The order a diffuse pattern measures

A diffuse pattern is not a smear to be subtracted. It is a measurement, and what it measures is how much one site of a crystal knows about another.

The quantity is a correlation: for each separation, the average of the product of the two occupations, which is one when the two sites always agree, minus one when they always disagree, and zero when knowing one says nothing about the other. The array of those numbers over separations is the whole of what a diffuse pattern reports, and the relationship between the two is a Fourier transform in both directions.

Where the maximum sits #

The single most useful reading of a diffuse pattern needs no transform at all: the position of the maximum says what the crystal is trying to become.

A crystal whose atoms prefer unlike neighbours is trying to alternate, and an alternating arrangement is a superstructure with a doubled cell — so its diffuse intensity piles up at the wavevector that doubling would produce, on the zone boundary. A crystal whose atoms prefer like neighbours is trying to separate into regions of each kind, and separation is a long-wavelength arrangement — so its intensity piles up near the origin.

Measured on the ensembles here: at a nearest-neighbour correlation of −0.40 the maximum is at the corner of the zone, at (½, ½); at +0.42 it is at the origin. Nothing between the two regimes has a maximum anywhere in particular, because a crystal with no preference scatters flatly.

That is the reading a crystallographer makes first, and it is qualitative in the right way: it does not need the intensity scale, the correlations, or a model, only where the pattern is brightest.

The transform pair, both ways #

The formal statement is that the diffuse intensity and the correlation array are a transform pair, and this collection computes both and compares.

Given the configurations, the intensity is computed directly — average |F|², subtract |⟨F⟩|². Given the same configurations, the correlations are computed by counting agreeing pairs of sites, and then transformed. The two agree to about 10⁻¹⁴ relative, at every wavevector, in every correlation regime.

The agreement is the content. It says the diffuse intensity contains the correlations and nothing else: no information is lost in either direction, so measuring the pattern everywhere determines the correlations completely, and knowing the correlations determines the pattern.

That is a stronger statement than it looks, because it fixes what a measurement of diffuse scattering can be for. It is a measurement of pair statistics, complete and exact, and it is not a measurement of anything else — which is a sharper version of the standing question of what diffraction cannot tell apart.

What the pattern cannot determine #

Many arrangements share a correlation function, and the diffuse pattern cannot tell them apart.

That is the same underdetermination this collection meets as two structures with one Patterson — where two different atomic arrangements have identical sets of interatomic vectors — restated for a disordered crystal. The pair statistics are fixed; the three-site and higher statistics are not, and different arrangements with the same pairs are indistinguishable to this measurement.

The practical consequence is that a model reproducing a diffuse pattern is not thereby the right model. It is a model with the right pair correlations, and there are others. What can be said with confidence is exactly the pair statistics; anything beyond them in a published model comes from an assumption, and the assumption should be visible.

There is a second and cruder limitation. The transform relationship is between the intensity and the correlations of the ensemble, and a measurement is made on one crystal. The two coincide when the crystal is large and its regions independent, which is an assumption that is usually right and is not checked by the measurement itself.

What the near neighbours say #

The correlations at the first few separations carry most of what a reader wants, and they have a standard normalisation worth naming.

The Warren–Cowley parameter for the i-th neighbour shell is the correlation there, scaled so that it is zero for a random arrangement and one for complete separation into like regions. A negative value means unlike neighbours are preferred; a positive one, like. The whole array is what the transform needs, but the first two or three values are what a description of a material usually quotes.

Two features of the arrays here are general rather than particular. The value at the origin is exactly one — a site always agrees with itself — which is not a measurement but a normalisation, and any computation returning something else has a bug. And the values alternate in sign for an alternating crystal: negative at the first neighbours, positive at the second, negative at the third, which is the arrangement propagating its preference outwards.

The rate at which they fall away is the correlation length, and it is the one number in the array that a transition changes qualitatively: it grows without bound as the crystal approaches ordering, which is what makes the diffuse maximum sharpen into a reflection — and, at the same moment, what makes the order parameter become non-zero.

The transition, seen from the diffuse side #

A crystal approaching an ordering transition is one whose correlations reach further and further, and the diffuse pattern narrows accordingly.

At the transition the correlations stop falling away at all: the crystal has long-range order, and the diffuse maximum has become a superlattice reflection at exactly the wavevector the maximum sat at. This collection has that reflection from the other side, as the consequence of an order parameter at a zone-boundary wavevector, and the two descriptions meet here.

