Five more clusters break the band

A four-parameter fit to a spin cluster’s magnetic susceptibility carries two nuisance parameters, the fraction of sample present as free monomer and a small temperature-independent term, and the curve cannot fix them separately. Their correlation changes sign as the exchange coupling changes, and it does so at a handful of separation couplings: every cluster with a ground spin has two below 150 cm⁻¹, and none with a singlet ground state has any.

Those separations came with a regularity that looked too clean to be an accident. Multiply the upper one by the ground spin plus a half and seven clusters — three doublets, two of spin one and stars of spin three halves and two — land between 114.7 and 129.3 cm⁻¹, a spread of under thirteen per cent across a factor of nearly three in the coupling itself. Varying the sample’s purity from half a per cent to sixteen widened that band and did not break it. A regularity that survives a factor of thirty-two in the one parameter a sample chooses is the kind that gets written down as a rule for where to expect trouble in a fit.

It was found on seven clusters, and seven clusters chosen for a different question. They were picked so that ground spin, count parity and frustration could be told apart, and they run from four centres to six. Nothing about that choice tests whether the ground spin places a separation. Two tests do, and both are made here. The first asks what the band would mean if it were a law, and checks the mechanism directly. The second builds clusters the band has never seen, predicts where their separations should fall, and computes where they do.

Seven curves at their separations do not look alike #

The obvious mechanism is the one the purity sweep left standing. A monomer impurity contributes a free-spin Curie tail, and the temperature-independent term contributes a constant. A cluster’s own susceptibility competes with both. At low coupling the cluster is paramagnetic across the whole window and looks like a sum of free spins; at high coupling it has frozen into its ground multiplet and looks like a single Curie paramagnet of spin S. Somewhere between, the fit’s two nuisance directions trade their correlation. The natural guess is that the upper separation marks the coupling at which the cluster’s curve has become enough like its ground multiplet’s Curie law — a fixed tolerance, the same for every cluster, beyond which the monomer tail and the cluster tail are the same shape and only their sizes differ.

If that were so, the band would follow almost for free. A ground multiplet of spin S carries a Curie constant proportional to S(S+1), and the excited multiplets above it lie at energies proportional to the coupling; the coupling needed to push them far enough out of a fixed window would then scale with some simple function of S. The guess makes a sharp prediction that does not depend on any of that algebra: at every cluster’s upper separation, the ratio of its χT at the top of the window to its ground Curie value should be one number.

It is not one number, or anything near one. At their separations the three doublets sit at 2.39, 2.01 and 2.37 times their ground Curie value, which is to say the top of the window still sees a cluster more than twice as paramagnetic as its ground doublet. The star of four and $K_{2,4}$, both of spin one, sit at 1.08 and 1.33. The star of five sits at 0.81 and the star of six at 0.63, below their ground Curie values, because in a star the first excited multiplet has a smaller spin than the ground one and populating it takes susceptibility away. The ratio runs from 0.63 to 2.39 across clusters that share a band product to within thirteen per cent. There is no tolerance that all seven separations sit at.

The whole curves make the point more plainly than the endpoints. Each cluster’s χT, divided by its ground Curie value and drawn across the 20–300 K window at that cluster’s own upper separation, starts within six per cent of one at the bottom of the window. From there the doublets climb steadily, the two spin-one clusters dip or hold and then rise a little, and the two larger stars dip and stay below. Seven curves, each at the coupling where the fit’s nuisance parameters decouple, are at no common stage of their approach to anything.

That rules out the mechanism, but not the band. A regularity can hold for reasons nobody has named, and the separations are defined by a four-direction singular value decomposition whose correlation depends on the whole curve and its derivatives with respect to all four parameters. A shape criterion on χT alone was always a simplification. The band itself has to be tested on its own terms, by prediction.

Five clusters the band had never seen #

The test clusters are chosen to push the band where the original seven could not. All twelve are built from spin-½ centres coupled antiferromagnetically with one exchange constant, and the ground spin of each follows from its graph: a star of n centres has one hub coupled to n − 1 leaves, and its ground spin is (n − 2)/2; a complete bipartite graph $K_{m,n}$ couples every centre of one set to every centre of the other, and its ground spin is |m − n|/2. So a star of seven has ground spin five halves and a star of eight spin three, both above anything in the band. $K_{2,5}$ has spin three halves and $K_{2,6}$ spin two, the same as the stars of five and six but with a different spectrum above them. $K_{3,5}$ has spin one, like the star of four and $K_{2,4}$, on eight centres rather than four or six.

For each, the band predicts a range for the upper separation: 114.7 to 129.3 cm⁻¹ divided by the ground spin plus a half. Every exchange Hamiltonian is then solved exactly, sector by sector in the total spin projection, and the fit’s two nuisance parameters are followed across the coupling at two per cent monomer until their correlation changes sign.

