Two waves measure the curvature one wave hides

A modulated crystal is one in which the occupation of its sites varies along it in a wave, and in which the spacing between sites follows the occupation: where a site holds more of the larger atom, the next spacing is longer. That dependence is the size effect, and in its simplest form it is a straight line, Vegard’s law. The second order cannot see a curved size effect alone allowed the line to curve. A quadratic term in the spacing gives the displacement a second harmonic, the second-order satellites should see it, and the question was whether they could measure it.

They could not, on their own. At every reflection a second-order satellite collects a term from the linear size effect and a term from the curvature, and the two grow with the scattering vector in the same way, so each satellite sees one fixed mixture of the two coefficients: the ratio of its sensitivities lay between 0.24 and 0.27 at every satellite of the first three reflections. Adding reflections added copies of one equation. Only the first-order satellites, which see the linear effect and not the curvature, broke the tie, and even with them and three reflections the curvature was known to about nine per cent of itself.

The essay ended with a suggestion. A crystal modulated by two waves at once has satellites at the sum and the difference of the two wavevectors, mixed satellites, and the curvature makes a direct term there too, from the product of the two occupation waves inside the squared composition. If that term behaves differently from the cross terms the linear effect makes at the same satellites, two waves might measure the curvature where one could not. This essay builds the two-wave chain and makes the calculation. They do, and the reason is a sign.

A chain with two waves #

The chain is the one-wave chain with a second wave added. Each of its 30,000 sites has an occupation that is a mean of one half plus two cosine waves of amplitude 0.3, one with the golden wavevector $q_1 = 1/\varphi^2 \approx 0.382$ and one with $q_2 = 2 - \sqrt3 \approx 0.268$, two irrational numbers whose sum and difference are well separated from each other and from the integers. The spacing after each site is one plus $\varepsilon_1 d$ plus $\varepsilon_2 (d^2 - \langle d^2 \rangle)$, with $d$ the mean of the two neighbouring occupations minus one half, $\varepsilon_1 = 0.03$ and $\varepsilon_2 = 0.15$, the same coefficients as before.

The square of $d$ is where the second wave changes things. With one wave it held a constant and a second harmonic. With two it also holds twice the product of the two waves, and a product of cosines is a sum of cosines at the sum and at the difference of their wavevectors. So the curvature puts displacement at four wavevectors where the linear effect puts it at two.

The amplitudes follow from summing the spacings, as for one wave. The first harmonics are $\varepsilon_1 a_i \cos \pi q_i / (2 \sin \pi q_i)$, and the mixed ones

$$u_\pm = \frac{\varepsilon_2\, a_1 a_2 \cos \pi q_1 \cos \pi q_2}{2\, |\sin \pi(q_1 \pm q_2)|}.$$

Fitted to the chain by least squares at each of the four wavevectors, the displacement matches all four formulas to three figures. The difference harmonic, $4.65 \times 10^{-3}$, is two and a half times the sum harmonic, $1.83 \times 10^{-3}$, although both come from the same product of waves. The difference wavevector is small, $q_1 - q_2 \approx 0.114$, so the displacement it drives accumulates over a long wavelength before the wave turns, and a small sine in the denominator makes a large harmonic.

Two mixtures of opposite sign #

The test of whether a set of satellites can separate the two coefficients is the one the one-wave essay used. For each satellite, compute how much its intensity changes per unit of the curvature and per unit of the slope, and take the ratio. Satellites with one ratio measure one combination of the two coefficients and cannot separate them however many are measured. Satellites with different ratios measure different combinations.

The sum satellites, at $h \pm (q_1 + q_2)$, behave like the one-wave second order: every one of the six at the first three reflections has a ratio between 0.20 and 0.22, so they too see one mixture. The difference satellites, at $h \pm (q_1 - q_2)$, also see nearly one mixture at each reflection, with ratios between $-1.02$ and $-1.94$. The two mixtures have opposite signs. A small increase in the curvature raises the intensity of a sum satellite in the same direction as a small increase in the slope does, and raises a difference satellite in the opposite direction. Two measurements whose sensitivities to two unknowns have opposite ratios are as far from redundant as two measurements can be.

