The model is what is fitted

A single coupled pair of spins can have its susceptibility computed exactly and a Curie–Weiss law fitted to it over four temperature ranges. The moments came back at 2.471, 2.535, 2.566 and 3.590 Bohr magnetons and the Weiss temperatures from −51 to −292 kelvin, from one sample with no noise in it — and three of the four fits agreed with their own data to better than a part in three hundred.

It ended with the standard advice, which is to fit the model the sample obeys: for a dimer that is the two-line Bleaney–Bowers expression with two parameters, and fitting it recovers both from any range. And it observed that this moves the difficulty rather than removing it, because a chain or a lattice has no exact expression and what gets fitted is a model chosen in advance.

This essay measures how much moving it costs, and the answer is a number rather than a caution.

The sample is exact and the model is not #

A Heisenberg chain of $n$ spin-½ centres is small enough to solve outright: write down the whole $2^n$-dimensional configuration space, block it by the total $z$ component, and diagonalise. Eight spins is 256 states and its largest block is seventy, which is nothing. Every susceptibility below is computed from the complete spectrum by Van Vleck’s formula, so the sample is exact in the sense that a real one never is.

What is fitted to it is the two-spin expression, which is what a chemist uses when the compound’s structure is not known or is known and is being idealised. It is the same substitution that a force field makes when it has more constants than frequencies: a model chosen for tractability, standing where a model chosen for the sample should be.

Before any of that is believed, the exact diagonalisation is checked against the closed form on the case where both exist. For two spins the computed χT and Bleaney–Bowers agree to $10^{-12}$ at every temperature tried, which is the check that the general calculation reproduces the special case rather than the special case being assumed.

What the wrong model reports #

The two-spin fit applied to a chain returns a coupling that is too large — by 5.6 per cent for four spins over 80–300 K, and by 33.3 per cent for eight spins over 150–600 K.

The error is systematic in two ways. It grows with the length of the chain, in every window, which is what identifies it as the model’s; and it grows with the temperature range, which is what makes it impossible to escape by choosing a window.

The reason for the length dependence is not mysterious. A chain has more ways of being partly excited than a pair does — a pair has one singlet and one triplet, a chain of eight has 256 states in a band — so its susceptibility rises toward its high-temperature limit more slowly than a dimer’s with the same coupling. At infinite temperature both reach the same place, which is the count of unpaired electrons and nothing else. The two-spin expression, asked to reproduce that slower rise, chooses a larger coupling, because a larger coupling is what makes a dimer rise slowly.

The reason for the range dependence follows: the shapes differ most at high temperature, so a window that reaches further up weights the disagreement more.

And the residual says nothing #

Every one of those fits has $R^2$ above 0.990, and the ones over the widest window — which are the worst — are above 0.990 too. The best of the wrong fits is 0.9984, at a coupling 21 per cent too large.

That is worth being blunt about. A residual measures how closely a two-parameter curve follows a smooth, monotone, featureless function of temperature. Almost any two-parameter curve does that well, and a susceptibility is exactly such a function: it rises, it flattens, it has no structure in it that a wrong model can fail to reproduce. A spectrum has lines that a wrong model misses; a susceptibility has one curve, and the shape of a curve is a weak constraint.

The distinction between the Curie–Weiss failure and this one is therefore worth stating precisely. There, the wrong model was visibly wrong: a Curie–Weiss law fitted to a dimer over a wide range has a residual that is bad enough to see, and the essay’s argument was about the windows narrow enough to hide it. Here the wrong model is not visibly wrong at any window, because it has the right functional form for a system of coupled spins and the wrong number of them.

Where the mismatch goes #

A two-parameter fit given data it cannot reproduce puts the discrepancy somewhere, and here it goes into the g factor.

The chains return g between 1.918 and 1.981 — every one of them a value a paper would report without comment, since a real spin-½ transition metal ion has g between about 1.9 and 2.2 for reasons involving spin–orbit coupling that are perfectly respectable — and which this collection computes rather than quotes. So the tell is present in every fit and is indistinguishable from a real effect.

The relation between the two parameters is what makes this happen. A smaller g reduces the whole susceptibility by a constant factor, and a larger |J| reduces it more at low temperature than at high; between them they can follow a curve of the wrong shape over a limited range, and the least-squares search finds the combination that does.