One of the five lands inside. The star of seven, predicted at 38.2 to 43.1 cm⁻¹, separates at 42.4. The star of eight, predicted at 32.8 to 36.9, separates at 40.1 — 8.5 per cent above the top of its range. $K_{2,5}$, predicted at 57.4 to 64.6, separates at 50.5, 12.0 per cent below the bottom. $K_{2,6}$, predicted at 45.9 to 51.7, separates at 36.0, 21.6 per cent below. $K_{3,5}$ misses by the smallest margin, predicted at 76.5 to 86.2 and separating at 75.9, 0.7 per cent below.

The failure is not a failure of the phenomenon. Every one of the five new clusters has exactly two separation couplings below 150 cm⁻¹, both poles of the third direction’s exponent at two per cent, exactly as a spinful cluster should. The lower ones sit at 4.5, 3.7, 7.0, 5.2 and 9.6 cm⁻¹ and fall with ground spin as the originals’ do. What the ground spin does not do is place the upper one to thirteen per cent. Across all twelve clusters the band product runs from 90.0 to 140.3 cm⁻¹, a spread of 1.56 where the seven gave 1.13.

The two families move in opposite directions #

Misses scattered around the band would say the band was noisy. These are not scattered, and the families figure at the head of this essay is the reason to believe the band was never a law. Put each cluster’s product against its ground spin and join the clusters of one kind.

The stars, from four centres to eight, run 129.3, 114.7, 117.3, 127.2 and 140.3 cm⁻¹ — down from the star of four to the star of five, and then steadily up. The $K_{2,n}$ graphs, from $K_{2,3}$ to $K_{2,6}$, run 114.8, 124.0, 100.9 and 90.0 — up from $K_{2,3}$ to $K_{2,4}$, and then steadily down. The two families cross inside the band, and the seven clusters that defined it sit at or near the crossing. Four and five centres, ground spins of one half to two, is exactly the range in which a rising curve and a falling one are close together. Extend either family by two centres and they are fifty cm⁻¹ apart.

The one member of neither family among the new clusters, $K_{3,5}$, lands just below the band at 113.9. That is the near miss that would have been reported as agreement had it been the only test, and it is worth noticing that the band’s apparent success on five-centre doublets — the ring of five, the trigonal bipyramid and $K_{2,3}$ at 126.1, 122.9 and 114.8 — is of the same kind. Three clusters with the same number of centres and the same ground spin agreeing to ten per cent is a statement about clusters of five, not about ground spin.

A ground spin shared across families makes the same point one row at a time. At spin three halves, the star of five separates at 57.4 cm⁻¹ and $K_{2,5}$ at 50.5. At spin two, the star of six separates at 46.9 and $K_{2,6}$ at 36.0. At spin one, the star of four, $K_{2,4}$ and $K_{3,5}$ separate at 86.2, 82.7 and 75.9. Same ground spin, same Curie constant for the ground multiplet, and separations as much as thirty per cent apart. The ground spin is a large part of where a separation falls, which is why a band appeared at all, but it is not the whole of it.

Nor does the first excitation place it #

If the ground multiplet’s size is not enough, the next candidate is the spectrum just above it. A separation is a statement about how the cluster’s curve changes shape across the window, and the shape is set by how many excited multiplets the window reaches. The first excitation is the one number that governs that at the bottom of the window, and it is the number a spectroscopist would reach for before any fit.

In units of the coupling it is fixed by the graph: one coupling unit in every star, two in $K_{2,4}$, $K_{2,5}$ and $K_{2,6}$, three in $K_{2,3}$ and $K_{3,5}$, 2.24 in the ring of five and two in the trigonal bipyramid. Multiplied by each cluster’s own upper separation, it gives the first excitation in cm⁻¹ at the coupling where the nuisance parameters decouple.

At the twelve separations the first excitation runs from 40 cm⁻¹ on the star of eight to 345 cm⁻¹ on $K_{2,3}$, a factor of more than eight. A 300 K window corresponds to about 209 cm⁻¹ of thermal energy, so some clusters decouple their nuisance parameters with their first excited multiplet well inside the window and others with it well above the top. The star of five and $K_{2,3}$ make the sharpest pair: band products of 114.7 and 114.8 cm⁻¹, a match to a tenth of a per cent, at first excitations of 57 and 345 cm⁻¹, a factor of six apart. Whatever sets the separation, it is not the gap alone either.

The within-spin rows do show something ordered, and it is the most useful thing the test leaves behind. At spin one, the star of four, $K_{2,4}$ and $K_{3,5}$ have first excitations of one, two and three coupling units, and their separations fall in the same order — 86.2, 82.7, 75.9. At spin three halves and at spin two, the star (one unit) separates above the $K_{2,n}$ graph (two units) both times. At fixed ground spin, the larger the first excitation in coupling units, the lower the coupling at which the nuisance parameters separate — but by much less than the ratio of the gaps, so neither the coupling nor the gap is what is held fixed. Those clusters also differ in how many centres they have, so the ordering cannot yet be assigned to the gap rather than the size, and three rows of two or three clusters are a pattern to test, not a result.

