The spots say the cell and little else

Every superstructure counted once enumerated the arrangements of two species on the square lattice, counting each structure exactly once whatever the shape of its cell. It found 2, 2, 4, 11, 16, 40, 48, 148, 188, 452, 496 and 2,012 structures for cells of one to twelve sites. That is the list a search for new ordered alloys walks through, and it was itself built on counting what a group cannot tell apart and on the sublattices how many sublattices enumerates. Its closing section asked a question the census makes possible and had not made: how many of those structures could be told apart by the positions of their superlattice reflections alone, before any intensity is measured.

The question matters because positions are the cheap part of a diffraction pattern. A spot’s position is fixed by the geometry of the cell and survives almost every experimental imperfection, including absorption, extinction, a poorly known scattering factor or a crystal that is partly disordered. Its intensity survives none of them well. The reflections a superlattice adds computed where the extra spots of one ordered structure fall. This essay asks the inverse question, over the whole census: given only what is seen, how many structures remain. The answer turns out to depend on arithmetic, and whether the number of sites is prime decides most of it.

Three things a pattern records #

A superstructure with $n$ sites in its primitive cell has a lattice of translations $L$ of index $n$ in the parent square lattice. Its reflections lie on the reciprocal lattice of $L$, which contains the parent’s reciprocal lattice. The reflections at the parent’s reciprocal lattice points are the fundamental ones, present for every arrangement. The others, one for each of the $n - 1$ non-zero elements of the finite group that the two reciprocal lattices differ by, are the superlattice reflections. Each has the structure factor

$$F(k) = \sum_{x} c(x)\, e^{2\pi i\, k\cdot x},$$

summed over the $n$ sites of the cell, with $c(x)$ one where an A atom sits and zero where a B atom does.

Three levels of measurement are compared. Positions: which superlattice spots are lit, meaning $F(k) \ne 0$. Positions and composition: the same set, and the number of A atoms, which the fundamental reflection’s structure factor $F(0)$ records. Intensities: $|F(k)|^2$ at every position. At each level two structures count as the same when their records agree up to an operation of the parent’s point group $4mm$, since a crystal can be mounted in any orientation the parent lattice allows. Structures with different cells are never confused. The lit positions of a structure with exact period $L$ generate the whole reciprocal lattice of $L$, or the structure would repeat on a larger lattice, so the spots always reveal the cell. The comparison is made cell by cell.

The census #

The picture at the head of this essay is the answer at every size up to twelve. At twelve sites the positions of the spots separate the 2,012 structures into 167 classes, twelve structures to a class on average. Adding the composition, which one fundamental intensity supplies, raises that to 504. Measuring every intensity raises it to 1,815. That leaves 197 structures indistinguishable from another by any diffraction measurement of a single domain. The structures were recomputed here from the census’s own definition and agree with it at every size, so the three columns are measurements on the same list.

The positions column is the surprising one, and it is small for a specific reason. At twelve sites, 1,122 of the 2,012 structures light every one of the eleven superlattice spots their cell has. For all of those, the pattern of positions is the reciprocal lattice of the cell, complete, and says which cell and nothing more.

The distribution of class sizes shows the same thing. Twenty-two sets of positions belong to a single structure each. These structures have so particular a set of missing spots that nothing else shares it. At the other end, the sets in which every spot is lit hold between 66 and 120 structures each, one set for each of the eleven cells. Missing spots are informative, and a lit spot is the default.

How much the positions resolve depends less on how many sites the cell has than on how many divisors that number has. At nine sites the 188 structures give only 11 sets of positions, at ten sites 452 give 21, and at eight 148 give 33. At twelve sites 2,012 give 167. Nine has one divisor between itself and one, ten has two, eight has two and twelve has four. Every divisor $d$ of the number of sites supplies superlattice positions of order $d$, and each such position is a place where a spot can be dark. A number with many divisors gives an arrangement many independent ways to hide a spot. A prime gives it none, which is the case the next section makes exact.

A prime number of sites #

The columns behave differently at five, seven and eleven sites than anywhere else. There the positions separate the structures into exactly as many classes as there are cells: three at five sites, three at seven and four at eleven. The positions say nothing about the arrangement in the cell, because every one of those structures lights every superlattice spot.

