The reflections a superlattice adds

An alloy of two metals in equal proportion can sit on a single lattice in two quite different ways. Above a transition temperature the two species are distributed at random over the lattice points, and every point is on average the same as every other. Below it the species sort themselves out: one kind takes a sublattice, the other takes what is left, and nothing has moved.

That is an ordering transition, and it is one of the few structural changes in which no atom changes position. The lattice of positions is exactly what it was. What has changed is the repeat of the contents, and it is now larger — by a factor which is the index of the sublattice.

What the reciprocal lattice does about it #

The parent’s reflections are indexed by integers because the parent’s repeat is the parent’s cell. Doubling the repeat halves the reciprocal spacing, so between every pair of the parent’s reflections there is now room for one more — and that is the whole statement, in the form it is usually met:

A superlattice of index n in real space is a reciprocal lattice n times as dense, so exactly n − 1 new reflections appear per parent cell.

The count is not an estimate or a typical case. The reciprocal lattice of the sublattice contains the parent’s reciprocal lattice with index n, and the classes of that containment are the reflections: one class is the parent’s own, and n − 1 are new. Nothing else can happen, at any index, for any sublattice.

These are superstructure reflections, and they are the direct evidence that ordering has happened. A diffraction pattern taken above the transition has the parent’s spots only; taken below, it has the parent’s spots and a set of new ones whose positions say which sublattice the crystal chose.

Why they are weak, and when they are not there at all #

The intensity is where the arithmetic turns interesting. Write f_A and f_B for the two species’ scattering factors. Summing over the atoms of one supercell, a parent reflection collects every atom in phase and its amplitude is proportional to n_A f_A + n_B f_B — a sum. A superstructure reflection collects the two species with opposite phases and its amplitude is proportional to f_A − f_B — a difference.

So superstructure reflections are weak by construction, and how weak depends entirely on how different the two species are. Copper and gold differ by fifty electrons and order visibly; iron and cobalt differ by one, and their ordering is nearly invisible to X-rays.

The vanishing is exact and the figure asserts it, which is worth doing because exactly zero and too small to see are different claims with different consequences. An exactly zero amplitude is a statement about the arrangement; a small one is a statement about the experiment. Here the arrangement is ordered and the reflections are absent, so the absence of superstructure reflections is not evidence of disorder — it is evidence of low contrast, which is a different thing and is often confused with it.

That is why neutron diffraction is the instrument for ordering problems in alloys. A neutron scatters from a nucleus, and nuclear scattering lengths do not follow the atomic number: iron’s is 9.45 fm and cobalt’s is 2.49 fm, so a pair of elements that X-rays cannot tell apart differ by a factor of four to neutrons. Nothing about the crystal has changed; the contrast has.

The sum, written out #

The computation behind those figures is the structure-factor sum this site has used from the beginning, with two scattering factors instead of one:

$$F(h, k) = \sum_j f_j \exp(2\pi i (h x_j + k y_j)),$$

taken over the atoms of one supercell, with the indices h and k measured in the parent’s reciprocal lattice so that a reflection’s indices say directly whether the disordered crystal already had it.

For a parent reflection — h and k both whole numbers — every exponential is 1, because every atom sits at integer coordinates in the parent’s basis. The sum is then simply the total scattering power of the cell, and it does not matter at all which atom is where. That is the arithmetic reason ordering leaves the parent’s reflections untouched: the fundamental reflections cannot see the arrangement, only the average.

For a superstructure reflection the exponentials are the n-th roots of unity distributed over the cosets, and they sum to zero over a complete set. So the term that survives is the departure of each coset’s scattering power from the average — which is why the amplitude is f_A − f_B rather than f_A + f_B, and why it is zero when the two are equal. The two statements in the figures above are the two halves of one sum.

The same arithmetic as centring, with the sign reversed #

Centring as a sublattice does this computation in the other direction, and the pair is worth putting side by side.

Centring adds lattice points: the centred lattice contains the primitive one with index two, so its reciprocal is sparser, and half of the reflections that the primitive description would have had are systematically absent. Ordering distinguishes lattice points that were equivalent: the contents’ repeat contains the parent’s with index n, so the reciprocal is denser, and n − 1 reflections that the parent did not have appear.

One is a lattice of positions becoming finer and its reciprocal becoming coarser. The other is a lattice of contents becoming coarser and its reciprocal becoming finer. The arithmetic is identical and the direction is opposite, which is a good demonstration that a systematic absence and a superstructure reflection are the same phenomenon seen from two sides — systematic absences are the sparse half, and this is the dense half.

There is a practical consequence in the confusion between them. A crystal whose superstructure reflections are weak but present, indexed on the parent cell, produces a pattern in which most reflections are strong and a scattered few are weak — which is exactly what a centred lattice’s pattern looks like if the weak ones are dismissed as noise. Choosing the smaller cell then produces a structure with the wrong composition.

