A moment counts electrons, not orbitals

Most chemical measurements return a number that has to be interpreted. A magnetic moment is one of the few that returns something close to an integer, and the integer is the number of unpaired electrons.

That makes it an unusually direct instrument. A compound’s moment does not depend on how the electrons are arranged among the orbitals, on which orbitals those are, on the geometry or on the ligands — except through the one thing all of those settle, which is how many electrons ended up unpaired.

The formula, and what is in it #

An unpaired electron has spin one half. Put $n$ of them in a high-spin arrangement and the total spin is $S = n/2$. The magnetic moment of a spin $S$ is

$$\mu = g\sqrt{S(S+1)}$$

in Bohr magnetons, with $g$ the electron’s g-factor, which is 2.0023 and is taken as 2. Substituting $S = n/2$ gives $\mu = \sqrt{n(n+2)}$ — the spin-only formula, which has one input.

That input is a count. It is not a measured quantity, not a fitted parameter and not a property of the ligand: it is the number of orbitals holding one electron in whichever filling the complex has, and everything the previous essays in this field computed feeds into it exactly once, at the point where the filling is decided.

The values are 1.73 for one unpaired electron, 2.83 for two, 3.87 for three, 4.90 for four and 5.92 for five. There is no sixth, because five orbitals cannot hold more than five unpaired electrons.

The comparison #

Nine first-row ions, with their measured moments:

The first five are as good as the measurements are. The last four are not, and the pattern of the failure is more informative than the successes.

Every discrepancy has the same sign. The measured moment is at or above the counted one in all nine cases, and the two small negative differences — vanadium at −0.08 and iron(III) at −0.02 — are within the spread of published values for those ions.

A model that was merely imprecise would miss in both directions. One-sidedness means a term has been left out and the term can only add, which is what the check requires: no measured moment falls meaningfully below the count, and at least one exceeds it by more than half a Bohr magneton.

What was left out #

Electrons have orbital angular momentum as well as spin, and orbital angular momentum is magnetic too.

In a free ion the orbital contribution is large. In a complex it is usually quenched: the ligand field fixes the orbitals in space, so an electron in dxy cannot circulate freely into dx²−y², and the orbital contribution largely disappears. That quenching is why the spin-only formula works at all, and it is a consequence of the ligand field rather than of anything about spin.

The quenching is incomplete in two circumstances, and the four failures above are exactly those.

An orbitally degenerate ground state. If the ground configuration allows an electron to move between degenerate orbitals by a rotation — as it does for cobalt(II), whose high-spin d⁷ arrangement leaves an unequal occupation of the lower set — the circulation is not blocked and a genuine orbital moment survives. Cobalt(II) is the worst case in the table, at +0.93.

Second-order mixing. Even where the ground state has no orbital degeneracy, spin–orbit coupling mixes in excited states with some orbital character, which raises the effective g above 2. Nickel(II) at +0.37 and copper(II) at +0.17 are this case, and the size of the excess correlates with how close the relevant excited state is — so the correction is larger for a small ligand field splitting, which is measurable independently.

How a moment is measured #

The instrument matters here because the quantity it returns is not a moment directly, and the conversion carries an assumption.

What is measured is a magnetic susceptibility: how much the sample’s magnetisation responds to an applied field. For a collection of independent moments the susceptibility follows the Curie law, falling as one over the temperature, and the moment is extracted from the slope of a plot of inverse susceptibility against temperature.

The assumption in that extraction is independence — that the moments do not talk to each other. It holds for a dilute solution or a magnetically dilute solid, and it fails when the moments interact, which is precisely the situation what couples two spins computes. A compound whose susceptibility does not follow the Curie law is a compound whose moments are coupled, and the departure from the law is a measure of the coupling.

So the “measured moment” quoted in tables is an effective moment inferred under a stated model. That is a familiar shape: a quoted experimental number is usually the output of a fit, and knowing which fit is part of knowing what the number means. Saying which model produced a number applies as much to a measurement as to a computation.

The count is not the electron count #

Two comparisons in the table make the essential point about what is being measured.

Iron(II) and iron(III) differ by one d electron — six against five — and their high-spin moments are 5.40 and 5.90, which is a difference of one unpaired electron in the same direction. So far the moment looks like an electron counter.

