Landé g-factor ($g_J$)

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 $g_s\simeq 2$. For $J>0$,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 $g_J$.

For a free ion with one $f$ electron in its weak-field ground multiplet, $L=3,S=1/2,J=5/2$ and $g_J=6/7$. The six magnetic substates have adjacent spacing $g_J{\mu }_{B}B$ and full spread $5g_J{\mu }_{B}B$. A crystal field can change the applicable effective low-energy description.