How many vectors of each length

Stand at a lattice point and count the others by distance. The nearest are all at the same distance and there are a few of them; then a gap; then a shell with more; then a length with nothing at all.

That sequence of counts is the lattice’s theta series, and it is the most natural measurement anybody makes of a lattice. It is also, unexpectedly, a question about divisors.

The counts are divisor sums #

Work in squared lengths, because they are integers. For the square lattice, the number of lattice points at squared distance n is the number of ways to write n as a sum of two squares — and Jacobi’s theorem says that is

$$r_2(n) = 4\bigl(d_1(n) - d_3(n)\bigr),$$

where d₁ counts the divisors of n congruent to 1 modulo 4 and d₃ those congruent to 3.

The 4 is not decoration. It is the number of units in the Gaussian integers — 1, −1, i and −i — because writing n as a sum of two squares is factoring n in ℤ[i], and every factorisation comes with four multiples of itself. The theorem is a statement about a ring, dressed as a statement about a lattice.

The hexagonal lattice tells the same story with different numbers:

$$r_{\text{hex}}(n) = 6\bigl(d_1(n) - d_2(n)\bigr) \bmod 3,$$

and the 6 is the number of units in the Eisenstein integers. Two lattices, two rings, two counts of units, and the same theorem twice.

Those Loeschian numbers are not new here. They are exactly the indices at which a hexagonal lattice has a sublattice keeping its symmetry, which is why seven turns up as the cell of the snub hexagonal tiling and as a sublattice index and now as a shell containing twelve points. One arithmetic, three appearances.

Which lengths occur at all #

The zeros in those tables are worth as much as the counts, and they are the older question.

Fermat’s two-square theorem says an odd prime is a sum of two squares exactly when it is congruent to 1 modulo 4: 5 = 1 + 4 and 13 = 4 + 9, while 3, 7, 11 and 19 are not sums of two squares at all. A general n is a sum of two squares exactly when every prime congruent to 3 modulo 4 occurs in it to an even power. So the square lattice has no vector of squared length 3, 6, 7, 11, 12, 14, 15 — and each absence is a prime congruent to 3 appearing an odd number of times.

Jacobi’s count is the same statement sharpened: when the excess of divisors is zero the length does not occur, and when it is positive it says exactly how many vectors there are. One formula answers both questions, which is why it is the theorem worth having rather than the existence statement. It also answers a third: how many essentially different ways there are, once the four signs and the swap of the two coordinates are divided out. That number is what a crystallographer would call the number of index families, and it is the multiplicity divided by the size of the orbit.

The hexagonal case runs identically with 3 in place of 4. A prime is a Loeschian number exactly when it is 3 or congruent to 1 modulo 3, so 2, 5, 11 and 17 never occur as squared lengths in the hexagonal lattice and 7, 13 and 19 do.

Both facts are about unique factorisation. ℤ[i] and the Eisenstein integers are both principal ideal domains, so a rational prime either stays prime in them or splits into a conjugate pair, and which of the two happens is decided by a congruence. A lattice’s shells are the norms of its ring, and the shells that are empty are the primes that stayed prime.

How much symmetry a lattice has, as a picture #

Reading those five rows together says something that is obvious in hindsight and not before.

A shell is a union of orbits under the holohedry. The point group of the lattice permutes the points at a given distance, so every shell’s size is a sum of orbit sizes, and an orbit’s size divides the order of the holohedry. The hexagonal lattice’s holohedry has twelve elements, so its shells come in sixes and twelves; the oblique lattice’s has two, so its shells come in twos and fours.

The consequence is that the sparsity and the height of a lattice’s theta series move together. A symmetric lattice puts many points on few circles; an oblique one puts few points on many. The total number within a radius is nearly the same for all of them — it is the area divided by the cell area, and that is the only thing a lattice’s density knows — but how it is distributed is the symmetry.

Only two of the five have a closed form, and which two is decided by the ring rather than by the symmetry. The square and hexagonal series are divisor sums because their quadratic forms are the norms of ℤ[i] and the Eisenstein integers, and both of those rings factor uniquely. A rectangular lattice with an arbitrary ratio of edges, or an oblique one, has a quadratic form belonging to no such ring, and its shell counts are computed by enumeration and by nothing else — there is no formula to check them against. That is worth naming because it inverts the expectation: the two lattices with the most symmetry are the two whose counts can be predicted, and the reason is not that symmetry makes counting easier but that those two forms happen to be norms.

The shortest vectors, and what they are for #

The first non-zero coefficient is a number with its own name: the kissing number, the count of lattice points at minimum distance. It is 4 for the square lattice, 6 for the hexagonal one, and 2 for a general oblique lattice with no coincidences.

