spin-only magnetic moment

Suppose you have weighed a paramagnetic complex in a magnetic field and worked out the strength of its magnetism, expressed as a magnetic moment. How do you turn that single measured number into an actual count of unpaired electrons inside the metal? The spin-only magnetic moment is the simple formula that does it, and it is one of the most useful back-of-envelope tools in coordination chemistry.

The formula is mu = square root of n(n+2), where n is the number of unpaired electrons and mu comes out in units called Bohr magnetons. It assumes the magnetism arises purely from electron spin, ignoring any contribution from the electrons' orbital motion (hence spin-only). Plug in the small whole numbers and you get a short table: one unpaired electron gives about 1.73, two gives 2.83, three gives 3.87, four gives 4.90, five gives 5.92. To use it in reverse, you measure the moment, find the nearest value in that table, and read off n.

This is exactly how chemists pin down the d-electron arrangement experimentally. A measured moment near 4.9 says four unpaired electrons; for an octahedral iron(II) (d6) that means high-spin, whereas a moment near zero would mean low-spin — so one measurement decides between the two and, combined with the oxidation state, confirms the geometry. The honest caveats are important: the spin-only value is an approximation that works best for the first-row transition metals where orbital contribution is largely quenched by the ligand field; for ions with significant orbital angular momentum (and especially for the lanthanides) the real moment departs from spin-only, and you must use the fuller treatment that includes orbital motion.