The excitation that raises the spin

A susceptibility curve fitted with the four-parameter model these fits have always used — a coupling, a g factor, a monomer fraction and a temperature-independent term — has couplings at which its two nuisance parameters become uncorrelated. Their rank-free reading makes them properties of the curve rather than of how the fit’s directions are named, and every cluster with a ground spin that has been examined has two of them, both poles.

The upper one was placed by a band: across seven clusters, the separation coupling times the ground spin plus a half sat between 115 and 129 cm⁻¹. Five clusters built to test it broke it, stars drifting above and complete bipartite graphs below, and neither the curve’s departure from its ground Curie law nor the first excitation explained where each landed. What that left was a lead with a confound in it. At a ground spin of one, three clusters with first excitations of one, two and three coupling units separated in that order — but they also had four, six and eight centres, so the gap and the size rose together.

Every cluster, not a handful #

Pairs of graphs with the same number of centres, the same ground spin and different spectra would separate the two. Rather than choose a few, this takes all of them. Every connected graph on four, five and six vertices, up to relabelling — six, twenty-one and a hundred and twelve of them — is made into a cluster of spin-½ centres with one antiferromagnetic coupling on every edge, and its exchange spectrum is found by exact diagonalisation.

Most have no ground spin. Of the six-centre graphs, ninety are singlets, and a singlet’s fit has no separations of this kind. What is left is forty-four clusters with a spin: the star of four; twenty doublets and the star of five among the five-centre graphs; twenty-one triplets and the star of six among the six-centre ones. The twenty doublets are clusters with the same size and the same spin that differ in every other respect — trees, rings, bipyramids, the complete graph — and so are the twenty-one triplets.

Every one of the forty-four has two poles. That is the first thing the census says, and it has to be read with a qualification.

The band was read below 150 cm⁻¹, because the clusters it was found on all put their upper pole there. Nine of the twenty doublets do; the other eleven put it between 152 and 413 cm⁻¹. Found over a range from 3 to 1500 cm⁻¹ instead, none is missing. So the statement that every cluster has two separations below 150 was a property of the twelve clusters it was read on, and the statement that every cluster has two poles survives the census. The sign rule that holds between the two poles is therefore a rule about every spinful cluster here, with the second pole sometimes much further out than it was ever looked for.

The first excitation does not order them #

With twenty clusters of one size and one spin, the lead can be tested directly: rank the doublets by their first excitation and by their upper separation and see whether the orders agree.

They do not. The rank correlation is 0.36 among the doublets and 0.15 among the triplets. Simply counting bonds does better, at 0.43 and 0.27, and nobody would propose the bond count as the mechanism. The clearest single case is four doublets that share a first excitation of exactly one coupling unit and separate at 118, 140, 307 and 413 cm⁻¹ — a factor of three and a half with nothing in the first gap to distinguish them.

The trouble with the first excitation is that it does not say what the excitation does to the magnetism. In several doublets the first excited level is another doublet: populating it adds states with the same spin as the ground state, which at the temperatures that matter leaves χT per centre almost where it was. In others it is a quartet, which raises χT as it fills. In the triplets the first excited level is often a singlet, which lowers χT. Three kinds of excitation that move the curve in three different ways were being ranked by one number.

The excitation that raises the spin does #

The quantity the fit’s nuisance parameters compete over is a curve that rises across the measuring window. The temperature-independent term contributes a χT that grows linearly with temperature; the monomer contributes a constant. A cluster’s own χT rises across the window only when it thermally populates multiplets of higher spin than its ground state. So the candidate that suggests itself is not the first excitation but the first excitation that raises the spin.

It orders them. Among the doublets its rank correlation with the separation is 0.86; among the triplets 0.68. On logarithmic axes the forty-four clusters fall along a line of slope minus one — the separation times the raising gap is roughly constant — and the median product is 359 cm⁻¹ for the doublets and 351 for the triplets: two groups of different size and spin landing within three per cent of each other. The three stars, of spin one, three halves and two, give 345, 287 and 282.

The spreads say the same thing as a single number each. The upper separation alone spans a factor of 3.59 among the doublets. Multiplied by the first excitation it spans 9.32 — the first excitation makes the clusters look more different, not less. Multiplied by the raising gap it spans 1.50. Among the triplets the three spreads are 5.70, 22.93 and 2.15.