Those numbers run through this collection. The kissing number is the coordination number of the corresponding sphere packing, so the hexagonal lattice’s six is why a close-packed layer has six neighbours; it is the number of shortest vectors reduction has to choose a basis from, which is why reduction is not canonical when the count is large; and it is the number of nearest faces of the Wigner–Seitz cell, since each nearest neighbour contributes one.

In three dimensions the same first coefficient is the packing story. The face-centred cubic lattice has twelve shortest vectors and the body-centred cubic has eight, which are the coordination numbers of the two close packings and the structure that is not close-packed. A theta series’ first term is a fact about how many spheres touch.

A powder pattern is a theta series #

The connection that makes this practical rather than pretty is in three dimensions.

A powder pattern collapses all the reflections of one spacing into one line. So the intensity of a line depends on how many reflections share that spacing, and that number — the multiplicity — is the number of reciprocal-lattice vectors of that length. It is a theta coefficient, by definition rather than by analogy.

For a cubic cell the multiplicity of the line at h² + k² + l² = n is r₃(n), the number of ways n is a sum of three squares. The check made here is that the two ways of arriving at it agree: counting vectors directly, and adding up the families. The line at n = 9 contains two families, {300} with six vectors and {221} with twenty-four, and its multiplicity is thirty — which is the accidental coincidence that indexing has to live with, appearing here as an addition.

And there is no line at n = 7. Nor at 15, 23 or 28. Those integers are not sums of three squares — Legendre’s theorem says the exceptions are exactly the numbers 4ᵃ(8b + 7) — so no cubic crystal, of any composition, at any wavelength, has a reflection there.

That is a gap in every cubic powder pattern ever measured, and it is not a property of any crystal. It is a property of the integers.

The practical consequence is that a missing line is not evidence. An absence caused by a screw axis or a glide plane is a fact about the structure and is how a space group is determined; an absence at n = 7 is a fact about arithmetic and says nothing at all. Indexing software distinguishes them without comment, and a reader working from a printed line list has to know which kind of gap is in front of them.

Who counted them, and when #

Gauss did the two-dimensional case, in the Disquisitiones of 1801 and in his work on the class number: counting representations by a quadratic form is exactly the question the theory of binary forms was invented to answer, and the divisor formulas fall out of it.

Jacobi gave the four-square and two-square counts in 1829 by a route that looks like nothing to do with lattices — manipulating theta functions, infinite products in a complex variable — and the name theta series is his. That a count of lattice points equals a coefficient of a modular form is the beginning of a subject that is still going: the theta series of a lattice in n dimensions is a modular form of weight n/2, and the space of such forms is finite-dimensional, so two lattices in the same dimension have theta series lying in a space with only a few coordinates. That is the reason the question of the next essay has the answer it does.

The Loeschian numbers carry the name of August Lösch, an economist, who used the hexagonal lattice in the 1930s to model the market areas of towns — the same arithmetic reaching crystallography, economic geography and the counting of shells, which is the sort of thing that happens to counts of integer points.

What the round trip checked, and how #

Every count is made twice. The enumeration counts points in a box whose size is derived from the form rather than guessed; the closed form sums divisors. They agree at every n for the two lattices that have a closed form, and a disagreement anywhere means the figure does not appear at all.

The wrong divisor rule must fail. Applying Jacobi’s four-and-modulo-four rule to the hexagonal lattice gives the wrong answer at n = 3, and the test requires it to.

An unreduced form must not count as a second lattice. 2x² + 2xy + y² is the square lattice in another basis, and the enumeration of lattices must not contain it.

And in three dimensions the multiplicity must agree with the family sum, which is the check that the powder machinery and the number theory are describing the same object.

Where else the same count appears #

Once the shape of the question is visible it turns up repeatedly in this collection, and the appearances are not analogies.

The Patterson function puts a peak at every interatomic vector, so the map that needs no phases has, at the origin’s neighbourhood, a peak whose weight counts vectors of each length — the theta series of the structure rather than of the lattice, but the same construction.

The Debye scattering equation for a powder is a sum over pairs of atoms of sin(Qr)/Qr, and grouping the pairs by distance turns it into a sum over shells weighted by their multiplicities. Every calculation of a powder pattern from a structure is a theta series being evaluated.

A superlattice’s extra reflections are the vectors the sublattice adds, and how many appear at each length is the difference of two theta series.

And the coincidence-site lattice of a grain boundary has an index Σ, which is a value of a quadratic form; every such index is odd for the cubic case, which is a statement about which integers a form represents — the same question as which lengths occur, asked of a different form.

Where the exactness stops #

A theta series is a series and everything here is a truncation. The counts are exact as far as they go and the box is sized from the form so that nothing inside the range is missed, but a statement about “the theta series” is a statement about infinitely many coefficients and no computation makes one.