Which sublattice, and how many there are to choose from #

How many sublattices counts the choices: σ(n), the sum of the divisors, in the plane. At index two there are three, at index three four, at index four seven, at index five six. Each is a different ordering pattern, each adds n − 1 reflections, and the positions differ — so the number of new spots gives the index and their positions give the sublattice, which is the whole of how a superstructure is identified from a diffraction pattern.

Not all of them are equally likely, and the reason is symmetry rather than counting. An ordering on a sublattice that keeps the parent’s rotations produces a daughter with high symmetry and few domain states; one on a sublattice that breaks them produces more domains and more interfaces, which costs energy. The sublattices that stay square computes which are which, and the answer at index five — two of the six — is Fermat’s theorem on sums of two squares doing work in a question about alloys.

What the ordered pattern’s group is #

There is a symmetry statement underneath all of this, and it is the one this site is built to make: the ordered arrangement is a different pattern from the disordered one, so it has a different group, and the group can be found by handing the point set to a detector rather than by reasoning about it.

Ordering onto a sublattice of index n can give a pattern whose group is a klassengleiche subgroup of the parent’s — the same operations, on a lattice of n times the area — and that is the case a textbook draws. It is not the general case, and which orderings manage it is decided by one integer test rather than by inspection. An operation of the parent survives only if it carries the chosen sublattice onto itself, and that happens exactly when H⁻¹MH has integer entries: no lengths, no angles, no tolerance. An ordering onto a sublattice the parent’s rotations do not preserve takes operations away as well as translations, so its descent is neither translationengleiche nor klassengleiche — which by Hermann’s theorem means it is not maximal, and the chain from it up to the parent passes through one step of each kind.

The counts are small and worth having. Of the three orderings at index two, exactly one keeps a four-fold rotation: the centred one, which puts the minority species at the corners of the cell and at its centre. Of the four at index three, none does. Of the seven at index four, one. Of the six at index five, two — and those two are the turned square sublattices whose existence is Fermat’s theorem about sums of two squares, so whether a four-fold crystal can order at a given index without losing its rotation is an arithmetic question and not a chemical one.

When the class does survive, nothing but diffraction notices. The optical properties, the piezoelectric coefficients, the shape of the crystal: all of them are decided by the point group, which has not moved. What has changed is the translation lattice, and the two things that see a translation lattice are a diffraction pattern and an electron travelling through the crystal. Both notice; nothing else does. When the class does not survive — which is most of the time — the ordering is a stronger event than a superstructure: it is a genuine change of crystal class, with everything a change of class implies for the properties, and the diffraction pattern reports it in the same currency of extra spots either way.

The domains ordering produces #

There is a second consequence of the index, and it is the one that shows up in an electron micrograph rather than in a diffraction pattern.

When the crystal orders, each region has to choose which coset of the sublattice the first species takes. There are n cosets and nothing prefers one, so different regions choose differently and the crystal ends up divided into n kinds of region. Where two of them meet there is an antiphase boundary: the ordering pattern is the same on both sides and out of step, by exactly a parent lattice vector that is not a superlattice vector.

That is antiphase domains, and the count of domain states is the same index n. So one integer decides three separate things — the number of new reflections, the number of possible orderings’ worth of domain states, and the factor by which the cell grows — and they are the same integer because they are all the index of the same containment.

Two alloys everyone meets #

The plane is where the arithmetic can be checked and the examples live in three dimensions, so the connection is worth making explicitly, with the claims kept to what follows from the counting.

β-brass, CuZn. Above about 460 °C the copper and zinc are distributed at random over a body-centred cubic lattice. Below it they order, copper taking the cube corners and zinc the cube centres — an ordering of index two, since the corners are a sublattice of index two in the body-centred lattice. The prediction from the counting alone is one new reflection per parent cell, and the reflections that appear are the ones with h + k + l odd, which are exactly the ones the body-centred lattice’s centring had been extinguishing. The ordered structure is called B2, and it is the CsCl arrangement.

Cu₃Au. Above about 390 °C, copper and gold at random on a face-centred cubic lattice; below it, gold on the cube corners and copper on the face centres. That is an ordering of index four, and the counting predicts three new reflections per parent cell — which are the ones with h, k, l of mixed parity, again the ones the F-centring had been extinguishing. The contrast is large, because gold and copper differ by fifty electrons, and the superstructure lines are easy. The ordered structure is called L1₂.

In both cases the new reflections appear precisely where the parent’s centring had made reflections absent, and that is not a coincidence: ordering onto a sublattice of the centred lattice is the removal of the centring translation as a symmetry, and the reflections it had been extinguishing come back. A superstructure reflection is a systematic absence undone.

