The sublattices that stay square

The square lattice has six sublattices of index five. Two of them are square.

That sentence contains the whole subject. A sublattice keeps one lattice point in five however it is chosen; whether the points it keeps still form a square lattice is a further question, with a different answer for every index, and the answer is a piece of number theory with nothing crystallographic in it. Index five has two square sublattices. Index seven has none. Index twenty-five has three.

The rule behind those numbers was proved by Fermat in the 1640s, in a letter about which primes are sums of two squares, and it decides which superstructures a square surface can form.

The test has no metric in it #

The natural way to ask whether a sublattice is square is to measure its basis vectors and compare their lengths and angle. That is a measurement, it needs a tolerance, and it is unnecessary.

A sublattice with basis matrix H is carried onto itself by an operation M exactly when H⁻¹MH has integer entries. The reasoning is one line: H⁻¹MH is the matrix of M written in the sublattice’s own basis, and an operation preserves a lattice precisely when its matrix in that lattice’s basis is integral — which is the fact the whole site rests on, applied one level down.

So the question is this sublattice square becomes: is H⁻¹RH integral, where R is the quarter-turn. Both matrices are integer matrices and the test is a divisibility check. There is nothing to measure and nothing to choose.

One subtlety makes the test legitimate. R is a quarter-turn of the plane, and it is a symmetry of the parent whatever it does to the sublattice — so asking whether the sublattice is invariant under it is asking about the sublattice’s own shape rather than about its relationship to the parent. A sublattice of a square lattice cannot be square in some other orientation: any quarter-turn about the origin is the same matrix, and it either preserves the sublattice or does not.

Sums of two squares #

Running the test over every index gives a list, and the list is famous.

An index has a square sublattice exactly when it is a sum of two squares: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29. The indices missing from it — 3, 6, 7, 11, 12, 14, 15, 19, 21 — are the ones divisible by a prime congruent to 3 modulo 4 an odd number of times, and no square sublattice of that index exists at any size or orientation.

The count is a divisor sum: adding +1 for each divisor one more than a multiple of four and −1 for each one less gives exactly the number of square sublattices of that index. Index five gives 1 − 0 + 1 = 2; index twenty-five gives three; index seven gives zero.

The identity behind all three columns is the arithmetic of the Gaussian integers, the complex numbers a + bi with a and b whole. A square sublattice of the square lattice is exactly the set of multiples of one Gaussian integer, its index is that number’s norm a² + b², and the number of sublattices of a given index is the number of ways of factoring that norm — which is why the count is multiplicative and why the primes congruent to 3 modulo 4, which stay prime among the Gaussian integers, contribute nothing.

Turned, and aligned #

The square sublattices divide again, and the second division is the one a crystallographer meets.

Some of them sit aligned with their parent — their basis vectors point along the parent’s, and every symmetry of the parent, mirrors included, carries the sublattice onto itself. Those are the indices that are a perfect square or twice one: 1, 2, 4, 8, 9, 16, 18, 25.

The rest sit turned. Index five’s two sublattices are the parent rotated by 26.57° in one sense and in the other. Each is square; neither is carried onto itself by the parent’s mirrors; and the mirrors exchange the two. So a superstructure at index five comes in a left-handed and a right-handed version, and a surface that forms one has broken a symmetry the substrate had.

Twenty-five, where all three cases occur at once #

Index twenty-five is the smallest index with three square sublattices, and it is worth drawing because the three are of two different kinds.

One is the parent scaled by five: aligned, related to nothing, the trivial 5 × 5 superstructure. The other two come from 25 = 3² + 4², are turned through 36.87° in the two senses, and are mirror images of each other. So a 5 × 5 reconstruction and a (√25 × √25)R36.87° reconstruction are both index-twenty-five square superstructures of the same surface, and they are not the same structure.

The arithmetic that separates them is the factorisation of 25 in the Gaussian integers: 5 = (2 + i)(2 − i), so 25 has the factors (2 + i)², (2 + i)(2 − i) and (2 − i)². The middle one is 5 itself and gives the aligned sublattice; the outer two are conjugates and give the turned pair. Three factorisations, three sublattices, and the pairing into one aligned and two turned is the pairing of the factors into self-conjugate and not.

