= Polarization-selective Zeeman splitting of an alkali D1 line
{title2=$\delta\nu_\pi=\pm2\mu_BB/(3h),\quad\delta\nu_\sigma=\pm4\mu_BB/(3h)$}
For the electronic $^2S_{1/2}\to{}^2P_{1/2}$ 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 $(\mu_BB/h)(g_em_e-g_gm_g)$.
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 $\pm2\mu_BB/(3h)$. Perpendicular linear polarization is an equal-amplitude combination of the two field-axis circular components, driving $\sigma^\pm$ transitions with $\Delta m=\pm1$ and shifts $\pm4\mu_BB/(3h)$. 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.
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