Landé g-factor

The Landé g-factor was introduced by Alfred Landé in 1921, even before the concept of electron spin was fully formulated, as an empirical way to fit the experimental data of the anomalous Zeeman effect. Its theoretical justification came later with the development of quantum mechanics. The formula arises from the vector model of the atom, where the orbital (\(\vec{L}\)) and spin (\(\vec{S}\)) angular momenta are considered to precess rapidly around their resultant total angular momentum vector (\(\vec{J}\)) due to spin-orbit coupling. The interaction with a weak external magnetic field is much slower. Therefore, the field effectively interacts with the time-averaged magnetic moment, which is the projection of the total magnetic moment (\(\vec{\mu}_L + \vec{\mu}_S\)) onto the direction of \(\vec{J}\).

The g-factor essentially accounts for the different ratios of magnetic moment to angular momentum for orbital motion (\(g_L=1\)) and spin (\(g_S \approx 2\)). When \(S=0\), then \(J=L\), and the formula correctly gives \(g_J=1\), corresponding to the normal Zeeman effect. When \(L=0\), then \(J=S\), and the formula gives \(g_J=2\) (using \(g_S=2\) instead of 1 in the derivation), corresponding to pure spin systems as in electron spin resonance. For all other cases, \(g_J\) takes on a rational value between 1 and 2, quantifying the complex interplay between spin and orbital contributions to the atom’s magnetism.