Magnetic Properties: When the Spin-Only Formula Works and When It Fails

The spin-only formula

where n is the number of unpaired electrons. It gives a fixed value for each n, so measuring the moment identifies n directly.

These five values are worth knowing, because a very large share of questions consists of matching a quoted moment to a number of unpaired electrons and hence to a spin state.

The chain of inference

Magnetic measurement is powerful because it starts a chain:

1. Measured moment gives the number of unpaired electrons.
2. That gives the electron configuration in the split d orbitals.
3. That identifies the complex as high spin or low spin.
4. Which tells you whether the ligand field splitting exceeds the pairing energy.
5. Which places the ligand in the spectrochemical series.

Many exam questions run some portion of this chain, in either direction. Recognising which link is being asked for is most of the work.

Why the orbital contribution is usually absent

An electron has orbital angular momentum as well as spin, and in principle both contribute to the magnetic moment. In most first-row complexes the orbital contribution is quenched — suppressed by the ligand field.

Where the formula fails

For lanthanides the full expression using the total angular momentum quantum number J must be used, which is precisely why term symbols matter for that series. The lanthanide case is therefore the standard example of the formula's limits, and it connects two topics.

Diamagnetism and paramagnetism

A substance with all electrons paired is diamagnetic and is weakly repelled by a magnetic field. One with unpaired electrons is paramagnetic and is attracted.

Diamagnetism is present in all substances, since all have paired electrons, but it is a much weaker effect and is masked wherever paramagnetism exists. So a measured moment of zero means no unpaired electrons, and any non-zero moment indicates paramagnetism.

Temperature dependence

For a simple paramagnet, susceptibility varies inversely with temperature — the Curie law. Plotting the reciprocal of susceptibility against temperature gives a straight line through the origin, and deviations from that line indicate magnetic interactions between centres rather than isolated ones.

Recognising that a non-zero intercept signals cooperative behaviour rather than experimental error is a higher-order point worth having.

Frequently asked questions

Why does the ligand field quench the orbital contribution?

Because orbital angular momentum requires electrons to circulate between equivalent orbitals. The field removes that equivalence, so the circulation is blocked.

Why does the formula fail for lanthanides?

Because the 4f orbitals are shielded by outer shells, so the ligand field barely reaches them and cannot quench the orbital contribution.

What does a moment slightly above the spin-only value indicate?

A residual orbital contribution, which occurs where the ligand field leaves suitable orbitals close enough in energy for partial circulation.

Can magnetic data distinguish tetrahedral from square planar?

Often yes for d⁸. A tetrahedral complex is paramagnetic with two unpaired electrons, while a square planar one is diamagnetic, so the measurement settles the geometry.