Iron(II) in a strong field has the same six d electrons and a moment of zero. Nothing about the metal changed; the ligand changed, the electrons paired up in the lower set, and the compound is diamagnetic.

That is the sharpest available statement of what the measurement returns. Not electrons, not d electrons, not oxidation state: unpaired electrons, and the ligand decides how many there are. Working out the condition is the subject of the pairing energy decides the moment, and the answer is a comparison between the splitting and a repulsion energy.

Why an integer measurement is worth so much #

Chemistry has few observables that return counts, and three have now appeared.

A spectrum counts environments: the number of bands is fixed by the symmetry before any of them is measured, which is what a spectrum counts environments, not atoms and how many frequencies, not how many modes establish.

A photoelectron spectrum counts occupied orbitals, which is what makes it decisive between bonding models, as what a photoelectron spectrum measures argues.

A magnetic moment counts unpaired electrons. The three have a common form: each converts a continuous instrumental reading into a small integer, and a small integer is the kind of quantity a model either gets right or does not.

That is why a moment settles arguments. A proposed structure with four unpaired electrons predicts 4.90 and one with two predicts 2.83, and those are not values a measurement confuses. The distinction between the two nickel compounds of VSEPR does not reach a transition metal — one tetrahedral and paramagnetic, one square planar and diamagnetic — is decided by a moment, and the shape follows from the magnetism rather than the other way about.

Unpaired is not the same as unfilled #

A caution about the word, because chemistry uses “unpaired” loosely and the formula does not.

Hund’s rule says electrons occupy degenerate orbitals singly with parallel spins before pairing, and it is a statement about repulsion — two electrons in the same orbital are closer together on average and repel more, and two with parallel spins keep further apart for reasons of symmetry rather than of charge.

That example is a warning attached to every count on this page. The number of unpaired electrons is read off a filling, the filling comes from a rule about which orbital is next, and the rule is about electron repulsion — which is the quantity a one-electron model does not contain. Where the rule holds, the count is right and the moment follows. Where it does not, the count is wrong and the moment says so, loudly, because it is an integer.

The integer, and how sharply it is held #

It is worth asking how close to an integer the count really is, because a formula returning 3.87 for three unpaired electrons is not returning three.

The values are $\sqrt{n(n+2)}$ rather than $n$ because a spin of $S$ has magnitude $\sqrt{S(S+1)}$ rather than $S$ — the same feature every quantum angular momentum has, and the reason a spin one half has magnitude $\sqrt{3}/2$ rather than a half. What is integral is the count, and the moment is a function of it that happens to be irrational.

The function is well spaced, which is what makes the measurement decisive: 1.73, 2.83, 3.87, 4.90, 5.92, with gaps of about one throughout. A measurement good to a tenth of a magneton — routine — separates adjacent counts by ten standard deviations.

The spacing narrows nowhere and the sequence has no ambiguity, so the only way to misread a moment is to be in one of the situations the previous section listed: a large orbital contribution, a mixture, or coupled moments. Each of those has its own signature, and none of them produces a value that could be mistaken for a different integer.

Where the formula stops #

The lanthanides. For f electrons the orbital contribution is not quenched at all: the 4f shell is buried inside the filled 5s and 5p shells and barely feels the ligands. The right formula there is $g_J\sqrt{J(J+1)}$ with the total angular momentum $J$, and the spin-only value is wrong by large factors — dysprosium(III) measures about 10.6 Bohr magnetons against a spin-only 5.92.

Second and third row transition metals. Spin–orbit coupling grows steeply with atomic number, so the second-order corrections that are a tenth of a magneton in the first row are considerably larger further down.

Anything cooperative. Everything here concerns an isolated ion. A compound in which the moments interact — ordering ferromagnetically or antiferromagnetically — has a measured susceptibility that depends on temperature in a way no single-ion formula describes, and the mechanism by which two moments come to interact at all is the subject of what couples two spins.

What a moment cannot say #

Three things it does not settle, each of which it is sometimes asked to.

It does not give the geometry. A tetrahedral d⁷ complex and an octahedral high-spin d⁷ complex both have three unpaired electrons and a spin-only moment of 3.87. What distinguishes them in practice is the size of the orbital contribution — larger in the tetrahedral case — which is a second-order argument requiring the spectrum as well.