The pairs the lead asked for #

The census contains the pairs that separate size from spectrum, and more usefully it contains pairs that separate the two spectral candidates from each other.

The four doublets with a first excitation of exactly one split in two by their raising gap. The two whose first excitation is itself a quartet, at one coupling unit, separate at 307 and 413 cm⁻¹; the two whose first excitation is a doublet, with the first quartet at three, separate at 118 and 140. The seven doublets whose first quartet sits at three coupling units, with first excitations of one, two or three, separate between 115 and 153 cm⁻¹ whatever the first excitation is.

So at fixed size and spin the separation follows the spin-raising gap, and follows the first excitation only when the two happen to be the same level.

The three spin-one clusters that suggested the lead are worth reading in this light. The star of four, $K_{2,4}$ and $K_{3,5}$ have first excitations of one, two and three coupling units, and every one of those is a singlet: each lowers the spin. Their first quintet, the level that raises it, sits at four coupling units in all three. So the raising gap says they should separate at nearly the same coupling, and they do: 86.2, 82.7 and 75.9 cm⁻¹, twelve per cent from first to last. The order the lead noticed is real, and it is a twelve-per-cent ordering riding on a rule that puts all three in the same place — the size of the residual the raising gap leaves everywhere else.

The triplets make the point harder #

The six-centre triplets are a stronger test than the doublets, because their first excitation is usually a singlet — a level that lowers χT as it fills — and it can be very low. One triplet has a singlet only 0.172 coupling units above its ground state, the smallest first excitation in the census; another has one at 0.214. If the first excitation placed the separation, those two would sit far out on the coupling axis, since a small gap needs a large coupling before its level is thermally significant. They separate at 119.8 and 97.9 cm⁻¹, in the middle of the triplets’ range.

What they share with the rest of the triplets is the level above: their first quintets sit at 2.438 and 3.432 coupling units, ordinary values, and their products with the separation, 292 and 336 cm⁻¹, are ordinary too. The singlet fills and empties across the window, bending the curve downward, and the fit absorbs that into the coupling and the g factor without its nuisance parameters noticing. A level that lowers the moment competes with the physical parameters; a level that raises it competes with the nuisance ones. That is the whole distinction the first excitation cannot draw, and the triplets are where it matters most, because their lowest level so often points the wrong way. Their rank correlation with the first excitation, 0.15, is essentially none.

The same distinction explains why purity renamed the poles rather than moving them. A monomer fraction adds a free spin’s Curie constant, which raises χT at the bottom of the window and changes nothing about which multiplets of the cluster fill where. The separations for spins of one and above barely moved with the fraction because the fraction does not touch the raising gap.

What the band was a shadow of #

The band’s variable was S + ½, and it worked for seven clusters and failed for five. The raising gap explains both. For a star of n centres the first spin-raising excitation is n coupling units, and its ground spin is (n − 2)/2, so the raising gap and S + ½ both grow with n and in nearly the same proportion; for the complete bipartite graphs the raising gap is also tied to the sizes of the two parts. In those two families S + ½ was a stand-in for the raising gap, correlated with it by construction, and the band held as long as the clusters were drawn from families where the correlation held. The stars’ products drifting one way and the bipartite graphs’ the other was the stand-in coming apart from the thing it stood in for.

The raising gap does not carry the whole answer. A spread of one and a half is not one, and it is largest among the frustrated doublets: the complete graph on five centres and one of the bipyramids share a first quartet at three coupling units with sixteen states in it and separate at 116 and 153 cm⁻¹. The number of states in the multiplet, and what lies above it, move the separation within the factor the raising gap leaves. Those are second-order effects on a first-order rule, and the rule is the part the census establishes.

A number worth carrying: 355 cm⁻¹ #

The median products, 359 and 351 cm⁻¹, say where the upper separation falls in physical terms. It falls where the first higher-spin multiplet lies about 355 cm⁻¹ above the ground state. At the top of the measuring window, 300 K, the thermal energy kT is 208.5 cm⁻¹, so that multiplet is about 1.7 kT up: close enough to start filling across the upper part of the window, far enough that it is only starting. A multiplet lower than that fills early and makes χT curve over the whole window; one higher barely fills at all. The separation is where the rise it produces is shaped most like the temperature-independent term’s straight line, which is where a fit can least tell the two apart.