Why the sign differs can be traced, at least in outline, to the terms each satellite collects. At a mixed satellite the linear size effect contributes through cross terms, one occupation wave times the other wave’s displacement, and the curvature contributes directly, through the mixed harmonic of the displacement. At the sum and at the difference the cross terms enter with the same sign, since both products of the two cosines carry a plus, but the direct term at the difference comes from a displacement wave whose phase is the difference of the two waves’ phases, and the occupation that multiplies it at that satellite carries the conjugate of one of them. On that reading the sign of the direct term relative to the cross terms flips between the two kinds of satellite. The outline is offered as a reading, not a derivation: the calculation measures the ratios, and the measured ratios carry the opposite signs at every reflection.

What each set of satellites can say #

The figure at the head of this essay turns the mixtures into precisions. For each set of satellites the Cramér–Rao bound, the smallest error any unbiased estimate can reach given the satellites’ intensities at two per cent error, is computed for the two size coefficients and the two amplitudes of the occupation waves. Two versions are drawn. In one the amplitudes are refined along with the size coefficients, as they would be if nothing else fixed them; in the other they are taken as known.

The sum satellites alone are useless: their one mixture leaves the curvature unknown to eight hundred times itself with the amplitudes free, and to a third of itself even with them known. The difference satellites alone are as useless with the amplitudes free. Taken together, at a single reflection and with the amplitudes known, the sum and difference satellites fix the curvature to 0.6 per cent, with the errors in the slope and the curvature almost uncorrelated, a correlation of $-0.07$. At three reflections that improves to 0.3 per cent.

With the amplitudes free the picture changes, and the reason is instructive. A mixed satellite’s intensity is proportional to the product of the two waves’ amplitudes as well as to the size coefficients, so a change in the amplitudes can imitate a change in either coefficient. With the amplitudes refined, the sum and difference satellites at three reflections give the curvature to only 47 per cent, and the two coefficients are again correlated at 0.9998. The opposite signs separate the slope from the curvature; they do not separate either from the amplitudes.

The first-order satellites supply exactly what is missing. They see the occupation waves directly, and so the amplitudes, and the slope through the first harmonics. With every satellite of the first three reflections, first order, second order, sum and difference, and nothing taken as known, the curvature comes out to 0.6 per cent. For comparison, the one-wave chain’s best, first and second orders at three reflections, gave about nine per cent, and the first-order satellites of this chain alone give eight. The second wave improves the measurement of the curvature by more than a factor of ten, and the improvement comes from the mixed satellites.

Two thin regions and where they cross #

The geometry of the measurement is the clearest picture of why it works. The sum satellites alone constrain one combination of the slope and the curvature very well and the other not at all, so their error region is a long thin band in the plane of the two coefficients. The difference satellites give another thin band, and because their mixture has the opposite sign, the band tilts the other way. Where two thin bands cross at a large angle the region consistent with both is small, and the figure shows it: a small, nearly round region at the crossing.

The one-wave essay drew the same plane and found a single band, the second-order satellites’ line, which only the first-order satellites could close, and then into a long ellipse. Here two kinds of satellite close it from opposite sides, and neither of them is a first-order satellite.

The second order of either wave, revisited #

One entry in the table was not expected. The second-order satellites of the two waves, at $h \pm 2q_1$ and $h \pm 2q_2$, which for one wave alone could not see the curvature at all, give it to 21 per cent with the amplitudes free and 3.4 per cent with them known. The second wave has changed what they see. Their ratios now spread from 0.18 to 0.32 instead of crowding between 0.24 and 0.27, because the mean of the squared occupation, which the curvature’s term is measured from, now depends on both waves, and each wave’s second harmonic is shifted by the presence of the other. The degeneracy the one-wave essay found was a degeneracy of one wave, not of second-order satellites as such, and the second order reads the strain again, which first put the second order to work on the slope, described the case in which it holds.

This matters for reading the earlier result. A real modulated crystal with a single sharp modulation wave is the case the earlier essay described, and its conclusion stands there: the first order is what makes the second order mean anything. A crystal with two modulations, or an intergrowth with two periodicities, is a different instrument, and in it the satellites that were blind become informative.

What this means for measuring a curved size effect #

The practical reading is a change in what to measure. In a crystal with two modulation waves, the satellites worth the beam time for the curvature are the weak mixed ones at the sum and difference of the wavevectors, and the first-order satellites that fix the amplitudes; the second-order satellites help and are not essential. Repeating measurements at one kind of satellite, as every reflection, several times over would suggest for random error, buys little against a degeneracy: what buys precision is a second kind of satellite with a different mixture.

