The spacing sets the constant

A four-parameter fit to a magnetic susceptibility curve has two nuisance parameters, a paramagnetic impurity fraction 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 couplings is ordered by the gap to the cluster’s first multiplet of higher spin, and under nine measuring windows the product of the two came to a constant that moved with the window: about 1.7 kT at the window’s top for a window from 20 to 300 K, 1.81 for a span — top over bottom — of 7.5, 1.64 for a span of 25. The model’s scale symmetry explained why the span was the variable: with no energy of its own besides the couplings and kT, it cannot care where a window sits, only how wide it is.

What the symmetry did not settle was why the constant falls as the span widens. Every window had forty temperatures evenly spaced, so a wider window is also sparser at its cold end, where the fit’s two nuisance parameters are best told apart. The window essay’s lead was to separate them: hold the span and change the number of temperatures, or space them evenly in the logarithm of the temperature, as many measurements are, and see whether a logarithmic design has a constant that does not lean at all.

It leans more. And the number of temperatures, which the lead suspected, turns out to be the one thing that does not matter.

One census, many designs #

The census is re-read exactly as the window essay read it: forty-one clusters, the four-parameter model’s logarithmic Jacobian with one per cent precision on every point, two per cent impurity, the correlation between the nuisance parameters from all four singular directions, and every sign change of that correlation between 3 and 4000 cm⁻¹ located by bisection. The upper separation is the second sign change, and the one cluster that rule misreads in windows starting at 40 K is set aside and named, as before.

What changes is the design. The window essay’s design was a list of forty temperatures evenly spaced in T between the window’s ends. Here the list is anything: twenty, forty or eighty temperatures; spaced evenly in T, in log T, or in 1/T; across windows from 10–75 K to 20–500 K. Twenty-six distinct designs in all, each a full re-reading of the census.

The refusal comes first. Forty temperatures evenly spaced from 20 to 300 K is the census’s own design, and every cluster’s upper separation under it must come back as the window census gave it, to a part in a billion. It does. The model is written out again beside the census rather than reused, so that a design can be any list of temperatures, and that check is what makes the rewritten model the same instrument.

Three ways to spend forty temperatures #

The three spacings are not three ways of drawing the same thing.

Evenly spaced in T, just over half of a 20–300 K design lies above 150 K. Evenly spaced in log T, a quarter does — a logarithmic design spends as many points between 20 and 40 K as between 150 and 300. Evenly spaced in 1/T, as a Curie plot’s horizontal axis would suggest, fewer than one in ten do, and the design is crowded into its coldest decade.

Practitioners choose among these for reasons that have nothing to do with fitting: a logarithmic sweep resolves the low-temperature structure where a susceptibility changes fastest, and a cryostat’s controller often steps in fixed fractions. So the question is not academic. It is whether the constant the window essay measured belongs to the clusters and the window, or also to a choice of spacing that is rarely reported.

The number of temperatures only converges #

The first test holds everything but the count.

Evenly spaced, twenty temperatures give 1.702, forty 1.696 and eighty 1.693. The steps halve as the count doubles, which is what a quantity converging on a limit as one over the count does, and the limit through the last two is 1.691. Logarithmically spaced, the same counts give 1.531, 1.490 and 1.470, a limit of 1.449 — a slower convergence, since a logarithmic design is coarse at the window’s top, but a convergence all the same.

The count is a discretisation, not a design choice. A fit’s geometry is a sum over its temperatures, and as the temperatures multiply the sum approaches an integral over the window with a weight set by the spacing. Doubling the count refines the integral; it does not change which integral it is. The two limits, 1.691 and 1.449, are as far apart as the forty-point values were.

The direction of the approach is itself a small confirmation of what follows. A list of evenly spaced temperatures that includes both ends of the window gives each end a full point’s weight, where the integral it approximates would give each end half — so a finite design slightly over-weights its own top and bottom, by an amount that falls as one over the count. Over-weighting the top is, as the next sections show, exactly what raises the constant. So a finite design should read slightly high and approach its limit from above, and at every count and under both spacings it does.