The reason is a fact about roots of unity. When the cell has a prime number of sites $p$, the group that the superlattice positions form is cyclic of order $p$. At each non-zero position, the structure factor is a sum of distinct $p$-th roots of unity, one for each A atom, since each $e^{2\pi i\,k\cdot x}$ is a $p$-th root of unity and distinct sites give distinct roots. A sum of distinct $p$-th roots of unity is zero only if it takes all of them or none. The polynomial $1 + z + \dots + z^{p-1}$ is irreducible over the rationals, so any combination of roots with coefficients zero and one that vanishes must be a multiple of the whole sum. A structure with at least one A atom and at least one B atom therefore lights every superlattice spot. At a prime size a missing spot is impossible, whatever the arrangement. The census confirms it for every structure at two, three, five, seven and eleven sites.

At a composite size the argument fails, and spots go missing without any symmetry forbidding them. For $n = 8$ the eighth roots of unity include $-1$, and two A atoms whose phases at one position differ by a half turn cancel there exactly, while summing to something at the other positions. The pattern on the right of the figure is one such structure. A spot is dark because the arrangement happens to be balanced against one character of the cell’s group, which is an accident of the arrangement rather than an absence a space group enforces. Systematic absences are the other kind, forced by a glide or a centring, and a crystallographer reading a pattern has to tell the two apart.

Why a spot goes dark #

The prime case is the extreme of a rule that decides every dark spot of a certain kind. A superlattice position $k$ assigns each site of the cell a phase, and the sites with the same phase form the cosets of a subgroup, the kernel of $k$. If $k$ has order $d$, there are $d$ cosets, and the structure factor is

$$F(k) = \sum_{j=0}^{d-1} a_j\, \zeta_d^{\,j}, \qquad \zeta_d = e^{2\pi i/d},$$

where $a_j$ is the number of A atoms in the $j$-th coset. When $d$ is prime the same irreducibility applies. A combination of the $d$-th roots of unity with whole-number coefficients vanishes only if all the coefficients are equal. A spot of prime order is dark exactly when the A atoms are spread evenly over the cosets of its kernel. At a prime number of sites every coset is a single site, so spreading evenly means all or none, and no spot is ever dark. At eight sites a spot of order two splits the cell into two halves of four sites, and it is dark whenever the A atoms divide equally between them.

The rule was checked rather than trusted. Over every structure with up to twelve sites and every superlattice position of prime order, 15,141 pairs in all, it predicts darkness exactly: 1,414 dark spots and no exception. Positions of composite order, four or six or twelve, go dark in other ways as well, since a vanishing sum of fourth or sixth roots of unity need not have equal coefficients: $1 + \zeta_4^2$ vanishes with coefficients one, zero, one, zero, and $\zeta_6 + \zeta_6^3 + \zeta_6^5$ vanishes too. There are 996 such dark spots up to twelve sites, and the rule for them is a statement about vanishing sums of roots of unity that is longer to state and not needed here.

The rule turns the pattern of dark spots into a list of balance conditions: this half of the cell holds as many A atoms as that half, these three thirds hold equal numbers. A crystallographer reading a superstructure pattern reads exactly those conditions and nothing finer, which is why the positions resolve so little. Each dark spot is one equation, the arrangement has as many unknowns as the cell has sites, and at twelve sites a typical structure has no dark spot at all.

This is the counting form of an old warning: the positions of reflections determine the lattice and the extinctions, and nothing else. What the census adds is a measure of how little else they can determine when there are no extinctions to read. At prime sizes it is nothing. At twelve sites it is about twelve structures per class.

What the intensities do not decide #

Intensities go most of the way. At five, seven and eleven sites they separate every structure, 16 of 16, 48 of 48 and 496 of 496, so at those sizes a complete intensity measurement identifies the arrangement uniquely. At eight sites they separate 143 of 148, at ten 433 of 452, at twelve 1,815 of 2,012. The coincidences are homometric structures: different arrangements with identical intensities everywhere, which two structures, one Patterson found on a ring of points and which reappear here as ordered alloys.

The first pair appears on the eight-site cell that is a chain of eight: A atoms at sites 0, 1, 3 and 4 in one structure and 0, 1, 2 and 5 in the other. The intensities at the eight points of the pattern are 16, 2, 4, 2, 0, 2, 4 and 2 in both, the fundamental included. The two arrangements have the same multiset of differences between their occupied sites: every separation from one to four appears the same number of times in each. That is exactly the condition for equal intensities, since an intensity is the transform of the pattern of differences. No translation or operation of the square lattice carries one arrangement to the other. Where the pairs come from explains pairs like this one by factoring the structure as a polynomial and reversing one factor. On a cyclic group of order eight that construction has room to work, and on one of prime order up to eleven it evidently does not.