The closed forms are for two lattices, not five. The square and hexagonal lattices have divisor formulas because their quadratic forms have class number one — every form of that discriminant is equivalent to the principal one, so representing n is a question about factorisation and nothing else. The rectangular, rhombic and oblique lattices here have no such formula available, and their counts are enumerations. That is a fact about the arithmetic of quadratic forms rather than a limitation of the code, and it is the first place in this collection where a lattice’s metric rather than its symmetry decides how hard a question is.

The bar chart compares series, not lattices. Five lattices drawn on one axis of squared length are five lattices at five different scales, since each form was chosen with its own coefficients. Scaling a lattice scales every squared length by the same factor, so the pattern of a row is meaningful and its horizontal position is not; comparing two rows at a fixed n is comparing nothing.

Multiplicity is not intensity. A powder line’s height depends on the multiplicity, the structure factor, the Lorentz–polarisation factor and the temperature factor. Only the first is counted here, and an essay that read a line’s height as a count of reflections would be wrong about every real pattern.

The generating function, briefly #

There is a compact way to write all of this that is worth naming even though nothing here computes with it.

Collect the counts into a power series, Θ(q) = Σ rₙ qⁿ, one term per shell. For the lattices in this collection that series is not an arbitrary sequence: it is a modular form, an object satisfying strong transformation rules, and the space of forms of a given weight and level is finite-dimensional.

That is the structural reason theta series are as rigid as they are. Knowing a modular form’s first few coefficients can determine all of them, because there are not enough dimensions in the space for two different forms to agree for long. It is also the reason the next essay’s counterexample lives where it does: in sixteen dimensions the relevant space of forms is one-dimensional, so any two even unimodular lattices of that rank have identical theta series, and there happen to be two.

Nothing in this collection computes with modular forms, and the counts here are all enumerations or divisor sums. The remark is here because a reader who meets the phrase “theta series” elsewhere will meet it in that context, and because it explains why a question that looks like bookkeeping has an answer at all.

Where the ladder goes next #

The theta series is a measurement of a lattice, and the obvious question about any measurement is whether it determines what it measures. Two lattices with the same lengths at every distance — are they the same lattice? In the plane the answer is yes and the search that confirms it is exhaustive. In sixteen dimensions the answer is no, and the counterexample is sixty years old.

Why some dimensions have formulas and others do not #

The plane gives a divisor sum and three dimensions gives nothing so tidy, which looks like bad luck. It is not luck, and the reason says which dimensions to expect a formula in.

Collect the counts into one function. Write $\Theta(q) = \sum_n r(n)\,q^n$, with $r(n)$ the number of vectors of squared length $n$. For the square lattice this is the square of a simpler series, for a lattice in $d$ dimensions it is a $d$-th power of it, and the function that results is not an arbitrary power series.

It is a modular form, of weight $d/2$. That is a strong constraint: the space of forms of a given weight is finite-dimensional, so a theta series is a combination of finitely many standard forms, and its coefficients are combinations of their coefficients.

In even dimensions the weight is a whole number, and the standard forms are the Eisenstein series, whose coefficients are divisor sums. That is exactly why the plane’s counts are divisor sums, and why four dimensions gives another one — the number of ways of writing $n$ as a sum of four squares is eight times the sum of the divisors of $n$ that are not multiples of four.

In odd dimensions the weight is a half-integer, and the theory is genuinely harder. There is no Eisenstein series to expand in, and the count of representations as a sum of three squares is instead tied to the class numbers of quadratic forms — the objects counted elsewhere in this collection, and not the kind of thing that reduces to a divisor sum.

So the shape of the answer alternates with the dimension, and crystallography is unlucky in living in three.

The counts are powder multiplicities #

There is a use for these numbers that a diffractionist meets before ever meeting a divisor sum, and it is the same table read as an experiment.

A powder pattern collapses direction. Grinding a crystal into randomly oriented grains means every reflection appears at a scattering angle set by its length alone, so all the reflections of a given length land on top of one another as a single line.

So the line’s intensity carries a factor of the count. A reflection with a multiplicity of forty-eight contributes forty-eight times as much to its line as one with a multiplicity of six, before any structure factor is considered — and getting that factor wrong is a straightforward way to misfit a pattern.

And the counts are exactly this essay’s numbers. The multiplicity of a powder line is the number of reciprocal-lattice points at that length, which is $r(n)$ for the reciprocal lattice.

The awkward case is when two different reflections share a length by accident. Indices unrelated by symmetry can have the same squared length, and their reflections then overlap in a powder pattern with no way to separate them. That is the loss a powder pattern suffers, and this table is where it can be predicted: any $n$ whose count exceeds the symmetry’s own orbit size has an accidental coincidence in it.