The satellite beside the zero-order reflection

The strain wave that tips the satellites showed that a size effect locks a displacement wave to an occupation wave a quarter of a period behind it, and that the first-order satellite pairs then lean. The inner one of each pair is stronger than the outer by an amount that reads off the size factor with the wave’s amplitude cancelled. The second order reads the strain again carried the same chain to second order. With every intensity divided by the square of its scattering vector, the lowest-order formula said the second-order pair should lean exactly half as far as the first. It did at the second and third main reflections. At the first it leaned about a third as far, with ratios between 0.30 and 0.45, and it did so on chains of 30,000 and of 120,000 sites alike.

That essay concluded that the departure was not the finite chain, since two very different lengths agreed. It suggested instead a term of higher order that happened to be large at this one reflection, and it left the question open. The conclusion was wrong and the suggestion unnecessary. The lowest-order formula is right at the first reflection too. The departure is the finite chain’s edge, and the two lengths agreed by coincidence.

The resolution takes three measurements, each able to fail. The exact structure factor of an infinite chain gives one half at every reflection. The error at the one reflection that misbehaved is the tail of the zero-order reflection, and its size can be predicted. And that tail swings with the chain’s length, not decaying steadily, in a way that made 30,000 and 120,000 give nearly the same wrong answer.

The formula with no chain at all #

The chain in question has sites at every integer, carrying an occupation wave: occupied with probability $c + a\cos 2\pi(nq + \phi)$ at site $n$, where $q = 1/\varphi^2 \approx 0.382$ is an irrational wavevector, so every reflection needs a second integer to index it. Each site is displaced by $u\sin 2\pi(nq + \phi)$, with the amplitude $u$ that a size factor $\varepsilon$ produces. For such a chain the structure factor at every satellite is a finite Fourier–Bessel sum. The occupation has three Fourier components and the displacement one, and the sum involves only Bessel functions of $z = 2\pi k u$, where $k$ is the satellite’s position. The size-coupled chain’s displacement is exactly of this form. Fitting its first harmonic leaves a residual of two parts in a hundred million.

So the ratio the earlier essay measured by summing over sites can also be computed with no sites at all, from the series for the infinite chain. The table does that. For size factors of 0.015, 0.03 and 0.06, the ratio at $h = 1$ is 0.5005, 0.5011 and 0.5024, and at $h = 2$ and $h = 3$ it is equally close. The lowest-order prediction holds at the first reflection. The only departure in the table appears at the largest size factor, 0.12, where the ratio climbs to 0.506, 0.512 and 0.523 from $h = 1$ to $h = 3$. That is the next order of the expansion, and it behaves as a next order should. It grows with the displacement and with the reflection, because $z$ grows with both. It raises the ratio above one half where the anomaly lowered it below, and it is smallest at exactly the reflection where the anomaly was largest.

A higher-order term was therefore never the explanation. Whatever pulled the ratio down to a third at $h = 1$ was in the sum over sites and not in the physics of the chain.

The chain’s edge scatters #

A sum over $N$ sites is the infinite chain’s structure factor plus the tails of every reflection the chain has. A finite chain’s reflections are not points but peaks, of width about $1/N$. Away from its centre, a peak of amplitude $F_0$ leaves a tail whose size at a distance $\Delta$ is

$$\frac{F_0\,|\sin \pi N\Delta|}{N \sin \pi\Delta}.$$

The tail falls only as $1/\Delta$ away from the peak, and it oscillates, vanishing whenever $N\Delta$ is a whole number.

The strongest reflection a chain has is the zero-order one, at $k = 0$, where every site scatters in phase and the amplitude per site is the mean occupation $c = 0.5$. The inner second-order satellite at $h = 1$ lies at $k = 1 - 2q \approx 0.236$, only a quarter of a reciprocal unit from it. It is also the weakest satellite in the calculation. Its amplitude grows with $k$ and $k$ is small here, so for a size factor of 0.03 its amplitude per site is $3.5 \times 10^{-4}$. The zero-order tail there has an envelope of $0.5/(N \sin 0.236\pi)$, which at $N = 30{,}000$ is $2.5 \times 10^{-5}$, seven per cent of the satellite. The picture shows the arrangement. Every other satellite in the calculation stands one to three orders of magnitude clear of the nearest tail, and this one stands one order clear. A seven per cent error in an amplitude is a fourteen per cent error in an intensity, and the tilt being measured is a difference of intensities of a few per cent. Nothing more is needed to lower a ratio of one half to a third.