So the diffuse maximum is a prediction of the transition. Its position says which superstructure the crystal will form — which wavevector will double the cell — before any long-range order exists, and it can be measured well above the transition temperature where no reflection is present at all.

What the diffuse pattern cannot say is whether the transition will actually happen. A crystal can have strong correlations at a wavevector and never order, and the intensity of the maximum grows with the correlation length rather than with the likelihood of anything.

A worked reading, from pattern to arrangement #

Taking one of these ensembles and reading it as an experimentalist would makes the chain of inference visible.

The pattern. Bright at the corner of the zone, dark at the origin, with a broad maximum rather than a spot.

The first inference. A maximum at the corner means the crystal prefers unlike neighbours in both directions — the arrangement it is trying to reach doubles the cell along each axis, which is a checkerboard.

The second. The maximum is broad rather than sharp, so the preference does not reach far: the correlation length is a few cells. Transforming the measured pattern gives the correlations directly, and they come out −0.40 at the first neighbours, positive and smaller at the second, and negligible by the fourth.

The third, and the one to be careful with. The crystal is not ordered. Nothing in it repeats with the doubled cell; there is no superlattice reflection; and a structure refinement would report a perfectly ordinary disordered alloy with half-occupancies at every site. Everything about its short-range preference is in the diffuse pattern and nowhere else.

That last step is where the diffuse scattering earns its place. The Bragg intensities of this crystal are identical to those of a completely random arrangement of the same composition, and the two crystals differ in every one of their local arrangements.

The correlations are a small array, and that is the point #

A structure refinement of an ordered crystal returns a handful of coordinates. A description of a disordered one returns a handful of correlations — and the smallness of both is what makes them useful.

The correlation array falls away with distance, so the number of parameters needed to describe the local order is set by how far the preference reaches: a few for a crystal with short-range order, more for one near a transition. That is a genuinely small description of an arrangement that has an astronomically large number of possible states.

And the parameters are measured rather than fitted, in the sense that each is an integral of the diffuse intensity against a cosine, which needs no model. That is unusual: most descriptions of disorder are model-dependent, and the pair correlations are the exception.

The limitation is the one already stated — pair statistics fix pair statistics — and it is what makes the small description honest rather than lossy. A parameter count that matched the number of possible arrangements would be a description of nothing.

What the diffuse intensity is in the ordered phase #

One boundary case is worth following, because it is where this essay and the transitions ladder share a wavevector.

Below an ordering transition the crystal has a superlattice reflection at the wavevector its diffuse maximum sat at. The diffuse scattering does not disappear: what remains is the scattering of the departures from perfect order — the antiphase boundaries, the remaining wrong-site atoms — and it sits around the superlattice reflection rather than in a broad hill.

So the same measurement made above and below the transition reports two different things in the same place: the tendency to order, and then the imperfection of the order achieved. Distinguishing them is a matter of the shape rather than the position, since both are centred on the same wavevector.

That is worth knowing because it is a standing source of confusion in the literature: diffuse intensity near a superlattice reflection is often described as short-range order surviving below the transition, and it can equally be domain structure. The two have different shapes — the first is broad and isotropic, the second is streaked along directions the walls permit — and telling them apart is a reading of the pattern rather than of its maximum.

Reading the symmetry of a diffuse pattern #

The diffuse intensity carries a symmetry, and it is not the symmetry of the arrangement.

Intensities are invariant under inversion of the wavevector, whatever the structure — the same statement as Friedel’s law for Bragg reflections — and the diffuse intensity of an ensemble is invariant under every operation of the average structure’s group. So the pattern has at least the symmetry of the average structure, and the arrangements it comes from need not have any of it.

That is the disordered case of the point this collection makes about the symmetry of an average: the average is more symmetric than what it averages, and the measurement reports the average.

The useful consequence is a check. A diffuse pattern with a symmetry lower than the average structure’s is evidence of something the model has not got — a second phase, a preferred orientation, a strain field — because the ensemble cannot produce one.

What is modelled here, stated plainly #

The configurations are generated by a rule and nothing about them is claimed to be a material.

Each is a block of ±1 produced by single-site updates at a fixed coupling from a fixed seed, which is an Ising arrangement. It is used because its correlations can be tuned and then measured, so that the transform relationship is tested against a range of cases rather than one. Nothing about a real alloy’s chemistry is in it.

The seeded generator matters for a reason beyond reproducibility: a figure whose numbers changed between builds could not be checked by a reader, and this site’s figures are meant to be checkable. Two runs of the same seed give the same configuration, and that is one of the assertions the machinery makes about itself.

