The moment a fit invents

The usual starting point is that a magnetic moment is one of the few chemical measurements that returns an integer: count the unpaired electrons, put the count into $\sqrt{n(n+2)}$, and nine first-row ions come out right.

The first way that breaks is known: sit an ion at its spin crossover and its moment runs continuously between two integers’ worth as the temperature is changed, so the same compound reports different numbers in the morning and the afternoon.

This essay is about the step that happens before either of those. A moment is not measured. What is measured is a susceptibility at a series of temperatures, and a moment is what comes out of fitting a model to it — so the number depends on the model, on the fitting, and on which temperatures the apparatus could reach.

The sample and the model #

Take two spin-½ ions coupled by an exchange interaction — the situation what couples two spins computes from hopping and the Pauli principle — with a singlet ground state and a triplet 2J above it. Their susceptibility follows exactly from the Boltzmann populations of those four states, and it is not a straight line in anything.

What is fitted to it is the Curie–Weiss law, $\chi = C/(T - \theta)$, by least squares on $1/\chi$ against T. Two numbers come out: an effective moment from the Curie constant, and a Weiss temperature from where the line crosses the axis.

The model is wrong. Not approximately wrong in a way that gets better with more data — wrong in kind, because the sample’s inverse susceptibility is curved and the model’s is straight. So the fit returns whatever straight line best matches the piece of curve it was shown, and different pieces have different tangents.

Four ranges, four answers #

The spread in the moment is forty-five per cent. For a quantity whose whole purpose is to count unpaired electrons — where the difference between two and three unpaired electrons is a moment of 2.83 against 3.87 — that is the difference between two answers.

The spread in the Weiss temperature is a factor of six.

And the fits are good. Three of the four reproduce their own data with $r^2$ above 0.996, and the best of them is 0.99999. A paper reporting any one of these fits would report an excellent one.

That is the whole difficulty in one observation: the quality of a fit measures how well a model matches the data it was shown, and the parameters are statements about behaviour outside that range. A straight line through a short arc of a curve fits the arc beautifully and points anywhere at all.

What the sample is actually doing #

The reason the fits disagree is worth drawing, because it makes the disagreement look inevitable rather than pathological.

At low temperature the singlet is the only state populated and the pair is nearly diamagnetic. At high temperature all four states are populated equally and the pair behaves like two independent spins, with a moment of $\sqrt{6}$ = 2.449 for g = 2. In between, it is somewhere between the two — continuously, with no plateau anywhere.

So the sample has no single moment to report, and the number a fit returns is a weighted summary of a range, weighted by whichever end of that range the fit happened to be most sensitive to.

The two limits, exactly #

The curve the fits are chasing has two ends and both are available in closed form, which is what makes the disagreement between the fits predictable rather than mysterious.

At high temperature every one of the four states is populated equally: one singlet and three triplet components, with the triplet carrying all the moment. The mean square moment is three quarters of the triplet’s, which for g = 2 gives √6 = 2.449 — the same as two independent spin-½ ions, because at high enough temperature the coupling has stopped mattering.

At low temperature only the singlet is populated and the pair is diamagnetic. The moment goes to zero exponentially, as $\exp(2J/kT)$, so it falls off a cliff rather than tapering.

Between those two the curve has an inflection at a temperature of order $|2J|/k$, and that is the temperature the fit is most sensitive to. A window entirely above it sees a nearly flat moment and returns something close to the high-temperature limit; a window straddling it sees a steeply rising curve and returns a straight line through the steep part, which extrapolates back to a Weiss temperature far more negative than the coupling.

So the six-fold spread in θ is not a numerical accident. It is the difference between the tangent to a curve at its steepest point and the curve’s own asymptote, and no amount of care in the fitting changes it.

There is a second system with the same difficulty and a different cause, and keeping the two apart is worth a paragraph. An ion at its spin crossover has a moment that runs between two integers’ worth as the temperature is raised, because two spin states of one ion are both populated. A Curie fit to any window of that curve suffers exactly what the fits above suffer — the fitted constant is a property of the window rather than of the sample.

The moment that counts electrons, and the one that does not #

It is worth separating the two claims now made about moments, because they are different and both are true.

A moment counts electrons, not orbitals is about the physics: for an ion with a well-isolated ground state, the spin-only formula works and works well, because the quantity being measured really is a count.