The other reflections explain why only $h = 1$ misbehaved. The inner second-order satellites at $h = 2$ and $h = 3$, at $k = 1.236$ and $2.236$, also sit a quarter of a unit from a main reflection. Their own amplitudes are five to ten times larger, because $k$ is, and the main reflections at 1 and 2 are weaker than the zero-order one. The earlier essay found those two within 0.024 of a half, and it now appears that the 0.024 was the same effect, smaller.

The prediction and the measurement #

The tail formula predicts the error, and the prediction can be checked chain by chain. For twenty-two chain lengths between 10,000 and 480,000 sites, the picture sets the predicted tail at the satellite beside the measured error in the satellite’s amplitude, the plain sum against the exact series. The points lie along or under the line of equality. The tail sets the size of the error. The phase between the tail and the satellite’s own structure factor decides how much of that size adds to the satellite’s modulus and how much turns it sideways, so the error can be smaller than the tail but never larger. At 10,000 sites the predicted tail is 18 per cent of the satellite and the measured error is 18 per cent.

Why two lengths agreed #

The oscillating numerator $|\sin \pi N\Delta|$ is the whole story of the coincidence. At the satellite, $\Delta = 1 - 2q = 0.2360679\ldots$, and what matters is the fractional part of $N\Delta$. For $N = 30{,}000$ it is 0.04, for 60,000 it is 0.08 and for 120,000 it is 0.16. All three are close to a whole number, and they are close in proportion, since each length is a multiple of the first. The tail’s envelope falls by a factor of four from 30,000 to 120,000, but the numerator rises by nearly the same factor, and the error at the satellite comes out almost the same. The ratio was 0.31 at 30,000, 0.30 at 60,000 and 0.32 at 120,000.

The picture at the head of this essay runs the length through twenty-two values. At 20,000 sites the fractional part is 0.36 and the plain ratio is 3.05. At 13,000 it is 0.88 and the ratio is slightly negative. At 480,000 it is 0.63 and the ratio is 0.63. The ratio swings further from one half as the chain gets shorter and closer to it as the chain gets longer, but not monotonically. Agreement between any two lengths says only that $N\Delta$ had similar fractional parts at both. The earlier test compared two multiples of one length and so compared two nearly equal tails.

Over many lengths the fractional part behaves differently. Because $\Delta$ is irrational, the fractional parts of $N\Delta$ for successive $N$ spread evenly over the unit interval, which is Weyl’s theorem on equidistribution, and the square of the oscillating factor averages to one half. A rational wavevector would behave the other way, its fractional parts cycling through a few values and never filling the interval. That difference between a fraction and an irrational is the one that gives every fraction a window of its own among a chain’s ground states. So over a random choice of length the error has a typical size of the envelope divided by $\sqrt2$, and it falls as $1/N$ in that averaged sense. What the averaged statement hides is the individual length. Any particular $N$ can land close to a zero of the oscillation or close to a peak, and multiples of one length land close to where the first one did, since the fractional parts of $N\Delta$, $2N\Delta$ and $4N\Delta$ are small together when the first is small. A test that doubles the length from a lucky starting point stays lucky.

Agreement at two sizes is not convergence when the error oscillates with the size. The earlier essay’s test had the right form, doubling or quadrupling $N$ and looking for a change, and it is the standard check for a finite-size error. It fails here for a reason particular to sums over a chain with an irrational wavevector. The error depends on $N$ through a fractional part, not through a power, and a fractional part returns close to its old value at every multiple.

A window removes the tail #

The cure is equally standard in signal processing, which meets the same problem whenever a weak spectral line sits near a strong one. The tail of a peak comes from the abrupt ends of the sum. Weighting each site with a function that rises smoothly from zero at one end and falls smoothly to zero at the other softens the ends, and the tails fall much faster. With the Hann window, $\sin^2(\pi(n + \tfrac12)/N)$ for site $n$, they fall as $1/(N\Delta)^3$ instead of $1/(N\Delta)$. At $N = 30{,}000$ the windowed zero-order tail at the satellite is about $5 \times 10^{-13}$, nine orders of magnitude under the satellite, and the picture of the tails shows every satellite standing far clear of it.

With the window, the ratio at $h = 1$ is one half to within 0.002 at every one of the twenty-two lengths, flat across the whole range. The windowed sum at 10,000 sites is closer to the infinite chain than the plain sum at 480,000.

The earlier essay’s figure has been redrawn from windowed sums. The same four chains, with size factors of $\pm 0.03$ and $\pm 0.06$, now give ratios between 0.499 and 0.502 at $h = 1$ and between 0.500 and 0.507 at $h = 2$ and 3. The points of all three reflections lie on the line of slope one half. The largest remaining departure is at $h = 3$ with the larger size factor, which is the next-order term the series table found, arriving where it should.