The second test goes straight at the lead’s hypothesis. If the wide windows’ lower constant came from their sparser cold end, then giving a wide window the narrow windows’ density should raise it. Holding the census’s own step of about 7.2 K — nineteen temperatures across 20–150 K, sixty-eight across 20–500 K — gives 1.808 and 1.639, against the forty-point designs’ 1.813 and 1.637. At the wide window the denser design does read higher, as the hypothesis wanted — by a tenth of a per cent, against the 3.5 per cent that widening the window from 20–300 to 20–500 K takes away. At the narrow window the sparser design reads lower by three tenths. Both are the count’s discretisation, and the span’s effect is not a sampling-density effect.

The spacing moves the constant, and the lean with it #

What does matter is how the temperatures are distributed across the window.

At a span of fifteen — the census’s 20–300 K — even spacing gives 1.696, logarithmic spacing 1.490 and inverse spacing 1.360. At every span tried the order is the same: even, then logarithmic, then inverse. The logarithmic design is twelve per cent below the even one at the window the census used, which is a larger change than moving the window’s top from 200 to 400 K made.

And the logarithmic design leans more. From a span of 7.5 to 25 the even constant falls from 1.809 to 1.637, by 9.5 per cent; the logarithmic one falls from 1.652 to 1.421, by 14 per cent. The hope in the lead was that a logarithmic design, which gives every decade of temperature the same number of points, would be the design with no lean. It is the design with more. Widening a logarithmic window adds whole decades at the cold end with their full share of points, where widening an even window adds only a sliver of points below its old bottom — so the logarithmic design’s centre of weight moves much further cold as the span grows.

The inverse design does something the other two do not: its constant falls from span 7.5 to 15 and rises again at 25, and its collapse loosens, the spread of the forty-one products reaching 2.5 where the other designs stay near 2.1 to 2.3. A design that puts nine tenths of its points in the coldest part of the window has almost nothing near the top, where the separation is measured, and the reading becomes less orderly for it.

What a practitioner would carry #

In the units a paper would quote, the difference is not small. For the census’s own window, 20 to 300 K, the median of the upper separation times the raising gap is 354 cm⁻¹ with an even design, 311 with a logarithmic one and 284 with an inverse one. A chemist who read the window essay and carried “about 1.7 kT at the top” to a logarithmic sweep would place the separation fourteen per cent too high — a larger error than the whole window dependence the window essay was written to correct.

The practical rule is short. A separation, like every other quantity read off this fit’s geometry, belongs to a design, and a design is three things: where the window sits, which the model’s symmetry removes; how wide it is; and how its temperatures are spread. A fit that invents a moment and a model that decides what is fitted were statements about the model; this is a statement about the measurement, and it is one that a methods section can satisfy in a single line, by giving the temperature list.

The symmetry holds under every spacing #

The window essay’s central argument was a symmetry: without the temperature-independent term, the model’s curve depends on J/kT alone, so a window and its scaled copy — every temperature doubled — must give the same constant in units of kT at the top. That argument never used the spacing, and it should hold for any design as long as the copy is spaced the same way.

It does. Logarithmically spaced, windows of 10–75, 20–150 and 40–300 K, all a span of 7.5, give 1.657, 1.652 and 1.652; windows of 10–150 and 20–300 K give 1.499 and 1.490. Within each group the values agree to under a per cent, as they did for even spacing. So the constant is a function of the span for each spacing — the symmetry makes that exact up to the temperature-independent term’s own small scale — but the function is the spacing’s. A report of a separation, or of anything read off this fit’s geometry, needs the spacing as well as the window.

The collapse is the clusters’ #

Through all of this the ordering the census found — the raising gap collapsing the forty-one separations, after a band that looked like a law broke — is untouched.

Under every design the product with the raising gap spreads by between 2.1 and 2.5 across the forty-one clusters, and the product with the first excitation by more than twenty. Which cluster separates where is a property of the clusters; the value the ordering collapses to is a property of the design, and the design now has three coordinates rather than two: where the window sits, how wide it is, and how its temperatures are spread across it. The first cancels by the symmetry, and the other two do not.