Where this decides something physical #

Surface science writes superstructures in a notation that is nothing but this arithmetic. A reconstruction of a square surface is called (√5 × √5)R26.57° — a cell √5 times larger on each side, turned through 26.57° — and what that name says is the index-five square sublattice. The √2 × √2 R45° reconstruction is the index-two one; the 2 × 2 is the aligned index-four.

The list above is therefore a list of the reconstructions that can exist on a square surface, and the missing indices are missing everywhere. There is no (√3 × √3)R° reconstruction of a square surface, at any angle, because three is not a sum of two squares. There is one on a hexagonal surface, and it is among the commonest structures in the whole field.

There is a notational consequence worth naming, because it is a case of a description being narrower than the thing described. Wood’s notation — the (√5 × √5)R26.57° form — can only name a superstructure whose cell is the parent’s scaled equally in both directions and turned. That covers every square sublattice of a square lattice and nothing else: the other four at index five have no Wood name at all, because there is no single scale factor and no single angle that produces them.

The alternative is to write the sublattice as its matrix, which is what surface science calls the matrix notation and what this page has been calling a Hermite form. It names every sublattice, including the ones with no symmetry, at the cost of two integers more. So the two notations in use divide exactly along the line this essay draws — the pretty one for the symmetric cases and the general one for the rest — and a reader who has only met the first has met only the sublattices that keep a symmetry.

The hexagonal lattice keeps a different list #

Running the same test on the hexagonal lattice, with the third-turn in place of the quarter-turn, gives a different sequence: an index has a triangular sublattice exactly when it is 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, …

Those are the Loeschian numbers — the values of m² + mn + n² — and they are to the hexagonal lattice what the sums of two squares are to the square one. The arithmetic behind them is the ring of Eisenstein integers rather than the Gaussian ones, and the structure of the argument is identical: a triangular sublattice is the set of multiples of one Eisenstein integer, its index is that number’s norm, and the primes that stay prime in that ring — the ones congruent to 2 modulo 3 — are the ones that never appear.

The coincidence indices are the same list, twice filtered #

There is a second place these numbers have already appeared on this site, and the connection is exact.

The coincidence site lattice of two square lattices turned against each other is a sublattice of both, and it is square, being the intersection of two square lattices. So every coincidence index in the plane must be on the list above — and every coincidence index is odd, which removes the even entries. What is left is 5, 13, 17, 25, 29: exactly the plane coincidence series this site computed from Pythagorean triples.

Two derivations, sharing no code and hardly any vocabulary, arriving at one list. The coincidence route asks which rotations are rational; this route asks which sublattices are square; and the answers agree because a rational rotation carrying a lattice into itself is precisely a Gaussian integer of the right norm.

Who proved it, and how long the proof waited #

Fermat stated the characterisation of the primes that are sums of two squares in a letter to Mersenne in 1640, and did not prove it in anything that survives. Euler proved it in 1749, after — by his own account — seven years of intermittent attempts, which is a useful measure of how much easier the statement is than the argument.

The connection to lattices came much later and from the other direction. Gauss introduced the integers a + bi in 1832, in a memoir on biquadratic reciprocity, and the fact that they form a ring with unique factorisation is what turns Fermat’s theorem into a statement about norms. Once it is a statement about norms it is a statement about sublattices, since a sublattice of the square lattice closed under the quarter-turn is an ideal of that ring — and the counting function follows without any further work.

None of the people involved had a crystal in mind. The application to surface reconstructions is twentieth-century and it consists entirely of reading a table that was complete before the question existed, which is a pattern this site has met in twinning and in the restriction and which is not a coincidence: the constraint a lattice imposes is arithmetic, so the mathematics that answers it is the arithmetic that was there already.

What the answer does not settle #

Three limits.