The first order was never at risk #

The same tail reaches every satellite, and it is fair to ask whether the first-order tilt, the reading the whole method rests on, carried a similar error. It did not, and the reason is the same arithmetic. The first-order satellites at $h = 1$ lie at $k = 0.618$ and $1.382$, with amplitudes near 0.2 per site, and the nearest main-reflection tails there are about $2 \times 10^{-5}$ at 30,000 sites, a part in ten thousand. Read with and without the window, the first-order tilt of every chain in the earlier figure agrees to the fifth decimal place: $-0.03693$ both ways at $\varepsilon = 0.03$, $0.07202$ both ways at $-0.06$. The second-order ratio went wrong because it divides a small difference of weak intensities by a quantity that was right. The whole of the error sat in the numerator, at the one satellite where the numerator was weakest.

That also says which of the earlier essay’s conclusions are safe. The phase sweep, in which the ratio of the two readings of the size factor follows $\cos\theta$, used the second reflection, where the inner satellite is five times stronger. The reduced second-order tilt at the first reflection changes, from an unexplained third to the predicted half. One comparison deserves a second look. The two readings of the size factor, from the first-order tilt and from the second-order strength, were found to agree within 7.6 per cent on chains of only 6,000 sites, using the first reflection among others. At that length the zero-order tail at the weak satellite is five times larger than at 30,000, and part of that 7.6 per cent may be the same tail. That comparison has not been repeated with a window.

What this says about measuring a weak reflection #

The error was in a calculation, but the arithmetic belongs to measurement too. A real crystal is a finite chain, or a mosaic of finite domains, and its reflections have tails set by the size of the coherent domains, the antiphase domains of an ordered crystal among them. As sharp as the sphere is wide met the same function in reciprocal space, as the ripple a truncated sum leaves around every peak of a map. A weak satellite near a strong main reflection is measured on top of that reflection’s tail. The inner second-order satellite of the first reflection is the worst case of that kind, weakest because its scattering vector is shortest and nearest to the direct beam, the strongest reflection an experiment has.

Two differences soften the warning for an experiment. A diffractometer integrates intensity over a peak rather than reading the structure factor at a point, and the integral of a tail over a peak’s width is smaller than its value at the peak’s centre. And real domains come in a range of sizes, so the oscillating factor $|\sin \pi N\Delta|$ averages to a constant instead of landing by chance near zero. What survives is the scaling. The contamination of a weak reflection by a strong neighbour is roughly the ratio of their amplitudes divided by $\pi$ times the number of cells in a domain times their separation in reciprocal units. For this satellite that ratio of amplitudes is well over a thousand.

What the account does and does not settle #

It settles the h = 1 ratio for these chains. The series, the windowed sums and the tail prediction agree, and each was computed separately. The ratio is one half at every reflection up to the size factors where the next order becomes visible.

It does not compute the next order in closed form. The series table shows the next-order departure at $\varepsilon = 0.12$ numerically, growing with $h$. The expansion of the ratio to the next power of $u$ has not been written out, and the table is the evidence for its sign and its growth.

The window is a choice. Other smooth windows give the same result with different constants. What matters is that the window goes to zero smoothly at both ends, and any window that does removes the zero-order tail by many orders of magnitude.

Who worked out the pieces #

The Fourier–Bessel description of a displacively modulated chain is the standard one for incommensurate structures, developed from the 1970s in the superspace work of de Wolff, Janner and Janssen, in which an extra dimension makes the chain periodic. The tail of a finite sum, the Dirichlet kernel, is Dirichlet’s. The windows that tame it are the everyday tools of spectral estimation, catalogued by Harris in 1978. Julius von Hann gave the Hann window its name through his work on meteorological series.

What the account must refuse #

The refused claim is the one the earlier essay made. The test asks the chains of 30,000, 60,000 and 120,000 sites to agree with one another within 0.05 while all three lie more than 0.1 from one half, and they do. Agreement without correctness is exactly the situation that must not be read as convergence. The second test is the one that settles the matter. A window that removes the tail must give one half at every length, and a correction that worked only at the lengths it was tried on would fail it.

Still open: the curved size effect #

With the anomaly gone, the second order can do what it was introduced to do. The second order reads the strain again left a second question, the curvature of the size effect. A quadratic term in the relation between occupancy and spacing gives the displacement a second harmonic. That harmonic makes second-order satellites directly, in proportion to the curvature and independent of the occupation wave’s amplitude. Separating that direct contribution from the cross term the linear size effect produces needs the second order at several reflections, and it needs them measured to within a few per cent. That is exactly the precision the plain sums could not deliver at the first reflection. With windowed sums it is available at every reflection, and whether two reflections are enough to separate the linear and quadratic size effects is now a calculation that can be trusted.