The window sets the constant

A four-parameter fit to a magnetic susceptibility curve has two nuisance parameters — the fraction of paramagnetic impurity and a temperature-independent term — and at two exchange couplings they stop being correlated. Across every spin-½ cluster on five and six centres, the upper of those two couplings was ordered by one property of the cluster: the gap to its first multiplet of higher spin. Multiplied by that gap, forty-one separations collapsed to a spread of about two, with a median near 355 cm⁻¹.

That number was then read as a position. At 300 K, the top of the window the fit was read over, kT is 208.5 cm⁻¹, so the raising multiplet sits about 1.7 kT above the ground state at the separation — close enough to start filling across the upper part of the window, far enough to be only starting. The reading made a prediction and said it had not been tested: a window ending at 200 K should put the constant near 240 cm⁻¹, and one ending at 400 K near 475, for every cluster at once.

It is tested here, and it very nearly holds — and the part that does not hold says what the constant is.

Nine windows, one census #

The census is re-read exactly as it was built: forty temperatures across the window, one per cent data, two per cent impurity, the four-parameter model’s logarithmic Jacobian, the correlation between the two nuisance parameters from all four singular directions, and every sign change of that correlation located by bisection on a grid of couplings from 3 to 4000 cm⁻¹. The upper separation is the second sign change, as in the census.

The windows move the top from 150 to 500 K with the bottom at 20 K, move the bottom to 10 and 40 K with the top at 300, and add two windows, 10–75 K and 10–150 K, that are scaled copies of others, for a reason the section on the span gives. Under the census’s own window, 20 to 300 K, every one of the forty-one separations comes back as the census gave it, to ten decimal places — the check that the re-reading is the same instrument.

The medians are 189, 243, 354, 463 and 569 cm⁻¹ for tops of 150, 200, 300, 400 and 500 K. The predictions of about 240 and 475 come out at 243 and 463: within three per cent, on the side of low at the higher top and high at the lower one. As a first test of a reading that was offered rather than proved, that is a pass.

The collapse does not care which window #

The finding the reading was built on survives every window. The product of the upper separation with the raising gap spreads across the forty-one clusters by a factor of 2.12 to 2.15 under all nine; the product with the first excitation spreads by 21 to 24.

So moving the window moves the constant and nothing else. The ordering of the clusters by their raising gap is a property of the clusters; the value the ordering collapses to is a property of the measurement. That division was implicit in the reading and is now measured, and it is the useful half of the result: a chemist who changes their cryostat’s range changes where their fit goes blind, and changes it for every cluster in the same proportion.

The same point one cluster at a time: doubling the window’s top multiplies every cluster’s separation by between 1.81 and 1.94. None doubles. The narrowness of that band is why the collapse is kept; its sitting below two is the next section.

But the constant leans #

If the separation sat at a fixed multiple of kT at the window’s top, the product in units of that kT would be the same number under every window. It is not. With the bottom at 20 K it is 1.81 kT at a 150 K top, 1.75 at 200, 1.70 at 300, 1.67 at 400 and 1.64 at 500 — falling at every step, by about a tenth across the range.

Fitted as a power law, the median rises as the top temperature to the power 0.92 rather than one. That is close enough to proportion that the reading’s predictions landed within three per cent, and far enough from it that “1.7 kT at the top” is a description of a 300 K window rather than a law. The factor a practitioner would carry is not 1.7 but a slowly varying function of the window.

The bottom moves it too #

The reading named one end of the window. A window has two, and the census had only ever varied neither.

With the top held at 300 K the median is 343 cm⁻¹ for a window starting at 10 K, 354 at 20 and 377 at 40. Raising the bottom by a factor of four raises the constant by ten per cent, in the same direction as raising the top. So the separation is set by the window as a whole: both ends pull it, the top a great deal and the bottom a little, and a reading in terms of the top alone was a reading of the dominant term.

The lean is the window’s span #

Raising the bottom moves the constant the same way raising the top does, and that points at what both are doing. With the bottom fixed, a higher top also makes the window wider; with the top fixed, a higher bottom makes it narrower. The candidate is that the constant, in units of kT at the top, depends on the window only through its width — the ratio of its top to its bottom.

There is a reason to expect exactly that, and it is a symmetry of the model rather than a pattern in the numbers. A cluster’s contribution to χT depends on its coupling and the temperature only through J/kT. The impurity term’s contribution to χT is a constant. So if every temperature in a window is doubled and every coupling with it, those two parts of the curve are unchanged, point for point — and a separation, which is a property of how the fit’s parameters trade against each other across the curve, moves by exactly the factor the temperatures did. In units of kT at the top it cannot move at all. Only the temperature-independent term carries a scale of its own, since its contribution to χT grows in proportion to the temperature, and so it is the only thing that can make a window and its scaled copy disagree.

The test is two windows that are scaled copies of others, added for it. Windows from 10 to 75 K, 20 to 150 K and 40 to 300 K — all a span of 7.5, at three temperatures a factor of four apart — give 1.819, 1.813 and 1.809. Windows from 10 to 150 K and 20 to 300 K, both a span of fifteen, give 1.706 and 1.696. Within each group the constant agrees to 0.6 per cent, and the residual runs the same way in both — the colder copy reads a little higher — which is what one broken scale produces: a small drift in a single direction with the window’s temperature, not scatter. Which way that term should push the constant is not derived here; that it pushes both groups the same way is measured.

So the lean is the span. A window from 20 to 150 K is not a narrower version of one from 20 to 500 K that happens to end lower; in units of kT at its top it is the same window as 40 to 300 K. The reading’s 1.7 kT at the top was the value of a function of the span at a span of fifteen, and a practitioner with any other span has a different number — 1.81 at 7.5, 1.67 at twenty, 1.64 at twenty-five — that the census can now supply.

