The sum whose answer depends on the shape

The halving a lattice will not permit puts two colours on a lattice: a subgroup of index two and its coset, alternating. Give one colour the weight +1 and the other −1, and ask for the sum of the weight over the distance, over every point but the origin.

The question sounds like it has an answer, and the first thing to establish is that it does not — not until something further has been said.

The arrangement is one this collection has already built. Colouring a simple cubic lattice by the parity of the sum of the coordinates leaves the even points forming a face-centred cubic lattice and the odd points forming its coset, which is the one halving of the primitive cubic lattice that keeps cubic symmetry — and it is the arrangement of sodium and chlorine in rock salt. So the sum below is not an arbitrary exercise; it is the simplest lattice sum a crystal actually has, and its convergence is a fact about the lattice rather than about the ions.

Two orders, two answers #

The terms are fixed. What is not fixed is which of them get added first, and for a sum with infinitely many terms of both signs that is not a detail.

The sphere curve is the one to look at, because a reader who has not met this before will expect it to be the better order. A sphere is the natural shape: it treats every direction alike, it respects the symmetry of the lattice, and every point at a given distance goes in at the same time. It is the order in which the terms are sorted by size. And it does not converge.

The cause is a surface effect and it is completely elementary. A sphere of radius R has a surface holding of order R² lattice points, and the signs on that surface do not balance — a sphere’s boundary cuts the two colours in a ratio that wobbles with R rather than settling. Each of those points contributes about 1/R, so the imbalance contributes about R²/R = R, which does not go to zero. The sphere is a worse shape than the cube precisely because it is rounder: a cube’s faces are lattice planes, and a lattice plane in this arrangement carries equal numbers of both colours.

There is a sharper way to say what goes wrong, and it is worth having because it explains why the cube works. Take the difference between the partial sum over a region and the partial sum over a slightly larger one. That difference is a sum over a shell, and its size is governed by how nearly the two colours balance inside the shell. A cubic shell whose faces are lattice planes contains equal numbers of the two colours to within its edges, so the difference falls off; a spherical shell contains whatever the sphere’s curvature happens to cut, and that varies with radius in a way governed by how many lattice points sit near a sphere of that radius — which is the error term in counting lattice points in a ball, and does not settle.

So the two orders are not two arbitrary choices. One of them expands a region that is nearly neutral at every stage and the other expands a region that is not, and neutrality of the boundary is the condition that matters. That is the criterion the classical treatments state as “sum over neutral cells”, and it is why a construction that adds whole formula units at a time converges much faster than the plain cube order drawn above.

What “conditionally convergent” means here #

The sum of the absolute values diverges, and that is the whole diagnosis. There are about 4πR² points at distance around R, each contributing 1/R, so the shells contribute about R each and the total grows without bound. Riemann’s theorem then applies in its usual brutal form: a series that converges conditionally can be made to converge to any value by reordering, and can be made to diverge.

So “the alternating 1/r sum over the cubic lattice” is not the name of a number. It is the name of a family of numbers indexed by a summation order, and the classical value 1.747565 is the one belonging to a particular family of orders — the ones that expand a neutral region. That is not a defect in the arithmetic; it is a statement about what the sum is, and it is the reason the number is usually presented with a construction attached rather than as a limit.

Which raises the question of what makes the classical value the right one, since the cube order is a choice too. The answer is not that the cube is natural. It is that the value it gives is the one that a different and absolutely convergent computation also gives, and absolute convergence is what makes a number well defined. Constructing that computation is the rest of this essay.

It is worth being precise about the strength of the failure, because “does not converge” covers several situations. The partial sums here are bounded — they do not run off to infinity — and they do not approach a limit either. They are dense in an interval, which is the generic behaviour Riemann’s theorem describes: enough freedom in the reordering to reach anything, and no preference for any point. A numerical experiment that stopped at radius twelve and reported the value it found would be reporting a number with no more meaning than the radius it stopped at.

