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 $g_u{m}_u-g_l{m}_l=\pm 2/3$, while the $q=\pm 1$ transitions give $\pm 4/3$. 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.

Neglect hyperfine structure and use a magnetic field weak compared with the spin-orbit interaction. The sodium D doublet has transitions $3p{}^2{P}_{1/2}\to 3s{}^2{S}_{1/2}$ and $3p{}^2{P}_{3/2}\to 3s{}^2{S}_{1/2}$. Their Landé g-factors are $2/3$, $4/3$ and $2$ respectively. With $m_u,m_l$ denoting upper and lower magnetic labels, the photon shift isThe first transition has four allowed shifts ${\mu }_{B}B\{\pm 2/3,\pm 4/3\}$; the second has six, ${\mu }_{B}B\{\pm 1/3,\pm 1,\pm 5/3\}$. The most blue-shifted component of the second is $m_u=1/2\to m_l=-1/2$, with shift $5{\mu }_{B}B/3$. The apparently larger shift $m_u=3/2\to m_l=-1/2$ is excluded by the electric-dipole selection rule. The $3p{}^2{P}_{3/2}$ level lies above $3p{}^2{P}_{1/2}$, so in this weak-field model the $5/3$ component also has the shortest wavelength of the whole doublet. ^[ref]

For the electronic ${}^2{S}_{1/2}\to {}^2{P}_{1/2}$ transition, neglect hyperfine structure and use weak-field LS coupling. The Landé g-factors are $g_g=2$ and $g_e=2/3$, so a transition from $m_g$ to $m_e$ shifts by $({\mu }_{B}B/h)(g_e{m}_e-g_g{m}_g)$.

With propagation perpendicular to the magnetic field, linear polarization parallel to the field drives the $\pi$ electric-dipole selection rule $\Delta m=0$. The two shifts are $\pm 2{\mu }_{B}B/(3h)$. Perpendicular linear polarization is an equal-amplitude combination of the two field-axis circular components, driving ${\sigma }^{\pm }$ transitions with $\Delta m=\pm 1$ and shifts $\pm 4{\mu }_{B}B/(3h)$. Each geometry therefore selects a doublet, with twice the separation in the perpendicular-polarization case. Equal line heights within a pair require equal lower-state populations and an unsaturating probe; line positions do not require that extra population assumption.