A precision is a spacing

A four-parameter fit to a magnetic susceptibility curve has two nuisance parameters, an 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 multiplied by the gap to the first multiplet of higher spin comes to a constant, and the constant moved with the measuring window. Then it moved with the spacing of the temperatures inside the window: 1.70 in units of kT at the window’s top for an even sweep across 20 to 300 K, 1.49 for a logarithmic one, 1.36 for one even in the inverse.

All of those designs gave every temperature one per cent, and a magnetometer does not. Relative precision is usually best where the signal is largest, at low temperature. That essay’s own framing supplied the prediction for what this should do. The fit’s geometry is a sum over its temperatures, and in the limit of many points the sum is an integral over the window with a weight. The weight comes from two places, how densely the temperatures are spaced and how well each is measured. If that is the whole story, then a sweep with better precision at low temperature is not a new kind of design: it is the same integral as one with more points there.

The obvious test, and what spoiled it #

The two can be made equal exactly. Take an even sweep and let its relative precision vary as a power, $\sigma(T)\propto T^p$. Its weight across the window is $T^{-2p}$. A logarithmic design has a density of temperatures proportional to $1/T$, so $p=\tfrac12$ gives the same integral, and an inverse design has a density of $1/T^2$, so $p=1$ does too. Those are predictions with no free parameter, about forty-one clusters’ medians.

Run as plain lists of forty temperatures, the first passes: an even sweep with $p=\tfrac12$ gives 1.446, close to the logarithmic design’s 1.490 and very close to the 1.449 that design was found to converge on at large count. The second does not look like a pass at all. An even sweep with $p=1$ gives 1.168 and an inverse design gives 1.360, sixteen per cent apart, which is more than the difference between any two spacings a reader would actually use.

The explanation is one temperature. A plain list of evenly spaced values counts each end of the window as a whole point, where the integral it is approximating, by the trapezoid rule, would count half. That error is what the earlier essay on the spacing saw falling as one over the count, and on an even design it is modest. But on an inverse design the top of the window is where the temperatures are sparsest, so the top point stands in for a wide stretch of the window, and counting it fully is a large overweighting of exactly the place that raises the constant. Fixing it needs nothing more than a factor of $\sqrt2$ on the sigma of the two end temperatures.

The size of this is easy to see from the design itself. Forty temperatures even in $1/T$ across 20 to 300 K have neighbours under half a kelvin apart at the cold end. At the hot end the last step runs from 221 K to 300 K, so a single reading is standing in for the upper quarter of the whole window, and a plain list gives it the same weight as each of those half-kelvin neighbours. It is a reasonable thing to do for a thermometer and it is not what the integral asks for. An even sweep has no such problem, because all its steps are the same, and its two ends are two points of forty. That also accounts for the odd behaviour the earlier essay noticed in this design, where the inverse constant fell from a span of 7.5 to 15 and rose again at 25, and the collapse loosened: much of what it was reading was the fortunes of one temperature.

An even flat design barely moves, 1.696 to 1.691, and that 1.691 is what the even design’s limit was computed to be by extrapolation in the count, here reached directly at forty points. A logarithmic one moves from 1.490 to 1.453, against its extrapolated limit of 1.449. And the inverse design falls from 1.360 to 1.243, by far the largest move in the set. So the published order of the three spacings stands, and the published distance of the third was largely the weight of its topmost point.

One measure, three ways #

With the ends fixed, every pair of designs that shares a weight can be compared directly.

For a weight of $1/T$ the even and logarithmic routes give 1.447 and 1.453 at 20–300 K, 1.627 and 1.632 at 40–300 K, 1.367 and 1.380 at 20–500 K. The widest window has the coarsest temperatures at its top under a logarithmic design and the largest difference, one per cent. For a weight of $1/T^2$ the even and inverse routes give 1.188 and 1.243 at forty temperatures and 1.194 and 1.208 at eighty, with their difference falling by a factor of nearly four as the count doubles, the behaviour of a trapezoid rule. A third route to that same weight, a logarithmic sweep with precision growing as $\sqrt T$, gives 1.198 at forty.