What orders the three spacings #

A single number that ordered all the designs would be a useful thing to report with a fit, and the obvious candidate is how much of the design sits near the window’s top, where the separation’s kT is measured.

The share of temperatures in the upper half orders the spacings — 0.53 to 0.57 for even designs, 0.23 to 0.35 for logarithmic ones, and from 0.15 down to 0.05 for inverse ones — and within each spacing the constant rises as the share does. But the designs do not fall on one curve: a logarithmic design at a narrow span and an even design at a wide one can have similar constants with quite different shares. The upper half’s share is the leading coordinate and not the whole of what sets the constant, which is an honest place to leave it; the fit’s geometry is an integral over the whole design, and no single moment of the design has to capture it.

How the claims can fail #

Each statement is checked where the figures are drawn. The even forty-point design across 20–300 K must return every cluster’s separation as the window census gave it, to a part in a billion. Under even and logarithmic spacing the constant must fall at every doubling of the count and move by under one and under five per cent between twenty and eighty. The fixed-step designs must match the forty-point constant to half a per cent. At four windows the constant must order even, logarithmic, inverse; at a span of fifteen the logarithmic constant must be more than ten per cent below the even one; and from a span of 7.5 to 25 the logarithmic design must lean by a larger fraction than the even one. Under both spacings scaled copies must agree to a per cent. Under every design all forty-one clusters must have an upper separation, the raising gap’s spread must be under 2.6 and the first excitation’s over twenty, and the only misreading may be the census’s own, in a window starting at 40 K.

Where the numbers stop #

A noiseless curve, at one impurity. As in every census reading, the monomer fraction is two per cent, and the separations move with purity in ways that were only mapped for one design. Beyond that, the data are ideal and the separations are properties of the fit’s geometry. A real measurement’s noise is not uniform across a logarithmic sweep — the cold points of a susceptibility are usually the best measured — and a design’s weight in the fit is its spacing times its precision, not its spacing alone. The precision here is one per cent at every point, in every design.

Three spacings. Even, logarithmic and inverse are three one-parameter families of designs among very many. A practitioner’s actual design — a dense sweep at low temperature and a sparse one above, say — is none of them, and the constant for it is a fourth number, computable in the same way from its own list of temperatures.

And a separation that is a construction. It is where two fitted parameters decouple, and whether a chemist needs to know where that is depends on whether they report the nuisance parameters at all. What the census establishes is that if they do, the design is part of the result.

A design is part of the instrument #

The window essay’s reading was that a fit’s blind spot is set by the window: both ends of it, through their ratio. The reading here is that the window was standing in for something more general — the distribution of the temperatures across it — and that the ratio of its ends was the only aspect of that distribution the census had varied. Varying the rest shows the count to be a discretisation that converges, and the spacing to be a genuine third coordinate that moves the constant further than the window’s top did.

How many parameters a curve is worth found that forty points reaching two kelvin were worth more than sixteen thousand starting at twenty, which was already a statement that where the points fall matters more than how many there are. The census makes the same statement about a different quantity. A susceptibility measurement’s temperature list is part of its instrument, like its field and its window, and a separation — or a correlation, or a product of parameters the fit actually determines — is a reading of that instrument as much as of the sample.

Still open: the precision profile, and the second-order spread #

The obvious open question is weight. Every design here gives every temperature the same one per cent precision, and a real sweep does not: a SQUID magnetometer’s relative precision is usually best where the signal is largest, at low temperature. A design’s effective weight is its spacing times its precision profile, and replacing the flat one per cent with a profile that improves at low temperature would say whether a realistic even sweep behaves like the even design here or like the logarithmic one — which decides which of the constants a practitioner should carry.

The nearer question is still the one the census left before any of this: the factor of about two that the raising gap does not remove. It survives every design — 2.1 to 2.5 under all twenty-six — which makes it even more clearly the clusters’ own. The candidate refinement is the population of the raising multiplet, its degeneracy as well as its energy, fitted on the doublets and tested on the triplets, and it can now be tested under three spacings as well as nine windows.