How the clusters were built and read #

The model is the four-parameter model these fits have always used: the exact susceptibility of the cluster, scaled by the g factor squared and by one minus the monomer fraction, plus a free-spin Curie term for the monomer and a temperature-independent term. The design is the logarithmic Jacobian of forty χT values from 20 to 300 K with respect to the four parameters at one per cent precision. Couplings are in wavenumbers throughout, the unit the exchange Hamiltonian is written in.

A separation is found without reference to how the fit’s directions are ranked. The correlation between the monomer fraction and the temperature-independent term, built from all four singular directions, is evaluated at a hundred couplings spaced logarithmically from 3 to 250 cm⁻¹; each change in its sign is bisected thirty times; and the third direction’s exponent at the result says whether the separation is a pole or a zero. That is the rank-free reading the purity sweep needed, where following the third direction by continuity reported poles vanishing that had only been renamed.

The ground Curie value is read off each cluster’s own curve at 2 K, at its separation coupling, where every excited multiplet is frozen out — so the normalisation carries no assumption about g or about how the susceptibility is scaled per centre. The first excitation is the lowest level above the ground multiplet in the exchange spectrum, in coupling units, times the separation coupling.

The checks, run wherever these figures are drawn: the seven original clusters reproduce the band, between 110 and 135 cm⁻¹ with a spread under 1.15; at their separations χT at the top of the window lies below 0.7 of the ground Curie value for at least one cluster and above twice it for another; the first excitation there varies more than eightfold across all twelve; at least three of the five new clusters fall more than five per cent outside the band; the star of eight lies above it and $K_{2,6}$ below; and every new cluster has exactly two separations below 150 cm⁻¹, both poles. The refusal is the instrument itself. The star of five is built a second time from its bond list inside the new calculation and must give the separation already found for it to within a per cent, or the new clusters would be read by a different routine from the one that found the band.

What the test does not reach #

Every cluster is built from spin-½ centres with one coupling. Real polynuclear complexes have unequal couplings and centres of higher spin, and a star with a slightly weaker hub coupling is not a star in the sense used here. The families are graph families, and the band’s failure is shown on idealised members of them.

One monomer fraction, one window, one g factor. The purity sweep showed the separations barely move with the fraction for spins of one and above, and the window was already known to rescale them. Nothing here tests whether the families’ divergence is itself window-dependent — whether a wider window moves the crossing to larger clusters.

Five new clusters are five. They were chosen as the smallest extension that could break the band in both directions, and they did; they do not map either family’s curve beyond eight centres. A spin Hamiltonian for a larger cluster is often replaced by a Curie–Weiss law at that point, and a model with a parameter for everything reports clean numbers for things the sample does not contain, so the extension has to be made exactly or not at all.

And the fit is to exact data. A real measurement’s background misbehaves with temperature, which moves where the nuisance parameters decouple. The separations here are properties of an ideal curve.

A band is the shadow of where two curves cross #

The general lesson is about testing a regularity on the sample it was found in. Seven clusters, chosen for a different purpose, gave a product that sat within thirteen per cent across a factor of three in coupling, and survived a factor of thirty-two in impurity. Every one of those was a genuine test that could have broken it, and none was the test that mattered, because all of them left the clusters the same. The regularity was a coincidence of range: two families of graphs whose products move in opposite directions with size happen to cross between four and six centres, and a set of clusters confined to that range sees a band.

The second lesson is about what the band had been standing in for. It made the ground spin look like the one number that places a fit’s trouble, and ground spin is still most of the answer — a star of eight and a star of four differ by a factor of two in their separations, and that is mostly spin. What the test adds is that the spectrum above the ground multiplet matters at fixed spin, in an ordered way, and that neither the curve’s distance from its Curie law nor its first gap is that order by itself.

Still open: what orders the separations at fixed spin #

The within-spin ordering is the lead. At spin one, three clusters with first excitations of one, two and three coupling units separate in that order, and at spins three halves and two the star sits above the $K_{2,n}$ graph. Those three clusters have four, six and eight centres, so gap and size rise together and cannot be told apart. Separating them takes pairs of graphs with the same number of centres and the same ground spin but different first excitations, and a handful of such pairs would say whether the order is spectral or merely size.

The nearer question is whether the families’ curves have a common form. The stars’ products fall and then rise, the $K_{2,n}$ products rise and then fall, and both turn between five and six centres. If each is a smooth function of the ground spin and the first excitation together, a fit that measures one combination of parameters and not another would have its own rule for where its nuisance parameters decouple — and a rule tested, this time, on clusters outside the range it was found in.