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Friedrich Hund's rules describe the usual ground-term ordering within an atomic electron configuration in the LS coupling regime. First maximize the total spin $S$ subject to the Pauli exclusion principle. Among the terms with that spin, maximize total orbital angular momentum $L$. Within the resulting term, the lowest level has $J=|L-S|$ for a less-than-half-filled open subshell and $J=L+S$ for a more-than-half-filled one. These are ordering rules, not universal replacements for diagonalizing an arbitrary atomic Hamiltonian operator.

For two equivalent $p$ electrons, parallel spins give $S=1$. Their spatial wavefunction must be antisymmetric; the highest allowed orbital projection with parallel spins is $1+0=1$, giving $L=1$. A $p^2$ subshell is less than half filled, so its lowest term symbol is ${}^3{P}_0$, followed within that term by ${}^3{P}_1$ and ${}^3{P}_2$. Closed subshells contribute zero total $L$ and $S$.

Polykarp Kusch and Henry Michael Foley compared hyperfine structure resonance frequencies for different electronic states of gallium, indium and sodium. Their 1948 paper reports mutually consistent departures from the ratios expected with orbital factor $1$ and spin factor $2$. Comparing several states constrained the alternative explanation of perturbed atomic levels.^[ref]

The Rabi molecular beam magnetic resonance experiment explains the readout principle: a resonant transition changes an atom's magnetic response, and hence its transmitted beam signal. Kusch describes rapidly interchangeable ovens, several radio frequency oscillators and careful field homogenization. Measurements taken during monotonic field drift were reduced to a common field.^[ref]

Real hyperfine structure lines are not simply bare electronic $g_J$ times field. Nuclear magnetic moments, known interaction constants and nonlinear field dependence enter the inference.^[ref] Field homogeneity controls line broadening; alternating species addresses drift. Neither control removes an error in the atomic model. The beauty of the comparison is the separation of two problems: cancelling a shared calibration scale and testing whether several physical systems imply the same electron spin g-factor.

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.

An original reduced model makes the cancellation explicit. Assume LS coupling, neglect nuclear spin, and write the electronic magnetic dipole moment as $-{\mu }_B(g_L\mathbf{L}+g_s\mathbf{S})/\hslash$. For $L=1$, $S=1/2$, the projection used to derive the Landé g-factor givesThese coefficients follow by substituting $J=3/2$ or $1/2$ into the angular-momentum projection formula; neither level measures pure spin alone. In a weak uniform magnetic field, adjacent electronic projections have spacing $h{f}_J=g_J{\mu }_{B}B$ when $g_J>0$. ThereforeThe common magnetic field and Bohr magneton cancel. The ratio determines $g_s/g_L$, not $g_s$ without further knowledge of $g_L$. For $g_L=1$, it gives $a_e=(R-2)/(R+1)$. Small departures from $R=2$ are magnified differently from small departures of $g_s$ from $2$; the two fractional changes should not be identified.

With the stipulated inputs $g_L=1$, $g_s=2.0024$ and ${\mu }_{B}B/h=100\mathrm{MHz}$, the two electronic frequencies are $66.5867$ and $133.4133\mathrm{MHz}$. Their ratio is approximately $2.0036043$, and inversion recovers $a_e=0.0012$. These are ideal electronic lines, not historical gallium data. Nuclear spin, mixing between levels and other deviations from LS coupling require a more complete model in an actual atom.

In LS coupling, an atomic term symbol specifies total electronic spin $S$, total orbital angular momentum $L$, and total angular momentum $J$. The superscript $2S+1$ is the spin multiplicity; the letters $S,P,D,F,\dots$ denote $L=0,1,2,3,\dots$. Coupling gives $J=|L-S|,\dots ,L+S$, and a fixed-$J$ level has $2J+1$ magnetic substates before an external field resolves them. For example, ${}^{3}P$ has $S=L=1$ and levels $J=0,1,2$, containing $1+3+5=9$ states altogether. A term symbol does not itself specify the radial electron configuration.