It does not give the oxidation state. Manganese(II) and iron(III) are both high-spin d⁵ with five unpaired electrons and a moment near 5.9. A moment constrains the count and the count constrains the pair (metal, oxidation state) jointly, so a second measurement is needed to separate them.

It does not distinguish a spin state from a mixture. A measured moment between two predicted values can mean an intermediate case, an orbital contribution, or a sample containing both spin states — and spin-crossover compounds, which convert from one to the other with temperature, exist precisely because the two states can be within a few kilojoules of each other. Temperature dependence separates those explanations, and a single-temperature measurement does not.

None of that reduces the value of a count. It means the count is one constraint among several, and it is the one that comes back as an integer.

What the count is worth to the rest of this field #

Two of the three claims the applied field makes about coordination compounds are settled by this measurement rather than by any calculation here.

Whether a complex is high spin or low spin is a claim about the unpaired count, and a moment reads it directly. Whether a four-coordinate d⁸ complex is tetrahedral or square planar is a claim about a shape, and a moment reads it too, because the two shapes force different counts. In both cases what is computed is a condition — a comparison between a splitting and a pairing energy, or between a ligand field gain and a repulsion penalty — and the moment says which side of the condition a real compound fell on.

That division is worth naming because it is how this site prefers to work. The computation supplies the structure of the answer and the boundary between two regimes; the measurement says where a particular compound sits. Neither replaces the other, and a computation with no measurable consequence would be the weaker half of the pair.

The family where counting spins is not even close #

The formula’s scope is usually given as transition-metal complexes, and the sharpest way to see what it depends on is to apply it to the other family of magnetic ions in the periodic table, where it fails by a factor of two.

The lanthanide ions are 4f, and the spin-only formula is not a small correction away from their moments — it is nowhere near them. Dysprosium(III) has five unpaired f electrons, so the count gives 5.92 Bohr magnetons. Its measured moment is 10.6.

What works instead is the free-ion expression, which keeps the orbital angular momentum rather than discarding it: $g_J\sqrt{J(J+1)}$, with the total angular momentum and the Landé factor of the ion’s ground term. Applied across the series it does very well:

So the two families need opposite formulas, and the reason is a comparison of two energies.

A 3d orbital is on the outside of its ion and the ligands reach it: the ligand field is ten to thirty thousand wavenumbers, spin–orbit coupling is a few hundred, and the field wins comfortably — it fixes the orbitals in place, the orbital angular momentum is quenched, and only the spin is left to be counted. A 4f orbital is buried beneath filled 5s and 5p shells: the ligand field reaching it is a hundred wavenumbers or so, spin–orbit coupling is a thousand or more, and the ordering reverses. Nothing quenches anything, the free ion’s $J$ survives into the compound, and the moment is the free ion’s.

The two exceptions among the lanthanides prove the rule by failing for a third reason entirely. Samarium(III) and europium(III) have low-lying states of different $J$ close enough to be populated at room temperature, so their moments are thermal averages over several terms — europium’s ground term has $J = 0$ and a predicted moment of exactly zero, and the measured value is about 3.4.

Which sharpens what the integer actually rests on. The count works because a ligand field is strong compared with spin–orbit coupling, not because a moment is intrinsically a count of spins. Where that inequality reverses the same measurement returns a number with no integer anywhere in it, and where both energies are comparable — the second and third transition series, and the two lanthanides above — it returns something that is neither.

It also explains a practice that looks arbitrary from outside. A chemist quotes a first-row complex’s moment as a count of unpaired electrons and a lanthanide’s as a number of Bohr magnetons, and never converts between them. That is not two conventions for one quantity; it is two quantities, measured by the same instrument, and the only thing they share is the apparatus.

Who found it, and when #

The spin-only formula belongs to the late 1920s, when the electron’s spin magnetic moment had just been established, and its application to transition-metal chemistry is largely John Van Vleck’s work through the 1930s — which is also where the quenching argument comes from, and why crystal field theory arrived in chemistry as a theory of magnetism rather than of colour.

The order is worth noticing, because it is the reverse of how the subject is now taught. Ligand field splittings were first inferred from magnetic susceptibilities, not from spectra: the susceptibility told experimenters how many electrons were unpaired, the unpaired count told them which side of the crossing the compound was on, and the splitting was bounded from that. Colour came later and turned out to measure the same quantity directly.