This essay and the spin-crossover one are about the cases where there is no well-isolated ground state. A spin crossover has two states of one ion within a few hundred wavenumbers; a coupled pair has a singlet and a triplet within a few tens. In both, several states are populated at once, the susceptibility is a Boltzmann average, and the number a fit extracts is a summary of a curve rather than a count of anything.

The distinction matters when reading a table of moments. A compound whose moment is quoted with no temperature is being reported as the first kind; a compound whose moment is quoted as at 300 K is being reported as the second, and the reader is entitled to ask over what range the fit was made.

Where the conversion does work #

The Weiss temperature is routinely converted into an exchange coupling, and the conversion is not wrong — it is a high-temperature statement being used without the qualification.

The mean-field conversion of a coupling of −50 cm⁻¹ is θ = −48.0 K. The fit over 300–600 K returns −51.4, which is agreement to seven per cent. The fit over 80–150 K returns −292.3, which is not agreement at all.

That is the honest structure of the result. The Curie–Weiss law is the leading term of a high-temperature expansion of the exact susceptibility, so a fit at temperatures well above the coupling recovers what the expansion says it should, and a fit at temperatures comparable with the coupling recovers a tangent to a curve. The natural first claim is that no window reproduces the mean-field value, and the high-temperature window refuses it — correctly, and the refusal is the more interesting result.

The practical statement is therefore a condition rather than a warning: the conversion holds when $kT \gg |2J|$ over the whole fitted range. For a coupling of 50 cm⁻¹ that means temperatures well above 72 K, and for the couplings of a few hundred wavenumbers that are common in bridged copper dimers it means temperatures no cryostat is asked to reach.

The fit that returns no moment at all #

At stronger coupling the arithmetic stops being merely misleading.

For a pair coupled at −200 cm⁻¹, the susceptibility over 80–300 K is rising with temperature — the triplet is still being populated — so the straight line fitted to $1/\chi$ has the wrong slope and the Curie constant comes out negative. The moment that would follow is the square root of a negative number.

Reported here as none rather than as an arithmetic failure, because it is not one: the fit has done exactly what least squares does, and the answer is telling the truth about the model. A negative Curie constant is a Curie–Weiss law’s way of saying that this sample is not a Curie–Weiss sample.

Why the fit is done on one over χ #

The choice to fit $1/\chi$ against T rather than χ against T is not neutral, and it is worth a paragraph because it is made automatically by everybody.

A least-squares fit weights every point equally in whatever quantity is being fitted. Fitting $1/\chi$ weights the high-temperature points, where $1/\chi$ is large; fitting χ weights the low-temperature points, where χ is large. The two therefore return different parameters from identical data, and neither is more correct — they are answers to which part of the curve should the straight line match.

The convention exists because a Curie–Weiss law is a straight line in $1/\chi$, so the plot is the natural way to see whether the law holds at all. That is a good reason, and it carries a consequence: the fit is being made in the representation that makes the model look best, and the high-temperature end that dominates the fit is the end where the model is a valid expansion.

Modern practice largely abandons both, and plots χT against T instead — a quantity that is constant for a Curie law, falls for an antiferromagnetically coupled pair and rises for a ferromagnetically coupled one, so the shape of the curve says immediately what kind of system it is. The fitted parameters still come from a model, but the reader can see what the sample did.

What is quoted, and what is computed #

The exchange coupling in the model is a parameter, set to −50 cm⁻¹ because that is an ordinary value for a bridged dimer. Everything else is computed from it: the four state energies, their Boltzmann populations, the susceptibility at every temperature, and the least-squares fit.

The mean-field conversion θ = 2J/3k is quoted from the literature. It is the standard route from a Weiss temperature to a coupling and it is what makes the comparison in this essay a comparison rather than a claim.

The measured side is not here at all, and the absence is deliberate. A real susceptibility measurement carries a diamagnetic correction, a temperature-independent paramagnetism, a small amount of uncoupled impurity that dominates at low temperature, and a g value that is fitted rather than known. Each of those adds a parameter, and each of them is fitted over the same range that this essay shows to be decisive. The point of computing the ideal case is that all of the spread above is present before any of that is added.

