 Spin-orbit g-factor ratio from two P states

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.