The same sample, fitted over four temperature ranges

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

Ten essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The moment a fit invents

The inverse susceptibility of a pair coupled at −50 cm⁻¹, which is a curve, with the four straight lines a Curie–Weiss fit produces over four temperature ranges. Each line is extrapolated back to the axis, and where it crosses is the Weiss temperature the experiment would report.

The same exact susceptibility fitted over four temperature ranges. The moment runs from 2.471 to 3.590 Bohr magnetons and the Weiss temperature from −51.4 to −292.3 K. The final column is the moment the sample actually has in the middle of each range, which agrees with none of them.

The moment the coupled pair actually has, against temperature, with each fit’s single reported number drawn across the range it was fitted over. The sample’s moment rises smoothly from nearly nothing at low temperature towards the uncoupled pair’s value; each fit replaces that curve with one number.

The model is what is fitted

Four samples, all coupled at −50 cm⁻¹, all fitted with the two-spin expression over the same 80–600 K range. The two-spin sample returns its own coupling exactly. The chains return −55.3, −58.6 and −60.5, and the error grows with the length — so it is the model’s rather than the fit’s.

The reported coupling divided by the sample’s, for four chain lengths over three temperature ranges. Every row runs to the right as the chain lengthens; every chain runs further right as the range widens. The two-spin sample sits on one in all three, which is the control.

The Curie–Weiss picture, which is the same failure one model further out. The exact inverse susceptibility of a dimer is not a straight line, and four straight lines drawn through different parts of it all fit well and all report different Weiss temperatures. Replacing the straight line by a curve with the right shape for the wrong system does not change the character of the problem; it changes how badly it fails.

How many parameters a curve is worth

What the two-spin fit established, and the reason this essay is about the right model rather than the wrong one: fitting a chain with the two-spin expression returns a coupling that is wrong by a definite factor and reports nothing. Everything here uses the correct model, and the difficulty is still there.

The visible half of the same difficulty, in the crudest fit there is. One exact curve for a pair coupled at −40, fitted with a Curie–Weiss law over four windows: the Weiss temperatures come back at 41.2, 1218.0, −69.0 and 203.2 kelvin, and three of the four cannot return a moment at all. Where a window carries no information about a parameter, what comes back is not a wrong number with a wide error bar — it is whatever the arithmetic happened to land on.

The product a curve measures

Above, each parameter alone; below, each combination. The best combination is twenty-five times better fixed than the best parameter.

The curve as fitted, the curve with all four parameters moved along the direction the fit does fix, and the curve with them moved by the same step along the free one.

The four combinations with their uncertainties, and the four numbers a paper would print instead.

The sign a frustrated ring changes

The power the temperature-independent term carries in the free combination, for an open chain and for rings of five to ten spins.

The size of the exponent among the frustrated rings, against the number of spins, with the even rings beside them.

The worst relative uncertainty among the four parameters, from a curve of one per cent precision.

It was the count, not the frustration

The free combination’s exponent against the number of spins, for open chains and for rings of four to nine.

Every odd system’s exponent, with the frustrated ones marked; the three chains are not frustrated and carry the same sign.

At each count, how far apart the ring’s exponent and the chain’s are, against the gap that separates the two parities.

The frustrated cluster with an even count

The chain-and-ring scan: two topologies, six counts, and the sign following the parity on both. Every case in it is a chain or a ring, and among those frustration and odd parity cannot be separated.

The four clusters that are frustrated and have an even spin count. The frustration account predicts a negative exponent for each; the parity rule predicts a positive one.

The magnitude of each exponent, grouped by spin count, with the frustrated clusters marked. Within a count the topology moves it by up to 38 per cent and never moves the sign.

The sign rule holds between two poles

Every cluster’s exponent at the coupling all earlier readings used, with its count’s parity, whether it is frustrated, and its ground spin. The two even clusters with a ground spin of one are negative.

Every cluster’s sign across couplings from 2 to 1000 cm⁻¹, with each change of sign marked. The dashed lines bound the only band on which the spin rule is right for all sixteen; the solid line is the coupling every earlier reading used.

The star of four passing through its upper pole: the monomer component of the third direction crossing zero, the correlation between the two nuisance parameters crossing zero at the same coupling, and the exponent passing through infinity.

Purity renames the poles

For each spinful cluster, the two couplings below 150 cm⁻¹ at which the nuisance parameters are uncorrelated, against the monomer fraction.

A ring of five’s third-direction exponent near its lower separation, at two and eight per cent monomer.

For each singlet cluster, its separation couplings below 150 cm⁻¹ at each monomer fraction.

Five more clusters break the band

The upper separation times the ground spin plus a half, against ground spin, for the seven clusters that defined the band and the five built to test it.

χT at 300 K over the ground multiplet’s Curie value, for each of the seven clusters at its own upper separation.

χT over the ground Curie value across the window, for each cluster at its own upper separation.