So the three ways of reaching a given weight converge on one constant, and a design has one coordinate where two seemed to be. Window aside, what a practitioner’s design is depends on where the information across the window lies, and it makes no difference whether that information arrived by taking more readings or by taking better ones. That is an expected fact about least squares, of course: a reading with half the variance is two readings. But the constant in this argument is a median over forty-one clusters of a bisected zero of a correlation between two nuisance parameters, some way downstream of the normal equations, and the clean agreement says nothing else is carrying it.

The refusal for all of this is on the same grounds. Scaling every sigma by three leaves every cluster’s upper separation unchanged, to a part in a thousand million. A correlation has no scale of its own, and only the shape of a precision profile can matter. So “one per cent” as such was never part of what these essays were reporting.

What a real sweep looks like #

A power of the temperature was a convenient test and not a description. A measured susceptibility has an error from the reading, roughly proportional to the signal, and an error that does not scale with the signal: a sample holder’s background, a diamagnetic correction, a magnetometer’s noise floor. So its relative precision is a floor plus a term that goes as one over the susceptibility.

$$\frac{\sigma}{\chi T}=0.01\sqrt{1+\left(\frac{T}{T_k}\,\frac{(\chi T)_{\rm free}}{\chi T}\right)^2}.$$

Here $T_k$ is a dial: the temperature at which the two errors would be equal for free spins, large where the absolute noise is small. And now the profile is not one shape, because it contains the cluster’s own curve.

For that cluster at a coupling of 100 cm⁻¹, a $T_k$ of 4800 K costs only 1.4 per cent at the window’s top, a $T_k$ of 1200 K costs 3.9 and one of 300 K costs 15. Those are not exotic: a few per cent of the room-temperature susceptibility is a familiar size for a background correction on a dilute sample. Nor do the curves follow any one power. A strongly coupled cluster’s moment collapses at low temperature, which keeps the relative precision there from improving as fast as a single exponent would.

The census of forty-one clusters was re-read at seven levels of this noise.

An even sweep’s constant falls from 1.696, through 1.598, 1.523 and 1.479, to 1.450, and then it stops falling, at about 1.44, for every $T_k$ from 600 K down to 75. A small background is enough to matter: 1.4 per cent at the top of the window moves the constant by a tenth of a unit, more than half of the whole range of windows in the essay that measured them, 1.81 to 1.64. And the place it stops is within a per cent of the logarithmic design’s limit, 1.453.

That answers the question the spacing essay left open, which was whether a realistic even sweep behaves like the even design or the logarithmic one: under a noticeable background, like the logarithmic one. The stop has a reason, and it is not obviously the same reason. With absolute noise dominating, the weight is proportional to $\chi^2$, and that is $1/T^2$ times the square of $\chi T$, which rises steeply with temperature across this range for a strongly coupled cluster. The two nearly cancel into something like $1/T$ for the clusters at couplings near their upper separation. That is a description of why they should be close, and the agreement is not exact in a way that would make it a theorem. The spread of the forty-one products says the same: it opens from 2.15 to 2.49, where the logarithmic flat design has 2.24, so the two designs give about one median and are not one design.

A logarithmic sweep under the same noise goes further, from 1.490 to 1.406, 1.299 and 1.264. There its noise is piled on a design already weighted to the cold end, and nothing has stopped it by the noisiest setting computed.

In wavenumbers #

The median of the upper separation times the raising gap, the number a chemist would quote, runs from 354 cm⁻¹ for an even sweep with no background to 300 with one and from 311 for a logarithmic sweep to 264. Someone who carried “about 1.7 kT at the top” to a noisy logarithmic sweep would place the separation about a third too high.

The earlier rule of thumb said a design was three things, where the window sits, how wide it is, and how its temperatures are spread. It is shorter than that and it asks for more. The third item is the distribution of information across the window, and it has to be computed from the sweep’s error model, which a paper does not always report. The rule in practice: if the error bars in a fit’s data are available, the design is known, and its constant can be read by re-running a calculation like this. If they are not, the constant is known only to between 1.44 and 1.70 even on an even sweep.

