Purity renames the poles

A magnetic susceptibility curve of a spin cluster is usually fitted with four parameters: the exchange coupling, the g factor, the fraction of the sample present as uncoupled monomer, and a small temperature-independent term. The last two are nuisances — nobody fits a susceptibility to learn them — and the curve cannot fix them separately. The fit’s least-determined directions lie almost entirely in their plane.

The sign of that plane’s structure was followed across the coupling for sixteen clusters, and it changes. It changes through poles, at couplings where the two nuisance parameters become uncorrelated, and every cluster with a ground spin has two of them below 150 cm⁻¹. Their positions were placed by the ground spin and the temperature window — couplings here are in wavenumbers, the unit the exchange Hamiltonian is solved in, so a coupling of 50 cm⁻¹ is 72 K — and multiplied by the ground spin plus a half, every upper pole landed between 114.5 and 129 cm⁻¹.

Every one of those poles was found with the monomer fraction held at two per cent. That is the one parameter a sample chooses rather than a chemist, and it is rarely known to better than a factor of two. If the poles moved with it, a compound of unknown purity would not know which side of its own pole it was on, and the sign of any correlation reported from its fit would be a property of its synthesis.

The separations stay where they are #

Swept from half a per cent monomer to sixteen — a factor of thirty-two — every cluster with a ground spin keeps exactly two separation couplings below 150 cm⁻¹ at every fraction, and they barely move. The star of five and the star of six, with ground spins of three halves and two, move by under 1.6 per cent. The star of four and the complete bipartite graph on two and four, both of ground spin one, move by under 3.1 per cent. The three doublets move more: the ring of five by 10.7 per cent at its lower separation and 9.5 at its upper, the trigonal bipyramid by 11.0 and 7.9, the graph on two and three by 9.9 and 9.4.

So the question has a clear answer. A sample’s position relative to its separation couplings is decided by its exchange coupling, and a factor of thirty-two in its purity moves the boundary by at most a tenth. For a cluster of ground spin one or more the boundary is, for practical purposes, a property of the cluster and the window alone.

The movement is also systematic in direction. The upper separations all rise with the monomer fraction and the lower ones all fall, so the band between them widens: a more impure sample has a slightly wider range of couplings on which its sign sits between its two separations. The doublets widen most, and they are also the clusters whose Curie tail most resembles the monomer’s own — a spin-½ cluster at low temperature and a spin-½ impurity both follow a Curie law with the same constant per spin, and a fit separates them only by how the cluster departs from that law as it warms.

But the same place gets a different name #

The obvious way to follow a pole across the monomer fraction is to do at each fraction what was done at two per cent: follow the third singular direction along the coupling and find where its monomer component changes sign. Done that way, the poles appear to vanish. At eight per cent monomer the ring of five reports no poles at all.

Nothing vanished; it was renamed. The third and fourth directions both lie in the nuisance plane and are orthogonal, so their exponents — the ratio of each direction’s temperature-independent component to its monomer component — are negative reciprocals. A pole in one is a zero in the other. Which of the two is called third is decided by which singular value is the larger, and near a separation the two singular values are close. Change the monomer fraction and they can trade places. At two per cent the ring of five’s lower separation is a pole of the third exponent at 20.1 cm⁻¹; at eight per cent the same separation, at 19.0 cm⁻¹, is a zero of it. The exponent passes smoothly through nothing where it used to run off to infinity.

The quantity that does not depend on the naming is the correlation between the two nuisance parameters, which the four directions together determine and which crosses zero at a separation whatever the ranks. Following that, every separation is found at every fraction, and each is then labelled by what the third exponent does there.

The labels change on a schedule set by the ground spin. Every doublet’s lower separation is a zero by four per cent; the two clusters of spin one keep both as poles through six per cent, and lose the first at eight and sixteen; the stars of spin three halves and two keep their lower separations as poles all the way to sixteen and see only their upper ones renamed there. Three of the hundred and twelve separations in the whole sweep are neither — the exponent within a factor of three of one — and all three sit at the fraction where their cluster’s two directions are trading places.

