orbital contribution to magnetic moment

The spin-only formula treats an electron as a magnet only because it spins. But an electron also orbits, and a circulating charge is itself a tiny electromagnet. Why, then, does the spin-only formula work as well as it does, and when does it break down? The answer is the orbital contribution to the magnetic moment — the extra magnetism from the electrons' orbital motion that spin-only deliberately leaves out.

Whether the orbital motion contributes depends on a subtle condition: there must be a way for an electron to circulate around the metal by moving between equivalent, degenerate d orbitals without changing anything else. When the ligand field arranges the orbitals so that no such free circulation is available, the orbital angular momentum is said to be quenched, and only spin is left — which is exactly why the spin-only formula is so often a good fit for first-row transition metals. When the configuration does leave a path for circulation (notably when the t2g set is partly filled in a way that lets an electron hop among dxy, dxz, dyz), an orbital contribution survives and the measured moment rises above the spin-only value.

Recognizing this keeps you honest when reading magnetic data. A cobalt(II) octahedral complex, for example, routinely shows moments well above its spin-only value of 3.87 (often 4.3 to 5.2) precisely because of an unquenched orbital contribution — so a moment that overshoots the table is a clue, not an error. The contribution grows with atomic number through spin-orbit coupling, which is why it is modest for the first row but large for the second and third rows and overwhelming for the lanthanides, whose buried 4f electrons feel almost no ligand field and keep nearly their full free-ion orbital moment. For those, spin-only is abandoned in favor of a moment formula built around the total angular momentum J.