Making it absolutely convergent #

The route runs through the previous rung. Write 1/r as an integral over Gaussians —

1/r = (1/√π) ∫₀^∞ t^(−1/2) exp(−r² t) dt

— and the lattice sum becomes an integral over t of a theta sum. Cut that integral at some t = α². The half with large t has narrow Gaussians and converges fast in real space; the half with small t has wide Gaussians, converges slowly in real space, and is exactly what the transformation turns into a fast sum over the dual lattice. Both halves are then absolutely convergent, so their order does not matter, and the total is a number.

The self term is the piece that looks like bookkeeping and is not. The Gaussian representation smears each point charge into a cloud, and a cloud has an interaction with itself that the original sum did not contain. Removing it is −2α/√π, and it is the only term in the whole construction that is not a lattice sum.

The word “damped” is doing precise work in that caption and it is worth unpacking once. The near sum is not the original sum truncated; every term is present, and each is multiplied by erfc(αr), which is one at short range and falls off like a Gaussian beyond r ≈ 1/α. So the near sum is a different sum that happens to agree with the original at short range, and the far sum supplies exactly what the damping removed. Neither half is an approximation to the whole; the two halves are an exact decomposition, which is why the total can be exact.

The far half deserves the same care. It is a sum over the dual lattice with a Gaussian factor exp(−k²/4α²) and a 1/k² from the Coulomb kernel, weighted by the structure factor of the cell — which is where the positions of the two ions enter. That the positions enter only through a structure factor is not a convenience; it is the same statement diffraction makes, and it is why this computation and a diffraction calculation share their expensive part.

The far half’s home is the reciprocal lattice, and it is worth naming that explicitly because it is the same object a diffraction experiment measures. The wide-Gaussian part of a real-space sum is a narrow sum over reciprocal space; the terms are indexed by the same hkl a reflection is indexed by, and the Gaussian factor is why only the first few shells of them are needed. A structure and its diffraction pattern are the two halves this construction is made of, and the split parameter decides how much of the work each half does.

There is a second structural point hiding in the same place. The 1/k² in the far sum comes from the Coulomb kernel and is singular at k = 0, and the term at k = 0 is simply omitted — which is legitimate exactly when the cell is neutral, since the structure factor vanishes there too. A cell that is not neutral has a genuine divergence sitting at the origin of reciprocal space, and no choice of α hides it. That is the arithmetic behind the rule that a periodic charged system has no energy without a compensating background, and it arrives here as one omitted term rather than as a physical argument.

The parameter that does not matter #

There is a free parameter in the construction — the place where the integral is cut — and a free parameter in a physical answer is normally a problem. Here it is the strongest available check.

That check is worth more than a comparison against a published value, because a published value can be matched by an implementation that is wrong in a way that happens to cancel at one setting. The independence has to hold at every α, and each α moves the work between two entirely different sums — one over the lattice, one over its dual. Nothing short of both halves being right produces a flat line.

It also says something about where to set α in practice. Large α makes the near sum cheap and the far sum expensive; small α does the reverse; and the cheapest total is at the balance, which for a lattice of unit spacing is around two. That is the same crossover the previous rung measures for the theta sum, arriving in a different currency.

There is one more consequence of α-independence that is easy to miss and is the practical one. Because the two halves are separately convergent for every α, the truncation radii can be chosen independently too, and the cost of the whole is the sum of two small numbers rather than the product of two large ones. That is the difference between a computation that scales with the cube of a radius and one that scales with the cube of a much smaller radius twice — and it is why this method is what every simulation of a periodic charged system actually runs.

Three structures #

Nothing above is specific to rock salt. What enters is a lattice, a set of positions in the cell and a sign for each, and the machinery does not care which arrangement of ions produced them.

The ordering of the three is the part worth reading. Caesium chloride sits highest at 1.7627, rock salt next at 1.7476, zinc blende lowest at 1.6381 — and that is the order of their coordination numbers, eight, six and four. More nearest neighbours of the opposite sign is a larger constant, which is the expected direction; what is not obvious in advance is how small the spread is. A factor of two in coordination number moves the constant by seven per cent, because the further shells partly compensate: a structure with fewer near neighbours of the opposite sign also has fewer near neighbours of the same sign pushing the other way.

