The sum that turns a lattice into its dual

How many vectors of each length counts the lattice points at each distance and gets a sequence — the theta series, which is a fingerprint, a set of divisor sums and a powder pattern’s multiplicities all at once. This essay does something else with the same object: it weights the points instead of counting them, and finds that the weighting can be moved to the dual lattice without changing the answer.

Put a Gaussian of width set by a parameter t on every point of a lattice L and add them all up:

Θ_L(t) = Σ exp(−π t |v|²)

over every v in L. The claim is that this equals the same sum over the dual lattice L* — the lattice of vectors whose inner product with every vector of L is an integer — with t replaced by 1/t, divided by the covolume and by t to the power of half the dimension.

The dual is worth pinning down before anything is summed, because there are two conventions and they differ by a factor of 2π. Here L* is the set of vectors w with w · v an integer for every v in L — no 2π — so if L has basis matrix B, then L* has basis matrix B⁻ᵀ and Gram matrix G⁻¹. That is the convention in which the covolumes multiply to one, and it is the one this whole essay is written in. The reciprocal lattice uses the other, because a diffraction condition is written with a 2π in the exponent, and the two differ by a scaling that changes nothing structural and every numerical factor.

The identity, checked rather than quoted #

The reason to check rather than quote is that the identity has three places to get a factor wrong, and each of them produces something that still looks like an identity. The power of t is −n/2 and not −n; the covolume enters once and not twice; and the dual is the lattice of the inverse Gram matrix rather than of the transposed one. A version with any of those wrong is exact for the cubic lattice at t = 1 and wrong everywhere else, which is exactly the shape of error that survives a spot check.

The proof is Poisson summation: the sum of a function over a lattice equals the sum of its Fourier transform over the dual lattice, divided by the covolume, and a Gaussian’s Fourier transform is a Gaussian with the width inverted. That is one line and it is not what this collection is for. What is worth having is the computation, and what the computation shows is that the statement is not approximately true or true in a limit — the same number arrives twice, by two entirely different routes over two entirely different sets of points.

There is one more thing the figure is doing that is easy to read past. The four values of t are not a sweep; they are chosen to straddle the point where the two sides swap roles. At t = 0.05 the left-hand sum is over Gaussians so wide that the answer is close to the covolume’s reciprocal and the individual terms are all of comparable size; at t = 3.2 it is within a rounding error of the single term at the origin. The identity has to hold across that whole change of character, and it does — which is a stronger statement than agreement at one setting, because the two sides are doing completely different things at the two ends.

It is also worth saying what the sum is at the two ends, since the numbers in the figure are otherwise opaque. As t grows the sum tends to one: every term but the origin has died, and what is left is the single Gaussian sitting on the lattice point at zero. As t shrinks the sum tends to 1/(covol · t^{3/2}), which is the number of lattice points inside a ball of the Gaussian’s width — the sum becomes a count of points weighted almost equally, and a count of lattice points in a large region is the region’s volume over the covolume. So the identity interpolates between “one term” and “count the points”, and the transformation is what says those two descriptions are the same function.

That reading also explains why the transformation looks like a statement about volumes. The covolume appears because a sum over a lattice, in the limit of a slowly varying summand, is an integral divided by the covolume — how many points a shape holds is the exact version of that statement, with the error term that Poisson summation makes precise.

Why anyone would want it #

An identity between two things that are equally hard to compute is a curiosity. This one is not, and the reason is visible in one dimension.

The parameter t is not physical and nothing depends on its value; it is a knob that decides how the same number is written down. That is an unfamiliar shape for an identity to have. Most identities relate two quantities that were defined separately and turn out to agree; this one relates a quantity to itself, described through a parameter that cancels. What it buys is that the description can be chosen after the fact, and the choice is made on cost.

A sum of wide Gaussians is expensive because nothing is negligible: every term is within a few orders of magnitude of every other, and the sum has to be carried out to a radius where the exponential finally bites. A sum of narrow Gaussians is cheap because everything beyond the first shell or two is already below the precision being asked for. The transformation converts one into the other, and the width parameter says which is which.

Radii understate it, because the number of lattice points in a box grows as the cube of the radius.

Three orders of magnitude is the whole argument for the identity. Nothing is being approximated and no error is being traded away — the same number is computed to the same precision, and one route reads a hundred and twenty-five numbers where the other reads twenty-four thousand.

There is a second reason the identity earns its place, and it is not about speed. A sum of wide Gaussians is numerically ill-conditioned as well as slow: it adds tens of thousands of comparable numbers, and the rounding errors add up as the square root of the count. The transformed sum adds a hundred and twenty-five numbers of rapidly decreasing size, and the answer is good to nearly full precision. So the two routes do not merely cost differently; the expensive one is also the less accurate one, which is the usual way with sums that refuse to converge quickly.

That is also why the radius reported in these figures is the radius at which the sum stops changing rather than a radius derived from a bound. A bound on the tail of a Gaussian sum is easy to write and always loose, and what is wanted here is the honest cost: how far the box actually has to go before the twelfth digit stops moving.

The counts in that figure are box counts rather than counts of terms that actually mattered, and the distinction is worth a sentence because it flatters neither side. A box of radius fourteen holds twenty-four thousand three hundred and eighty-nine points, and at t = 0.05 most of them contribute something above the twelfth digit — that is what “the sum has not settled” means. A box of radius two holds a hundred and twenty-five, and at the corresponding dual setting the outermost shell of it is already below the precision asked for. So the ratio is if anything an understatement of the difference in useful work.