What would have told the truth #

Three things would, and none of them is a better fit.

A wider range. The fits get worse as the range widens, which sounds like the wrong direction and is exactly the right one. The disagreement between a chain and a dimer is largest at high temperature, so a measurement that goes there sees more of it — and a fit that goes there returns a worse-looking residual, which is information. Narrow windows conceal.

The high-temperature limit. Every one of these samples has the same number of spins per formula unit and therefore the same Curie constant at infinite temperature. Checking that a fit’s implied Curie constant matches the known spin count is a check that does not depend on the coupling model at all, and it is what the drifting g factor above would fail.

Or a second measurement. The same shape of problem elsewhere in this collection is resolved by isotopic substitution, which supplies data the first measurement could not. Here the analogous move is a magnetisation curve at low temperature, whose saturation and steps depend on the number of coupled centres directly rather than through a fitted parameter.

The Curie–Weiss numbers, for comparison #

The Curie–Weiss fit is available on the same samples, and putting the two side by side shows which of the two failures is worse and in what way.

Fitting a Curie–Weiss law to the eight-spin chain over 80–600 K gives a Weiss temperature of −122.6 K and a moment of 2.560 Bohr magnetons; over 80–300 K it gives −174.8 and 2.769. The two-spin fit over the same two ranges gives couplings of −60.5 and −57.0, a difference of six per cent.

So the wrong-shape model moves by forty per cent between windows and the right-shape-wrong-size model moves by six. That is the improvement the Curie–Weiss advice buys, and it is real. What it does not buy is accuracy: the six per cent spread is around a value that is itself twenty per cent wrong.

Why this is not an argument against fitting #

It would be easy to read these two failures as saying that magnetic fits are worthless, and that is not what the arithmetic says.

The two-spin fit on a two-spin sample is exact: J to five decimal places and g to six, over every window tried. When the model matches the sample, a two-parameter fit to a noiseless susceptibility recovers both parameters and does not care about the temperature range. The whole complaint about the Curie–Weiss law is that it is the wrong model for a dimer; the complaint here is that a dimer expression is the wrong model for a chain. Both are instances of one statement and it is not fitting is unreliable.

What it is, is that the fitted number reports on the model, and the model is an assumption about the structure. So the sentence this compound’s exchange coupling is −66.7 cm⁻¹ is a compressed way of saying if this compound is a pair of coupled spins, its coupling is −66.7 cm⁻¹, and the conditional is doing more work than the number.

The same structure turns up in four places, and it is worth naming them together. A resonance energy is measured from a chosen reference. A mode’s percentage is a property of a chosen coordinate set. A point group is a group at a chosen tolerance. And an exchange coupling is a coupling in a chosen model. In every case the number is quoted as a measurement, the choice is not quoted at all, and the size of what the choice moves is the quantity nobody reports.

A different magnetic quantity makes the contrast: a moment that is a thermal average over two spin states is a real number that moves with temperature, and nobody is tempted to call it a constant. The quantity fitted here moves with temperature too; the difference is that the fit reports it as a constant, and the reporting is the whole of what this essay is about.

How the exact calculation is done, and why it stops where it does #

The method is worth a paragraph because its limits are the essay’s limits.

A chain of $n$ spin-½ centres has $2^n$ states. The Hamiltonian conserves the total $z$ component, so it blocks: for eight spins the blocks are of size 1, 8, 28, 56, 70, 56, 28, 8, 1, and the largest is seventy. Diagonalising them all gives every eigenvalue with its $S_z$, and the susceptibility follows from Van Vleck’s zero-field formula — three times $g^2$ times the Boltzmann average of $S_z^2$ — which reduces to the two-spin expression exactly when $n = 2$.

The cost doubles with every spin added, and it doubles in memory as well as in time, so twelve spins is the practical end of it and eight is where the figures stop. That is not a limitation peculiar to this calculation: it is why nobody fits a real chain to an exact answer, and it is the reason the wrong model gets used in the first place.

It is worth being clear about which kind of object the fitted number is compared against. An exact many-body spectrum is every state of a small system written down and solved, with nothing variational and nothing self-consistent in it — a susceptibility computed from such a spectrum has no fitted parameter anywhere, which is what makes it a fair test of a fit rather than another fit.

