= Landé g-factor
{c}
{title2=$g_J$}
{wiki}
The Landé g-factor is the dimensionless electronic magnetic splitting factor relating a level's <magnetic dipole moment> to its total <angular momentum>. In the approximation where total <orbital angular momentum> $\mathbf L$ and total <spin> $\mathbf S$ couple to $\mathbf J=\mathbf L+\mathbf S$, take the orbital factor as one and the <Electron> <spin> factor as $g_s\simeq2$. For $J>0$,
$$
g_J=1+(g_s-1)\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.
$$
This follows by projecting $\mathbf L+g_s\mathbf S=\mathbf J+(g_s-1)\mathbf S$ onto $\mathbf J$ and using $2\mathbf S\cdot\mathbf J=\mathbf J^2+\mathbf S^2-\mathbf L^2$. At $S=0$, it gives $g_J=1$; at $L=0$ and $J=S$, it gives $g_J=g_s$. A $J=0$ level instead has zero first-order electronic shift; the formula's zero denominator must not be used. This $g_J$ is distinct from the <statistical weight of an atomic level> $2J+1$, sometimes also written $g_J$.
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