What the span does not settle is why the function falls. Every window here has forty temperatures evenly spaced across it, so a wider window samples its cold end more sparsely, and the cold end is where the fit’s two nuisance parameters are best told apart. Span and sampling density move together in this design, and the symmetry argument above says only that the constant cannot depend on anything else — not which of the two it is responding to.

A reading that depends on the window it reads #

One of the nine windows exposed a weakness in the census’s own rule, and it should be named rather than smoothed over. The upper separation is taken as the second sign change of the correlation, whatever kind of sign change the first one is. With the window’s bottom raised to 40 K, one cluster — a five-centre doublet, g5-3 in the census’s labels — acquires an additional sign change at 3.2 cm⁻¹, a coupling about nine times smaller than the kT at the window’s bottom and far below anything that window resolves. Every index after it shifts by one, and the rule reads that cluster’s lower separation as its upper.

It is set aside in that one window and named, and it is found rather than assumed: a cluster whose product falls below half its window’s median is flagged, and exactly one is flagged across all nine windows. With it set aside the spread under that window is 2.12, like the others. With it kept, the spread is 5.87 — a single misread cluster would have made the collapse look as though it failed on a window that raised its bottom, and nothing about the physics would have changed.

The lesson is about rules that count. A rule that takes “the second event” is a rule about the events’ order, and an event anywhere in the range — including a part of it the measurement cannot see — renumbers everything after it. The rule worked on the census because the census used one window; under nine it needs a condition on where an event may fall to count at all, and that condition is a choice this essay reports rather than makes.

What the forty-one are, and are not #

The clusters are every connected graph on five centres with a doublet ground state and every one on six with a triplet — twenty and twenty-one — with one antiferromagnetic coupling on every edge, spin-½ centres, and exact diagonalisation. The data are ideal: noiseless, one per cent precision assumed in every point’s weight, two per cent impurity. Each window has forty temperatures evenly spaced across it, as the census’s were, so a wider window is also a sparser one, and that is part of what the lean may be measuring.

The checks, made wherever these figures are drawn: every cluster has an upper separation under every window; the raising gap’s spread stays under 2.2 and the first excitation’s over 20 under every window; the median rises with the top as a power between 0.85 and 0.97; the predictions of 240 and 475 land within three per cent; the median in units of kT at the top falls at every step; with the top at 300 K the median rises with the bottom; windows that are scaled copies agree in units of kT at the top to a per cent, with the colder copy higher in both groups; exactly one cluster in one window is misread by the census’s rule. The refusal is the census’s own window, under which every cluster’s separation must come back as the census gave it, to 10⁻⁹ — and does.

Where the numbers stop #

A noiseless curve. The separations are properties of the fit’s geometry on ideal data. A real measurement has a drifting background, and a fit with a parameter for everything absorbs it wherever the geometry is weakest. The window dependence found here is the ideal case’s, and a real one can only be less orderly.

Forty points in every window. A practitioner’s window has however many points the instrument took, spaced however it was set. Density and span are tied together here, and a study that varied them separately would say which of them the lean belongs to.

And the separation itself is a construction. It is where two fitted parameters decouple, and which product of parameters a curve actually fixes is a separate question about the same fit. The collapse says the raising gap orders it; it does not say that a chemist needs to know where it is. What it does say is that if they do, they need to know their window, both ends of it.

A position is relative to its instrument #

The earlier reading turned a median into a mechanism: 355 cm⁻¹ became the multiplet a little under two kT above the ground at the window’s top. The mechanism predicted a proportionality, and the proportionality is nearly there — 0.92 rather than 1, a constant that drifts by a tenth across a factor of three in temperature, and a second end of the window that was never in the mechanism at all.

The symmetry is the part worth keeping. A model whose only energies are the couplings and kT cannot care where a window sits on the temperature axis, only how wide it is, and the census could have said so before a single window was re-read. What the measurement added was the size of the one thing that breaks the symmetry — the temperature-independent term, at 0.6 per cent across a factor of four in temperature — and the confirmation that nothing else in the fit does.

That is the ordinary fate of a good first reading, and the discipline it calls for is the one the census used on the raising gap: once a rule has been read off one configuration, vary the configuration and see which part of the rule moves. Here the ordering stayed and the constant moved, which separates what the clusters decide from what the measurement decides more cleanly than any single window could.

Who measured what #

Temperature-independent paramagnetism and paramagnetic-impurity terms have been fitted beside exchange couplings for as long as susceptibility has been used to measure them, and the practice of reporting the fitting window alongside the fitted constants is common but not universal. The four-parameter model’s geometry, the separations and the census of clusters are this argument’s own, built up across the essays that read what a susceptibility curve can support.

What is computed here is the census under nine windows, the collapse under each, the median’s dependence on each end of the window, the reduction of both to the window’s span by the model’s own scale symmetry, and the one reading the census’s rule gets wrong.

Still open: density against span, and the second-order spread #

The obvious open question is the one the span leaves. The constant is a function of the window’s span, and in this design a wider window is also a sparser one at its cold end. A census that holds the span and changes the number of temperatures, or spaces them evenly in the logarithm of the temperature instead of the temperature, would say whether the fall from 1.81 to 1.64 belongs to the width or to where the points fall — and whether a practitioner who spaces their measurements logarithmically, as many do, has a constant that does not lean at all.

The nearer question is the one the census left: the factor of about two that the raising gap does not remove. It is window-independent — 2.12 to 2.15 under every window read — which says it belongs to the clusters rather than the measurement, and the candidate refinement is still the population of the raising multiplet rather than its energy alone. A correction of that form fitted on the doublets and tested on the triplets would now be tested under nine windows at once.