Three things a diffuse pattern is often asked for and cannot give #

The composition. The average occupancy is in the Bragg intensities, not the diffuse ones; the correlations are normalised so that composition drops out. A crystal’s composition and its short-range order are measured by different parts of the same pattern.

The arrangement. Pair statistics only, as above.

The energetics. It is tempting to read a correlation as an interaction energy — a preference for unlike neighbours as an attractive interaction between unlike atoms — and the inference needs a model of the statistics as well as the measurement. The relationship between correlations and interactions is temperature-dependent and is not the identity; treating it as one is a standard error, and this collection computes neither side of it.

The same measurement, made on positions instead of species #

The occupations here are species — which of two atoms sits at a site — and the identical machinery reads a different kind of disorder if the variable is a displacement instead.

Let each site carry a small displacement rather than a label. The average structure is the atoms at their mean positions; the variance is the correlation of displacements between sites; and the diffuse intensity is again its transform, with one difference that matters: the intensity now grows with the square of the wavevector, because a displacement’s effect on a phase does.

That factor is the standard way the two are told apart in a measurement. Occupational diffuse scattering has an intensity roughly independent of how far out in reciprocal space it is measured; displacement diffuse scattering grows, and thermal diffuse scattering — the same thing with the displacements uncorrelated in time rather than in space — grows in the same way.

So a single pattern measured over a wide range of wavevectors separates the two kinds of disorder by their trend, without any model of either. This collection computes only the occupational case, and states the displacement case as the standard result it is.

What the pictures cannot show #

The pattern is drawn at the resolution the block permits. An N × N block reports intensity at multiples of 1/N and nothing between, so the figures are coarse where a measurement is smooth. That is the finite-block restriction again, and refining it means a larger block rather than a better drawing.

Nothing here is three-dimensional. Diffuse scattering in space can concentrate on planes or on lines — the shape says how many dimensions the correlations extend in — and the plane has no room for that distinction.

And no scattering factors appear. Every site here carries ±1, standing for the difference between two scattering powers. In a real measurement that difference is a number that depends on the radiation, which is why the same alloy’s diffuse pattern is strong for neutrons and weak for X-rays when the two elements are neighbours in the periodic table.

The other thing that scatters diffusely #

Everything computed here is occupational: which species sits where, with every site on its ideal position. Real crystals also have atoms off their ideal positions, and the two kinds of disorder scatter differently enough to be separated.

Occupational disorder scatters with no strong dependence on scattering angle. The intensity is a difference of scattering factors times the transform of the correlations, and apart from the slow fall of the form factors it is the same at high angle as at low.

Displacement disorder grows with angle. A small displacement changes a phase by an amount proportional to the wavevector, so its contribution to the intensity rises roughly as the square of the distance from the origin in reciprocal space. At low angles it is nothing; far out it dominates.

And it is antisymmetric about the Bragg positions. Displacements correlated with occupancy — a large atom pushing its neighbours outwards — produce diffuse intensity that is stronger on one side of a reflection than the other, and the asymmetry reverses from one side of the pattern to the other. Occupational disorder alone produces intensity symmetric about each reflection.

So the two are separable by where they are measured. Collect at low angle and the pattern is essentially the chemical correlations of this essay. Collect far out and the size effect appears, with its own information in it: how much each species distorts its surroundings, which is a quantity no average structure records.

Which is why diffuse data are collected over a large volume of reciprocal space. A single plane through the pattern shows the correlations; the volume shows the correlations, the displacements and the coupling between them.

Reading the correlations without a model #

The transform pair on this page suggests a method, and it has become the standard one.

Transform the diffuse intensity directly. Take the measured intensity, subtract the Bragg contributions, and Fourier-transform the remainder into real space. What comes back is a map whose value at a separation is the correlation at that separation — the array this essay computes, obtained from the data with no model in between.

That map is what is fitted, not the intensity. A candidate arrangement is judged by whether its correlations match the measured ones, which is a comparison of small numbers rather than of a noisy three-dimensional pattern, and it makes the underdetermination explicit rather than hiding it inside a fit.

The complementary approach builds configurations. Start from a random arrangement in a large box and swap atoms, accepting swaps that improve agreement with the measured intensity, until the box reproduces it. The result is one arrangement consistent with the data — not the arrangement, since many are, but a concrete object whose other properties can be inspected and whose plausibility can be judged chemically.

Both methods take the underdetermination as the starting point. The pattern fixes the correlations exactly and fixes the arrangement not at all, so a method that returns correlations is complete and a method that returns a structure is offering an example.