That matters for anything reported as a sign. A published correlation between monomer fraction and temperature-independent term, from a fit of a doublet at five per cent impurity, would carry the opposite sign convention from the same fit at one per cent — with the physics unchanged and the naming of two nearly equal directions swapped. The sign reading itself was already known to be ambiguous between a pole and a zero; purity is what decides which ambiguity a given sample gets.

Singlets acquire separations of their own #

The ground-spin account had a second half: clusters with a singlet ground state have no separation coupling in the low regime at all, because there is no ground-state Curie tail for the monomer’s to compete with. At two per cent that is exactly true for all nine singlets.

It stops being true at a few per cent. The octahedron and the fan of six acquire two separations each at six per cent. The open chain of four, the square with a diagonal, the tetrahedron, the ring of six, the triangular prism and the balanced bipartite graph acquire theirs at eight. The ring of four, alone, holds out to sixteen. Every separation a singlet acquires is a zero of the third exponent rather than a pole.

The mechanism is the one the ground-spin account named, running the other way. A singlet cluster’s susceptibility vanishes at low temperature, so a monomer impurity’s Curie term dominates the bottom of the curve, and once the impurity is large enough its tail plays the part a ground spin plays in a spinful cluster. The monomer fraction becomes, in effect, the sample’s ground spin, and the fit’s two nuisance directions acquire separations where the cluster itself has none.

The order in which the singlets acquire them is not the order of their gaps. The octahedron’s first excited level is two exchange units up and the fan’s 1.24; they go first together. The ring of four’s gap is two units too and it goes last. Whatever sets the onset, it is not the one number a spectroscopist would reach for first.

The regularity widens and does not break #

The product that made the ground spin look like the thing placing the upper pole was the upper coupling times $S + \tfrac12$. At two per cent it lies between 115 and 130 cm⁻¹ for the seven clusters. At half a per cent the range is 113 to 129; at sixteen per cent, 116 to 138. The doublets rise by about a tenth across the sweep and every higher spin by one or two per cent, so the band widens upward and keeps its members.

That is a useful constraint on what the regularity can mean. Whatever makes the upper separation scale as the reciprocal of $S + \tfrac12$ is a property of the cluster’s spectrum measured against the window, and it is nearly indifferent to how much free-spin paramagnet is mixed in. The most natural candidate — that the upper separation is where the cluster’s own curve becomes indistinguishable from a Curie law of its ground multiplet — cannot be tested by replacing the cluster with such a Curie law, because a pure Curie paramagnet has no coupling and so no separation in the coupling to find. What would test it is the departure from that law, which is a different calculation and is not made here.

The movement ordered by ground spin is the clearest summary. The three doublets move eight to eleven per cent; the two spin-one clusters under three; the two highest spins under one and a half. A larger ground spin gives a Curie tail that dwarfs a free spin’s, so the monomer term perturbs the separations less, and the same size ordering governs how late their names change.

How the separations were followed #

The model is the one these fits have always used: the exact susceptibility of the cluster from its exchange Hamiltonian, scaled by the g factor squared and by one minus the monomer fraction, plus a free-spin Curie term for the monomer fraction and the temperature-independent term. The design is the logarithmic Jacobian of forty χT values between 20 and 300 K with respect to the four parameters, at one per cent precision, and its singular value decomposition gives four directions.

At each of six monomer fractions — 0.5, 2, 4, 6, 8 and 16 per cent — the coupling is swept over 110 points spaced logarithmically from 3 to 250 cm⁻¹, and at each point the correlation between the monomer fraction and the temperature-independent term is computed from all four directions and their singular values. A change in its sign is bisected thirty times. At the result the third direction’s exponent is read: larger than three in size, the separation is a pole of it; smaller than a third, a zero; otherwise neither.

The spectra behind every curve come from the exchange Hamiltonian built over each cluster’s bond list and diagonalised exactly, sector by sector in the total spin projection, so nothing in the susceptibility is approximated before the fit analysis begins. The clusters are the sixteen of the pole calculation: rings, a chain, polyhedra, stars and complete bipartite graphs of four to six spin-½ centres, chosen there so that ground spin, count parity and frustration could be separated, and the count and not the frustration turned out to be what an earlier sign had tracked.

