and <magnetic dipole moment>. The last unpaired <nucleon> supplies the nuclear <spin> $j$ and the stretched-state moment $\mu=g_jj\mu_N$, where $g_j$ is the <Landé g-factor> and $\mu_N$ the <nuclear magneton>.
Using <addition of angular momentum> with $s=1/2$ gives
$$
g_j=g_\ell+(g_s-g_\ell)\frac{j(j+1)-\ell(\ell+1)+3/4}{2j(j+1)}.
$$
For $j=\ell+1/2$, this reduces to $\mu/\mu_N=g_\ell\ell+g_s/2$. For $j=\ell-1/2$, it gives
$$
\frac\mu{\mu_N}=\frac{2\ell-1}{2\ell+1}\left[g_\ell(\ell+1)-\frac{g_s}{2}\right].
$$
The free-nucleon orbital factors are $g_\ell=1$ for a <proton> and $0$ for a <neutron>. Residual nuclear interactions and configuration mixing can shift measured moments away from this ideal prediction. A single hole in an otherwise closed $j$ shell has the same stretched-state moment as a particle in that shell: removing the $m_j=-j$ particle leaves total $J_z=j$ and moment $g_jj\mu_N$.
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