That is offered as the mechanism rather than proved. What is measured is the product; the reading of it as a position relative to the window is consistent with the window rescaling every separation, which the purity sweep had already found, and it predicts that a window ending at a different temperature would move the constant in proportion.

What was computed, and how #

The graphs are every connected graph on four to six vertices, generated from edge subsets and kept once per isomorphism class by canonical relabelling. Each is a Heisenberg cluster of spin-½ centres with an equal antiferromagnetic coupling on every edge, diagonalised exactly in every sector of total $S_z$; multiplets are levels grouped to 10⁻⁷, each labelled by its largest $|S_z|$. The first excitation is the lowest level above the ground multiplet; the first spin-raising excitation is the lowest multiplet with a larger spin than the ground state.

The separations are read exactly as the band’s were: the four-parameter model’s logarithmic Jacobian over forty temperatures from 20 to 300 K at one per cent precision and two per cent monomer, the correlation between monomer fraction and temperature-independent term built from all four singular directions, and every sign change located by thirty bisections — here on a logarithmic grid of 160 couplings from 3 to 1500 cm⁻¹ rather than up to 250, so that no upper pole is cut off. The upper separation is the second, and in every spinful cluster it is a pole.

The claims are stated where they can fail: that the census holds six, twenty-one and a hundred and twelve graphs; that every spinful cluster has an upper separation and that it is a pole; that fewer than half the doublets have it below 150 cm⁻¹; that at each of the two sizes the product with the raising gap spreads less than half as much as the separation and less than a third as much as the product with the first excitation, and ranks the clusters better than the first excitation or the bond count. The refusal is the star of six, found among the hundred and twelve by enumeration: its upper separation has to be the 46.93 cm⁻¹ the band’s own calculation gave it, to a per cent, and it is.

Where the census stops #

One coupling on every edge, spin-½ centres, six at most. Real polynuclear complexes have unequal couplings and centres of higher spin, and seven or more centres are where a fit with a parameter for everything starts reporting things the sample does not contain. The census is complete inside its bounds and says nothing outside them.

One window and one monomer fraction. The product’s value is tied to a window ending at 300 K, and the reading above predicts it moves with the window; that prediction is not tested here.

And ideal data. The separations are properties of a noiseless curve. A real measurement with a drifting background decouples its nuisance parameters wherever its background says, and the moment a fit invents a parameter to absorb it the separation stops being a property of the cluster.

Two numbers that looked like one #

The habit this calls for is to ask of a spectral predictor what the level does, not only where it is. The first excitation was a natural candidate, and in the families it was noticed in, it was the right level. It was right because in stars and complete bipartite graphs the first excited multiplet always raises the spin. A census that included clusters whose first excitation lowers the spin, or leaves it alone, separated the level that happens to be lowest from the level that does the work, and the rank correlation went from convincing to worse than counting bonds.

It is the same mistake the band made one level up. S + ½ and the raising gap move together in stars and bipartite graphs, so a rule stated in one looked like a rule stated in the other, and only a sample outside those families could tell them apart. Both times the fix was the same: stop choosing clusters to test the rule, and take every cluster there is.

Still open: the second-order spread, and the window #

The obvious open question is the factor of one and a half the raising gap leaves. It is largest among frustrated doublets whose first quartet is highly degenerate, and the candidate refinement is the population the multiplet carries — its number of states times the change in Curie constant it makes — rather than its energy alone. A one-parameter correction of the form “energy minus kT times the logarithm of the weight” can be fitted on the doublets and tested on the triplets, which is the discipline the band lacked.

The nearer question is the window. If the separation sits where the raising multiplet is about 1.7 kT above the ground at the window’s top, then a window ending at 200 K should put it at about 240 cm⁻¹ and one ending at 400 K at about 475, for every cluster at once. That is one re-run of the census at two windows, and it would turn the reading of 355 cm⁻¹ as a position into a measurement or refute it.