What was checked #

With no coupling the fit is exact. At J = 0 the susceptibility really is a Curie law, and the fit returns θ = 0 and a moment of √6 to six decimal places, from every window. That is the check that the fitting code is right and the disagreement below is the model rather than the arithmetic.

With coupling, the fitted parameters depend on the window — the Weiss temperatures spread by more than ten kelvin and the moments by more than five per cent, which is what the table shows in full.

The high-temperature window reproduces the mean-field conversion and the low-temperature one does not, by more than a factor of three. Both halves are required, because a claim that a fit is simply unreliable would be neither true nor useful.

The same competition at a different d count differs only in size: two states one unpaired electron apart rather than four. What matters is the shape of the statement rather than the number. Every quantity in this essay is a Boltzmann average, and a fitted parameter is a summary of an average — two operations away from anything measured directly, and the second of them is the one nobody writes down.

What a careful measurement reports instead #

The remedy is not to distrust susceptibility measurements, which are among the most reliable in chemistry. It is to report the thing that was measured and to say what was assumed.

Three practices follow from everything above and all three are standard in the careful literature.

Report the curve. χT against T over the whole measured range is the data; a moment is a summary of it. A paper that shows the curve lets a reader see whether the sample is a Curie system at all.

State the range. A moment quoted without the temperatures it was fitted over is not reproducible, for the reason this essay is about: the same data over a different range gives a different number.

Fit the model the sample obeys. For a dimer the exact expression is two lines long and has two parameters, and fitting it recovers both from any range. Fitting a Curie–Weiss law to a dimer is fitting the wrong model to data that supports the right one.

None of this is a new complaint — it is what a magnetochemistry text says on the page where the Curie–Weiss law is introduced. What the arithmetic above adds is the size of the error when the advice is ignored, on a sample with no noise, no impurity and no instrumental drift in it at all.

What the Weiss temperature is a temperature of #

The four fits return two numbers each, and the second is usually treated as a nuisance parameter — a correction that absorbs whatever the Curie law does not describe. It is not a nuisance parameter in general, and knowing when it is one is the sharpest way to see what has gone wrong here.

In a lattice of coupled spins the Weiss temperature has a derivation and a meaning. A mean-field treatment gives it as the sum of the couplings each spin feels from all its neighbours: the number of neighbours times the coupling times a spin factor. So θ measures the total interaction per site, its sign says whether the interactions are ferromagnetic or antiferromagnetic, and its magnitude can be compared against an ordering temperature measured on the same sample — with mean field known to overestimate, typically by a factor of one and a half to three.

That is a quantity: derived, interpretable, and checkable against a second measurement.

For a dimer none of it applies. There is no lattice, no coordination number, and no ordering temperature; the exact susceptibility is not of Curie–Weiss form at any temperature, so nothing in the sample corresponds to the parameter being fitted. The θ that comes out is the amount by which a curve of the wrong shape has to be shifted to resemble the data over the chosen window, and it changes when the window does — from −51 to −292 kelvin across four fits of one perfect sample.

So the four Weiss temperatures are not four estimates of a quantity. They are four descriptions of a mismatch, and the range between them measures how badly the shapes differ rather than how uncertain any physical parameter is.

That is worth separating from the moment problem, because the two failures have different characters. The fitted moment at least corresponds to something — it is the sample’s χT at the top of the window, converted — so its four values can be understood as one quantity evaluated in four places. The fitted θ corresponds to nothing at all, and quoting it for a dimer is quoting a fitting residual under the name of a physical temperature.

The practical form of that is a short rule. A Weiss temperature is worth reporting when the sample is an extended lattice, where it can be compared against a coordination number and an ordering temperature. For a discrete molecule it is a shape mismatch with a unit attached.

Still open: fitting the exact expression, and models without one #

The obvious open question is what a spectroscopist does instead: fit the exact expression rather than the Curie–Weiss law. The Bleaney–Bowers form has two parameters, J and g, and fitting it to data over any range recovers both — which is the standard practice for a dimer and works precisely because the model being fitted is the model the sample obeys.

That moves the difficulty rather than removing it, and the direction it moves is the interesting one. A dimer has an exact expression; a chain, a square lattice or a triangular one does not, and what is fitted is a high-temperature series or a numerical result for a model chosen in advance. So the question what is this compound’s coupling becomes which model was assumed, which is a familiar shape of question and is a fact about the field rather than about magnetism.