Existence is not preference. The list says which square superstructures are geometrically available; it says nothing about which one a surface forms, which is an energy and is not computed anywhere here. Silicon’s (111) surface makes a 7 × 7 reconstruction whose unit cell contains 49 atoms and whose structure took twenty-five years to determine — geometry admitted it and geometry did not predict it.

Symmetry of the lattice is not symmetry of the pattern. A square sublattice is a statement about points, and a superstructure is a statement about what sits on them. An adsorbate arranged on a square sublattice can easily have a lower symmetry than the sublattice does, in which case the reconstruction’s group is smaller than this arithmetic allows — the motif matters, here as everywhere on this site.

The list is about lattices and not about what a real surface does to itself. A reconstruction usually moves the atoms of the top layer as well as adding a longer period, and the periodicity is the part this arithmetic sees. Two reconstructions with the same sublattice and different atomic arrangements are one entry here and two entries in a surface-science catalogue, which is the same relation a space group has to a structure.

And the two hands are indistinguishable in a diffraction pattern. The turned sublattices come in mirror-image pairs, and a measurement that obeys Friedel’s law cannot tell which one a given domain is. What it sees is both, because a real surface forms domains of each in equal numbers, and the resulting pattern has the substrate’s full symmetry restored by averaging over them.

How many domains a reconstruction has #

A turned sublattice has a mirror image which is also a sublattice of the same index, and the pair is not a bookkeeping detail: it is the number of distinct regions a real surface breaks into.

The parent lattice’s point group acts on the set of sublattices of a given index, and the sublattices fall into orbits. The size of a sublattice’s orbit is the number of domains a reconstruction on it can form, because each element of the orbit is an equally good arrangement and nothing decides between them — a surface nucleating its reconstruction in several places at once will pick different members in different places.

An aligned square sublattice is fixed by the whole of the parent’s point group, so its orbit has one member and the reconstruction has one domain. A turned one is carried to its mirror image by any of the parent’s reflections, so its orbit has two members and the reconstruction has two domains, related by a reflection and by a rotation through twice the turn angle.

That is directly observable. A diffraction pattern from such a surface is the superposition of the patterns of both domains, so it shows more spots than either domain produces and it carries the full symmetry of the substrate even though neither domain does. A reader who reads the symmetry of such a pattern as the symmetry of the structure will conclude that the reconstruction is aligned when it is not, which is the same error a twinned crystal produces one dimension up and for the identical reason.

Square sublattices become rare #

The list of indices with a square sublattice looks generous at the start — 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20 — and the impression that most indices are on it does not survive going further out.

The sums of two squares thin out. The count of them up to N grows like N / √(log N) rather than like N, so their density tends to zero, slowly. That is the Landau–Ramanujan theorem, and the constant in front of it is a specific number about 0.7642 whose only role is to make the estimate accurate.

The practical reading is that a square superstructure of large index is not merely one option among many; it is a rarity. Up to a hundred, forty-odd indices admit one. Up to a million, the fraction has fallen below a tenth. So a surface wanting a long-period square reconstruction has few periods available, and the ones it has are the ones whose prime factorisation avoids 3 mod 4 primes to odd powers — which is a strong constraint of a kind nothing physical imposes.

The hexagonal case thins the same way and to a different constant, since the Loeschian numbers are the ones whose 2 mod 3 primes appear to even powers. Both lists have density zero and both are infinite, which is the usual shape of an arithmetic condition: never exhausted, and never common.

Where the ladder goes next #

The arithmetic here is about which sublattices keep a symmetry. The companion question — which groups have subgroups on those sublattices, and how many — is the subgroup ladder, where the same sublattices reappear as the second half of a klassengleiche descent.

What is genuinely unfinished is the three-dimensional case. The sublattices of a cubic lattice that are themselves cubic are counted by an analogous rule involving sums of four squares, the coincidence indices in space are the odd ones among them, and this site has computed the second list without computing the first.

Both observations are about the same gap between what exists and what is available. The arithmetic says which indices have a square sublattice, and it says so for every index at once; what a surface can actually build is a short initial segment of that list, because a reconstruction of index forty is a long period and a real surface has a correlation length. The theorem is about the infinite list and every measurement is about its first dozen entries.