What this cannot say #

Eight spins is not a chain. The whole point of exact diagonalisation is that it stops. A real chain has an infinite-length limit that this calculation is heading toward and cannot reach, and the error grows monotonically with $n$ across the sizes computed, so the numbers here are lower bounds on the error a real chain would produce.

No anisotropy and no interchain coupling. The Hamiltonian here is isotropic Heisenberg with nearest-neighbour coupling only, which is the model a chain compound is usually assumed to obey and is itself an assumption.

The g factor is isotropic and temperature-independent, which for a real transition metal ion it is not.

And there is no noise. Every point is exact. A real dataset has scatter, and scatter would make every disagreement above harder to see rather than easier — which is the direction that matters, since the whole difficulty is that a wrong model already fits well.

What was checked #

The control, twice over. The two-spin fit recovers J and g exactly from a two-spin sample over three windows; and the exact diagonalisation reproduces the Bleaney–Bowers closed form to $10^{-12}$ at four temperatures, which is what lets the first control be trusted.

Every wrong-model fit above $R^2 = 0.99$, and every one reporting a coupling out by more than five per cent.

The worst out by more than a third, with its residual quoted beside it.

The error growing monotonically with the chain length in every window, which is what makes it the model’s fault rather than the search’s.

And the g factor drifting by more than two per cent, so that the tell is present and is inside the range a real measurement would call ordinary.

The tell nobody looks at is the one that can be checked #

Three numbers of which only the last says anything is wrong, and it is the one nobody looks at is the conclusion, and the g factor deserves more than being named as the tell — because unlike the other two it can be measured independently on the same sample.

Electron paramagnetic resonance returns a g factor directly, from the position of a resonance line, to four decimal places. It is a different instrument, a different physical quantity and a different systematic error from a susceptibility measurement, and for a paramagnetic transition-metal compound both are routine.

So the check is available and costs one spectrum. Fit the susceptibility, read off the fitted g, and compare it with the resonance value. A fit that returns 1.973 for a compound whose resonance says 2.05 has been refused by a measurement rather than by an argument.

Why the drift happens at all is worth stating, because it makes the tell reliable rather than incidental. In a susceptibility fit the g factor and the coupling are correlated: g sets the overall scale of the curve and the coupling sets its shape, and when the assumed shape is wrong the fit compensates by adjusting the scale. A wrong model therefore does not leave g alone and get the coupling wrong; it drags both, and the direction of the drag is set by which way the shapes differ.

That makes the two failures a matched pair rather than two independent errors. A coupling that is too large comes with a g factor that is too small, in the case here, and neither could be corrected without the other.

Two consequences follow for reading a published fit.

A quoted g in the ordinary range is not evidence that the model is right, because the drift here — two per cent — lands inside what a real sample could plausibly have. Only a comparison against an independent measurement makes it evidence.

And a fit that holds g fixed at a measured value is a much stronger test than one that fits it. Fixing g removes the compensation, so a wrong model can no longer absorb its shape error into the scale, and the residual has to carry it — at which point the $R^2$ of 0.998 that hid the problem would have had somewhere to go.

That last is a recommendation the arithmetic here supports directly. The most informative version of this fit is the one with the fewest free parameters, because every free parameter is a place for a wrong model to hide.

Still open: odd and even rings, and how much a curve can support #

The natural open question is the ring rather than the chain, and it is interesting because the answer changes sign. A ring of an even number of spins has no ends, so its susceptibility is closer to a chain’s bulk behaviour at the same size — and a ring of an odd number has a frustrated ground state whose low-temperature susceptibility diverges rather than vanishing. Fitting a dimer expression to that would fail visibly, which is the first case where the wrong model would announce itself.

The nearer question is the one that would make all of this operational. Every fit here reports two numbers, and the sample has one parameter and a known structure. The quantity that would be worth computing is how many parameters’ worth of information a susceptibility curve actually contains — a curve with no features cannot support four fitted parameters, and the sense in which it cannot is a statement about the conditioning of the fit that could be measured rather than assumed. It is the same question an underdetermined structure raises, asked about a curve instead of about a set of constants.