The zone folds, and the electrons notice #

The reciprocal-space statement has a consequence beyond diffraction, and it is the reason ordering transitions matter to a physicist rather than only to a crystallographer.

A repeat n times as large gives a Brillouin zone with one nth the volume, and every electronic state that lived in the large zone is folded into the small one. Where the new zone boundary cuts through occupied states, a gap opens — the ordering has made a new set of Bragg planes, and electrons at those wavevectors are now diffracted by the crystal’s own periodicity. So an ordering transition changes the electrical resistivity, the specific heat and sometimes the colour, all through a change of repeat with no atom moving.

The cell nobody chose draws that folding directly: the zone of the parent, and the zone of the superlattice at one nth the area, with the ratio asserted against the index. The same integer that counts the new reflections counts the folding, because they are the same containment of reciprocal lattices read two ways.

Where the exactness stops #

Three limits, and the first two are about what has been assumed rather than computed.

The ordering here is perfect. Every atom of one species sits on the sublattice and every atom of the other does not. A real ordered alloy has a long-range order parameter below one — some fraction of sites hold the wrong atom — and the superstructure amplitudes are proportional to that parameter, so they fall smoothly to zero as the transition is approached from below. Nothing in this essay computes an order parameter, and the intensities here are the fully ordered limit.

The scattering factors are constants. A real atomic scattering factor falls off with angle, and the two species’ factors fall off differently, so the contrast changes across the pattern. Everything computed here is at the level the structure factor is defined on this site: point scatterers of fixed strength, which is exact for the arithmetic and an idealisation of the experiment.

Nothing here says the transition happens. Whether an alloy orders, at what temperature, and into which of the σ(n) available patterns, is thermodynamics. This essay says what the diffraction pattern of each possibility looks like and how many possibilities there are. The site’s applied field keeps that line carefully and it is worth keeping here: a symmetry argument states a permission and predicts nothing.

Where the ladder goes next #

The sublattices anchor now has three rungs: how many there are, which of them keep the parent’s symmetry, and what ordering on one does to the diffraction pattern. The rung above is the one an alloy metallurgist would ask for next — the order parameter and the way the superstructure intensities track it, which needs statistical mechanics this site does not have and does not intend to acquire.

The rung sideways is more attractive: a superstructure reflection appears at a fractional index of the parent cell, and so does a satellite of a modulated structure. The two are told apart by whether the fraction is rational, and the satellites that need a second integer is where that distinction is made. A superstructure is the rational case of a modulation, and the lock-in transition between them is a crystal deciding to be one rather than the other.

Why neutrons see what X-rays cannot #

The vanishing above is a statement about a difference of scattering factors, so the instrument that solves the problem is the one whose scattering factors differ — and the reason neutrons oblige is worth stating, because it is not a matter of them being better.

X-rays scatter from electrons, so an atom’s scattering factor is essentially its electron count. Two neighbouring elements in the periodic table differ by one electron in twenty or thirty, and the difference that produces a superstructure reflection is that same one part in twenty, squared.

Neutrons scatter from nuclei, and a nucleus’s scattering length has no orderly dependence on atomic number at all. It varies erratically from element to element, because it is decided by nuclear structure rather than by counting. Two elements adjacent in the table can have scattering lengths differing by a factor of two.

Some are negative. Hydrogen, manganese and titanium scatter with the opposite sign to most elements, so an ordered arrangement of manganese and iron gives a difference of scattering lengths larger than either — the superstructure reflection is stronger than a naive estimate, not weaker.

And isotopes differ. Two isotopes of one element are chemically identical and can have quite different scattering lengths, so a sample can be prepared with the contrast dialled to any value including zero. That is the most direct possible test of the arithmetic on this page: make the two species scatter identically and the superstructure reflections must vanish exactly.

What the width of the new reflections says #

The superstructure reflections carry a second piece of information that the parent’s reflections cannot, and it comes from their being the ones that ordering created.

Ordering begins in many places at once. Separate regions of the crystal order independently, and two regions that chose different starting points meet at an antiphase boundary — a surface across which the ordering pattern is out of step, with no change of orientation. How many such choices there are is an index, and it is usually small.

The parent’s reflections do not notice. Every atom is still on the parent lattice, on both sides of the boundary, so the fundamental reflections are as sharp as the crystal’s own perfection allows.

The superstructure reflections do. Their scattering comes from the ordering pattern, which changes phase at every boundary, so the coherent regions are the antiphase domains rather than the whole crystal — and a reflection scattered coherently from a region of finite size is broadened in inverse proportion to that size.

So the two families of reflection measure two different lengths. Sharp fundamentals and broad superstructure peaks in the same pattern say that the crystal is a good crystal which is imperfectly ordered, and the ratio of the widths gives the average domain size directly. A pattern in which both are equally broad says something quite different — that the crystal itself is small or strained.