It also says something about which crystals can tell a curved size effect from a straight one at all. A departure from Vegard’s law is a statement about how spacing depends on composition over a range of compositions. A single modulation wave samples that range along one cosine and sees the curvature only through a harmonic the slope’s cross terms imitate. Two waves sample it along a two-dimensional pattern of compositions, and the product terms are where the curvature shows itself without imitation. The strain wave that tips the satellites found the slope’s signature in the satellites’ asymmetry; the curvature’s is in the mixed satellites’ sign.

How large the mixed satellites are #

Weak reflections are only useful if they can be measured, and the mixed satellites are weak. At the first reflection the sum satellites have intensities near $8 \times 10^{-5}$ and $4 \times 10^{-6}$ of the per-site scale, and the difference satellites $5 \times 10^{-5}$ and $3 \times 10^{-5}$; the main reflection is of order one quarter, so the mixed satellites are three to five orders of magnitude below it. The second-order satellites of either wave are of the same order, so the mixed satellites are no harder to measure than the reflections the one-wave essay relied on. The difference satellites are in fact among the stronger of the weak ones, because the large difference harmonic feeds them.

The precision quoted above assumes a two per cent error on each intensity, the same assumption the one-wave essay made, and that is generous for a reflection five orders of magnitude below its neighbour. The ranking of the sets does not depend on the size of the error, since the Cramér–Rao bound scales with it, but the absolute precisions do. A real measurement at ten per cent error on the weak satellites would give the curvature to about three per cent rather than 0.6, still several times better than one wave could do at the same error. A bound is a statement about the best possible, and a measurement that reaches it is the exception.

What the calculation leaves out #

One chain, two waves of equal amplitude. The amplitudes, the wavevectors and the coefficients are one choice, made so that every kind of satellite is resolved from every other. Other choices change the numbers, and a difference wavevector near zero would put the difference satellites under the main reflections, where they could not be measured separately. The sign argument does not depend on the choice, since it comes from the phases of the product terms, but the precisions do.

A bound, not an estimate. The Cramér–Rao bound is the best any unbiased estimate can do at the stated intensity error, and it is reached only when the model is right and the errors are as assumed. A real refinement adds the systematic errors a smooth error introduces, and the weak mixed satellites are the ones most exposed to them.

One dimension. The chain is a line of sites. A real modulated crystal is three-dimensional and its modulation waves may run in different directions, so that the sum and difference satellites lie off the line of the main reflections. The calculation here is the one-dimensional section of that, and the sign of the mixtures is what would need checking in the full geometry.

Windowing as before. Intensities are computed from a Hann-windowed sum over the chain, the window the satellite beside the zero-order reflection found necessary to keep a weak satellite from being swamped by the leakage of a strong neighbour. The difference satellites sit close to the main reflections, at a separation of 0.114, and without the window they would be the first casualties.

What the calculation must refuse #

The refused claim is the natural first guess, that the sum satellites, the ones the earlier essay pointed at, separate the two coefficients by themselves. They do not: their errors in the slope and the curvature are correlated at $-0.9997$ even with the amplitudes known. Only the difference satellites, with their mixture of the opposite sign, make the sum satellites useful.

Who worked out the pieces #

The description of modulated crystals by satellites and their orders is de Wolff’s, with Janner and Janssen’s superspace formalism of the 1970s supplying the bookkeeping for several independent wavevectors, in which the mixed satellites at the sum and difference of two wavevectors are reflections with two non-zero satellite indices, the satellites that need a second integer, a case beyond the single wave whose groups superspace groups in the plane counts. Vegard stated his law in 1921, and departures from it are a standard object of measurement in alloys and solid solutions. The Cramér–Rao bound is from the 1940s. What the calculation adds is the observation that the mixed satellites carry the curvature with mixtures of opposite sign, so that two waves separate what one wave confounds.

Still open: three waves, and whether the cubic term shows #

Two waves put the square of the size effect into the displacement at their sum and difference. Three waves would put a cube there too: a cubic term in the spacing, $\varepsilon_3 d^3$, has a product of three waves, and makes displacement at the sums and differences of all three wavevectors taken together, where neither the slope nor the curvature puts any directly. Whether those triply mixed satellites see the cubic coefficient with a mixture distinct from everything else, so that a three-wave crystal could measure the third derivative of the spacing against composition, is the same calculation with one more wave and one more coefficient. It has not been made.