= Anomalous Zeeman effect
The anomalous Zeeman effect is <Zeeman splitting> whose pattern is not the simple orbital triplet. <Spin> and <spin-orbit interaction> make the upper and lower <Landé g-factors> differ. The same weak-field formula still applies when its assumptions hold: “anomalous” does not mean nonlinear in the <magnetic field>.
For example, take an electronic transition with $L_u=1,S_u=1/2,J_u=1/2$ and $L_l=0,S_l=1/2,J_l=1/2$, neglecting <hyperfine structure>. With $g_s=2$, the formula above gives $g_u=2/3$ and $g_l=2$. The two allowed $q=0$ transitions have dimensionless shifts $g_um_u-g_lm_l=\pm2/3$, while the $q=\pm1$ transitions give $\pm4/3$. Thus there are four <frequencies>, even though each level contains only two <magnetic quantum numbers>. This is an independent worked deduction; it should not be substituted for the unresolved widths in the 1896 experiment.
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