This is also the place to say what the number is not. A Madelung constant is arithmetic: a lattice, a sign pattern and a distance scale. Turning it into an energy needs a charge, a permittivity and an assumption that the ions are points, and none of those is here. What counts as a bond makes the same division from the other side — the geometry is decidable and the chemistry attached to it is not.

A note on the scaling, since the three constants are only comparable because of it. Each structure is scaled so that the nearest neighbour of opposite sign sits at distance one, which is the convention the published constants use. Without that convention the numbers are not comparable at all: a Madelung constant is dimensionless only after a length has been chosen, and choosing the cell edge instead of the nearest-neighbour distance changes all three by different factors. That is a small thing and it is exactly the kind of small thing that makes two correct computations disagree.

One more comparison is worth drawing, because it puts the three constants in the collection’s own terms. Coordination number is the first shell of the theta series, and the Madelung constant is a signed weighted sum over every shell. So the ordering of the three constants by coordination number is the statement that the first shell dominates — which it does, but only just: the first shell contributes more than the whole constant in each case, and the remaining shells subtract. Rock salt’s first shell alone gives six, against a total of 1.75.

What the halvings have to do with it #

The sign pattern is not free either, and that is the connection to the rung this essay’s prerequisite establishes.

Alternating signs on a lattice means a homomorphism to the two-element group: a subgroup of index two carrying +1 and its coset carrying −1. There are exactly seven of those on any lattice, and for the sum to describe a crystal rather than an arbitrary decoration, the pattern has to be one the lattice’s own symmetry preserves. On the primitive cubic lattice exactly one of the seven does — the one that colours a point by the parity of the sum of its coordinates — and that one is rock salt. The arrangement whose Madelung constant everybody knows is the only two-colouring of the simple cubic lattice that keeps cubic symmetry, and it is not a choice.

That is a satisfying closing of a loop, and it comes with a caution attached. A sum over a lower-symmetry colouring is perfectly well defined and the same machinery computes it; what it is not is a cubic crystal. The symmetry condition decides which sums describe a Bravais arrangement, not which sums converge.

The first refusal is the one that would be easiest to get away with. A version of this essay that reported only the cube order would be arithmetically correct and would leave a reader believing the sum has a value in the ordinary sense. Measuring the sphere order and printing what it does is the difference between a computation and a demonstration, and it costs nothing but the willingness to draw a curve that does not go anywhere.

One further reading of the sphere failure, and it is the one that connects this rung back to the collection’s usual subject. The reason a cubic shell is neutral and a spherical one is not is that the cube’s faces are lattice planes of the coloured lattice and the sphere’s surface is not a plane at all. Which planes are neutral is a question about the halving: a plane whose normal is a direction along which the colours alternate carries one colour, and a plane whose normal is a direction along which they do not carries both equally. So the shapes that work are the ones bounded by the second kind, and the classification of those is a question about the same functional the halving is. The convergence of a lattice sum is a fact about which planes the sign pattern is even on, which is not where one expects a question about convergence to end up.

That last figure carries three of its five rows from the previous rung, and the sharing is deliberate rather than economical. The two essays are one piece of machinery used twice: the transformation is what makes the far half of the split converge, and a check that the transformation is right is a check that this sum is right. Splitting the refusals between them would have made each look better tested than it is.

Where this stops #

Two things this does not do.

It does not handle a cell whose charges do not sum to zero. A non-neutral cell has an infinite sum with a term that grows as the volume, and no ordering rescues it; the machinery refuses such a request rather than returning the finite-looking part, because that part is an artefact of the truncation and not a value. Real treatments of charged cells add a compensating background and say so, which is a modelling decision rather than an arithmetic one.

And it does not compute anything at finite temperature or with the ions anywhere but at their ideal positions. Displace one ion and the sum is still well defined and the machinery still computes it, but the constant is no longer a property of a structure type — it is a property of a configuration, and the whole reason a Madelung constant is quoted as a single number is that the configuration is the symmetric one. The average that knows the atoms is where this collection treats displacement properly, and it treats it in reciprocal space for the same reason this essay ends up there.