The ambiguities that remain are small and almost all pairs. At eight sites the five coincidences are five pairs, and at ten the nineteen are nineteen pairs. At twelve sites 191 intensity records are shared by more than one structure: 187 by a pair, two by three structures and two by four. No record at any size up to twelve is shared by five structures or more. So a complete intensity measurement on a single domain leaves an ordered alloy either identified or narrowed to two candidates, with four exceptions at twelve sites, and the choice between two candidates is of the kind where the pairs come from explains on a ring, a reversal of one factor of the arrangement. Whether every pair here arises that way has not been checked. That is a much stronger statement than the positions allow, where a typical twelve-site pattern is compatible with a hundred structures. The step from positions to intensities, not the step from a powder to a single crystal, is where most of the information enters.

There is a second, cheaper kind of coincidence. Exchanging the two species changes $F(0)$ and multiplies every other structure factor by $-1$, which leaves every superlattice intensity unchanged. Without the fundamental reflection’s intensity, a structure and its species swap are indistinguishable, and at twelve sites the intensity classes fall from 1,815 to 1,061 when the composition is dropped. Swapping the species does not halve the count counted structures that are their own swap, and those are the ones that do not merge.

Positions and composition are not the whole of the arrangement #

Between positions and full intensities lies a large gap: at twelve sites 504 classes against 1,815. Two structures with the same cell, the same composition and the same lit positions can still be different arrangements, and not merely one arrangement turned. The pair drawn here shares a cell of eight sites, two A atoms in each, and the same six of seven superlattice positions. The intensities differ, and so do the arrangements. For a powder pattern the loss is worse again. What a powder pattern loses showed that a powder merges every reflection at the same spacing, so positions that differ only by an operation of the lattice’s holohedry cannot even be told apart.

What the census assumes #

Long-range order, perfectly achieved. Every structure here is fully ordered. A real alloy quenched from high temperature keeps only part of its order, and the order a diffuse pattern measures shows what remains: the superlattice spots weaken by the square of the order parameter while their positions stay fixed. Partial order therefore never lights a dark spot or darkens a lit one, and the position census holds for it unchanged. The intensity census does not.

One domain, and a parent that stays square. The census compares patterns of single crystals up to the parent’s point group. A real ordered alloy usually grows in several orientation domains, and a pattern from all of them superposes their spots and can light positions no single domain lights. The parent lattice is also held exactly square, with no strain accompanying the ordering, which moves spots but does not change which are lit.

Point scatterers with a scattering contrast and nothing else. The two species differ only in scattering factor. Atomic displacements from the ideal sites, which every real superstructure has, would light spots that pure occupational order leaves dark, and would move some of the cancellations at composite sizes away from exact zero.

The comparison is up to the parent’s point group, not the structure’s. Two structures whose records agree after a rotation of the pattern are counted as one class. A crystal mounted in a known orientation relative to the parent would separate some of them, and the counts here are then lower bounds on what an oriented experiment could distinguish.

Who counted what #

Enumerating derivative superstructures is a standard computation in the search for ordered alloys. Gus Hart and Rodney Forcade published the algorithm most codes use in 2008, for lattices in any dimension. That superlattice reflections reveal the cell and the extinctions and no more is a textbook statement of crystallography. The observation that at a prime index no superlattice reflection can vanish is the arithmetic of cyclotomic polynomials, which Gauss used for another purpose. Homometric structures were first described by Lindo Patterson in 1939 and 1944.

What the census has to refuse #

The test that carries the argument about primes is the direct one: every structure at every prime size up to eleven lights every superlattice spot, checked structure by structure rather than inferred from the theorem. The refused claims are both ones a crystallographer might make in a hurry. A full intensity measurement leaves 197 of the 2,012 twelve-site structures ambiguous. The same positions and composition leave far more than a choice of orientation.

Still open: a three-species alloy and a cubic parent #

The census here is two species on a square lattice, because that is what the enumeration of structures supplies. An alloy designer’s usual question has three species on a face-centred cubic lattice. Three species change the prime argument. The structure factor at a superlattice position becomes a sum of roots of unity weighted by two different scattering contrasts, and a vanishing combination no longer needs all the roots or none. Whether a prime number of sites still forbids missing spots for three species depends on the contrasts being rationally independent, and whether the intensities still separate every prime-size structure is not known here. Both are finite computations on the same records, and neither has been made.