Where the crossover is #

The two costs cross somewhere, and where they cross is a property of the lattice rather than a universal constant.

The reason the crossover is not always at t = 1 is worth stating, because it is the same fact that makes reduction theory necessary. A lattice with one short direction and two long ones has a dual with one long direction and two short ones, and a sum over a box does not notice that: the box is isotropic in the indices, not in the metric. So the cost of each side is set by the worst direction, and a lattice whose worst directions are unbalanced has an unbalanced crossover. The shortest basis is the machinery that fixes this in general, by making the basis as close to orthogonal as the lattice allows before anything is summed.

The product being exactly one is the smallest statement in the essay and it is the one that makes the rest legible. A lattice and its dual are not two independent objects with a relation between them; they are one object described from two sides, and the covolume is the conversion factor. The reciprocal lattice is the same construction wearing a factor of 2π and a diffraction experiment, and every plane lattice is its own dual is the case in two dimensions where the two sides coincide up to a rotation.

One consequence of the crossover being lattice-dependent deserves stating, since it is what makes the identity usable rather than merely true. Nobody has to know where the crossover is. Both sides are available at every t, so a program can compute the cheaper one by comparing two radii, or — better — can split the work between them, which is what the next rung does with a genuinely divergent sum. The identity is not a rule about which sum to write down; it is the fact that the choice exists.

One more property of the crossover is worth extracting from the table. The two radii at the two ends of the range are nearly mirror images: what the direct sum needs at t is what the dual sum needs at 1/t, for every lattice. That is not a separate fact — it is the identity itself, read as a statement about cost rather than about value — and it means the crossover is at the t where a lattice and its dual are “the same size” in the only sense that matters here, which is that their boxes run out at the same rate.

What this is doing in a collection about symmetry #

Three things, and it is worth being explicit because the identity is analysis rather than group theory.

The first is that it is where the reciprocal lattice comes from. A crystal’s diffraction pattern is the Fourier transform of its density, the density of a perfect lattice is a comb of deltas, and the statement that the transform of that comb is another comb on the dual lattice is this identity, in the limit t → 0. Everything the collection says about the reciprocal lattice, about zones and about which reflections a centring removes rests on it, and it is usually asserted rather than shown.

The second is that a great many quantities a crystal has are lattice sums of exactly this shape, and they are conditionally convergent when the weights alternate — which is the subject of the next rung.

The third is a caution about fingerprints. The theta series is a sequence of integers and it very nearly decides the lattice; the lengths do not name the lattice is where it fails, in four dimensions and above. This identity says something stronger and much less useful about the same object: the function Θ_L determines Θ_{L*} completely, so the two fingerprints carry the same information and comparing a lattice against its dual can never distinguish anything that comparing it against itself could not.

A fourth thing, smaller and worth having: the identity is the cleanest available proof that the dual of the dual is the original lattice. Apply the transformation twice and t goes to 1/t and back, the covolume factors multiply to one, and the result is the identity Θ_L(t) = Θ_L(t). That is not how one would prove L** = L — the direct argument is three lines of linear algebra — but it is a check that the whole apparatus is consistent, and it is the kind of check a numerical implementation can actually run.

There is one figure this essay deliberately does not have, and the omission is the point. A picture of a three-dimensional lattice with Gaussians on it would be a picture of a fog, and it would show nothing: the whole content is in how fast a sum settles, which is a number and not a shape. The one-dimensional panel above is a picture of the mechanism rather than of the object, and it is drawn in one dimension precisely so that the two regimes can be seen at once. Every other figure here is a measurement, because a measurement is what the claim is.

What the machinery has to get right #

The covolume test is the one worth running because dropping the covolume is the natural mistake. On the cubic lattice the covolume is one, so the wrong identity and the right one agree; on the triclinic lattice they differ by a factor of nearly two, which is what the refusal reports. A construction checked only on the symmetric case would have passed.

The other check is subtler and is inside the figure rather than beside it. Each side of the identity is summed over a box large enough for itself rather than over one fixed box, because at large t the dual sum is the wide one and a radius that is ample for the direct sum leaves it seven digits short. Truncating both at the same radius makes an exact identity look approximate, and the first version of the figure did exactly that and reported a relative difference of one part in a million — which is neither the identity failing nor the arithmetic being imprecise, but a box being too small on one side only.

There is one honest limitation to record. Everything above is about a lattice with no basis in it: one point per cell, all weights equal. A real crystal has several atoms per cell and the sum acquires a structure factor, which is where the average that makes it finite and the rest of the diffraction machinery start. The transformation survives that untouched — the structure factor rides along in the reciprocal sum as a phase — but nothing here computes it, and a reader coming from diffraction should read this as the empty-lattice case rather than as the general one.

One further use, and it is the one that makes the identity part of crystallography rather than of analysis. A Gaussian on every lattice point is what a crystal’s electron density looks like when every atom is identical and its thermal motion is isotropic — which is the model behind the Debye–Waller factor. The transformation then says that broadening the atoms in real space narrows the pattern in reciprocal space by exactly the reciprocal factor, with no approximation and no expansion in small quantities. That is a statement crystallography uses constantly and states as a rule of thumb, and it is this identity with the parameter named after a temperature.

Where this goes #

The next rung takes the same machinery to a sum that has no value at all until the order of summation is named. Weight the lattice points by alternating signs instead of by nothing, replace the Gaussian by 1/r, and the sum over expanding cubes and the sum over expanding spheres disagree — the first creeps towards a limit and the second does not converge. Writing 1/r as an integral over Gaussians and cutting the integral in two, with this transformation applied to the wide half, is what makes it a number.