And the lean with the window’s span gets worse.

Where the flat even design falls by 9.5 per cent between spans of 7.5 and 25, one with a fixed $T_k$ of 600 K falls by 12.1 per cent, approaching the logarithmic design’s 14. It should be read carefully, since a fixed $T_k$ is not a scale-free thing. The window’s scale symmetry was exact only without a temperature-independent term, and a fixed absolute noise is a second quantity with the units of a temperature in it, so these three windows are no longer scaled copies of anything.

What was computed, and how #

Every five-centre doublet and six-centre triplet is read as in the census: the four-parameter model, the logarithmic Jacobian, two per cent impurity, every sign change of the correlation between the two nuisance parameters from 3 to 4000 cm⁻¹ by a scan and bisection, the upper separation the second change. The change here is the weighting. The correlation is read from the inverse of the Fisher matrix, each row of the Jacobian divided by its own sigma, scaled to unit diagonal before inversion, since the four parameters’ sensitivities differ by orders of magnitude. Thirty designs in all were read, including controls: the flat even design at forty returns 1.696 as before, and a scaled one returns the same cluster by cluster. A cluster falling under half the median is set aside as the census’s rule does, and in the noisier designs one is. The weight comparisons are stated where they can fail: under one per cent or so at three windows, a closing factor of more than three as the count doubles, and a settled value within one and a half per cent of the logarithmic limit.

Where it stops #

Everything here is the geometry of a noiseless fit. The precision profile sets the weights and no noise is added, so a separation is what a fit’s covariance says, in the first-order approximation, at the point where the fit is exact. Real noise in a real fit would scatter around this.

The noise model is two terms. A floor and a constant on $\chi$ are the two usual contributions, not a measured instrument. A background correction is also a systematic: it does not average down across temperatures, and a fit may absorb it into the temperature-independent term, which is one of the two parameters being read. Treating it as independent noise at each temperature is a choice, and it favours independence.

The factor of two is still there. The raising gap collapses the clusters to a spread of 2.14 to 2.66 under every design read here. That spread has now survived windows, spacings and precision, and it stays the clusters’ own to explain.

The stop at 1.44 is a near-coincidence with a reason, not a proof. Other clusters’ profiles differ, the median smooths over them, and a sample of forty-one clusters does not show that the same near-cancellation holds in other families.

Who said what #

Weighted least squares goes back to Gauss, and optimal experimental design has long treated the allocation of readings and their precisions as a single measure over the design space, Kiefer’s work in the 1950s and 1960s being the standard reference. Fitting susceptibilities with impurity and temperature-independent terms is common laboratory practice, and how many parameters a curve is worth has been a concern of this argument for some time, alongside the moment a fit invents. The trapezoid’s half-weights are in every textbook of numerical analysis. What is measured here is the three routes to one weight agreeing on this census’s constant, the end point’s share of the inverse design’s offset, and the effect of a background on the constant.

A report of a design #

The practical habit this leaves is to ask a fit for its weights. The usual report of a susceptibility measurement gives a temperature range and perhaps a step. It says little about where the information lies, though a model decides what is fitted and a design decides what can be told apart, and for a quantity read off the fit’s geometry those are two halves of one statement. A report of the same sweep with its error model has told the reader which constant applies, and a report without one has left it open by as much as the whole dependence on window, spacing and count that these essays set out to measure.

Still open: the population of the raising multiplet, and a background that is fitted #

The obvious open question has stayed the same through three essays: the factor of about two that the raising gap does not remove. It has survived thirty more designs here, which leaves little room for it to belong to the measurement. 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 the cheap test is whether the same refinement reduces the spread under every weight, or only under some.

The nearer question is the background itself. Treating absolute noise as independent at each temperature is the mild version. A constant offset in $\chi$ that the fit must carry is a fifth parameter, nearly degenerate with the temperature-independent term, since a constant $\chi$ is exactly what that term is. Adding it would say whether the upper separation survives at all once the fit is honest about the thing most likely to be wrong.