Two earlier versions of this sweep are worth recording because both reported something false. The first followed the third direction along the coupling by keeping its dot product with the previous step positive, which is how the poles were located at two per cent; at higher fractions, where the two nuisance directions swing past each other within a step, it reported pairs of poles disappearing. The second fixed the direction’s orientation by the sign of its temperature-independent component, on the ground that this component never changes sign at two per cent; at higher fractions it does, and the sweep reported odd numbers of poles on singlets, which no smooth function of the coupling can have. Both were reading the renaming as a change in the fit, and the correlation was chosen because it cannot be.

The checks, run wherever these figures are drawn: at two per cent every spinful cluster’s two separations are poles, and the upper one is the published pole to within a per cent — not exactly, because the correlation involves all four directions and the published pole was located on the third alone; every spinful cluster keeps exactly two separations at every fraction; they move by under thirteen per cent; the upper one times $S + \tfrac12$ stays between 110 and 145 cm⁻¹; every doublet’s lower separation is renamed by four per cent and no higher spin’s before eight; no singlet has a separation at two per cent and every singlet acquires one between six and sixteen. The refusal is the naming itself: at two per cent every separation must be decisively a pole or a zero, by a factor of ten, or the published poles were never poles — and the labels that are neither must appear only above it.

What the sweep leaves fixed #

One g factor, one temperature-independent term, one window. Only the monomer fraction was varied. The window was already known to move the separations in proportion; the other two nuisance-adjacent values were held at 2 and 2×10⁻⁴.

The impurity is modelled as free spin-½. That is the same kind of idealisation that lets a Curie–Weiss fit invent a moment: whatever the real impurity is, a parameter with a physical name will absorb it, and the fit will report a clean number for something the sample does not contain. A real impurity can be a fragment with its own coupling, or a different oxidation state with a different spin, and neither is a Curie term of the monomer’s constant.

Six fractions are six points. Where a label changes between four and six per cent, or a singlet’s onset lies between six and eight, the true threshold is somewhere inside that interval and was not bisected; the schedule above is a schedule of which side of each sampled fraction a cluster is on.

And the fit is to exact data. The design says how well an ideal curve determines the parameters; it does not include noise that is correlated with temperature, which is how a real measurement’s background usually misbehaves. What a curve measures was established under the same idealisation.

Two names for one place #

The general lesson is about naming by rank. A singular value decomposition sorts its directions by size, and sorting is discontinuous wherever two sizes cross: an object followed by its rank can change identity without changing anything physical. A quantity that is to be tracked across a parameter should be defined without reference to an ordering — here, a correlation that vanishes at a separation whatever the ranks — and then described in whichever ranked language is convenient at each point.

The second lesson is about what an impurity is to a fit. A small monomer fraction is a nuisance; a large one is a paramagnet in its own right, and at a few per cent it gives a singlet cluster the separations a ground spin would. The sign rules found for these fits are statements about pure samples, and the boundary of pure is a few per cent — which is inside the range a real sample’s purity is usually known to.

Still open: the departure from Curie, and a second impurity #

The obvious open question is the one the band keeps pointing at. If the upper separation is where a cluster’s curve stops being distinguishable from its ground multiplet’s Curie law, then the fractional departure of the cluster’s χT from $S(S+1)$ at the top of the window, measured at the upper separation, should be nearly the same number for every cluster. That departure is available from the spectrum without any fit, and computing it at each cluster’s upper separation would either turn the $S + \tfrac12$ band into a law or show it to be a coincidence of seven clusters.

The nearer question is an impurity that is not free. A dimeric fragment with its own coupling contributes a susceptibility that is itself temperature-dependent, and the fit would then be separating two clusters rather than a cluster and a paramagnet. Whether the singlets still acquire separations — and whether they do so at the same few per cent — is one more term in the model and the same sweep.