Where the boundary goes when the atoms differ

The cell nobody chose builds the cell that belongs to a lattice rather than to a convention: assign each point of space to the nearest lattice point, and the region that results is a convex polyhedron whose faces are perpendicular bisectors, whose symmetry is the lattice’s own, and which tiles space by translation.

Every step of that argument assumes the lattice points are interchangeable. In a crystal of two elements they are not, and the cell it produces is wrong in a way that is easy to state: the boundary between a caesium ion and a chlorine ion lands halfway between their centres, and a caesium ion is nearly twice the size, so part of it is in the chlorine’s cell.

The problem is worth sizing before repairing it. In caesium chloride the two ions sit at the corner and the body centre of a cube, and their radii differ by nearly a factor of two — so the halfway plane sits well inside the larger ion. In the cell nobody chose’s terms the arrangement is not a lattice at all but a lattice with a basis of two, and the Voronoi construction applied to the whole set of sites treats the two kinds as interchangeable, which is exactly the assumption that is false.

The obvious repair does not work #

Split the distance in the ratio of the two radii instead. That is the rule anybody writes down first and it fails, for a reason that has nothing to do with atoms.

The set of points closer to p than to q in the ratio of their radii — points with |x − p|/r₁ ≤ |x − q|/r₂ — is bounded by a sphere, not a plane, when the radii differ. That is an Apollonius circle in two dimensions and its three-dimensional analogue in space. Regions bounded by spheres do not fit together into a tiling by convex bodies, the intersections are conic curves rather than straight edges, and the whole apparatus of faces, edges and vertices that makes a cell computable is gone.

It is worth being clear that the failure is not aesthetic. A partition of space into regions bounded by spheres is perfectly well defined and one can compute with it; what it is not is a partition into polyhedra, and every downstream use in this collection wants polyhedra. A face count is a coordination number, an edge is where three cells meet, a vertex is an interstitial site — all of those are combinatorial notions that a curved partition does not supply. Losing them to gain a more plausible-looking boundary is a bad trade, and the point of what follows is that the trade is not necessary.

Measuring to a sphere instead of to a point #

Give each site a weight w and measure to it by

|x − p|² − w.

For w = r² that quantity is the squared length of the tangent from x to the sphere of radius r about p — the power of the point with respect to the sphere, which is where the name comes from. Assign each point of space to the site of least power distance.

The boundary between two sites is now where |x − p|² − w₁ = |x − q|² − w₂, and the |x|² on each side cancels. What is left is linear in x: a plane, perpendicular to p − q as before, but displaced towards the smaller site by an amount fixed by the difference of the weights. Setting the weights equal puts it back at the halfway point, so the ordinary cell is the special case rather than a different construction.

Everything that made the ordinary cell tractable survives. The cells are intersections of half-spaces, so they are convex polyhedra; every point belongs to exactly one; and they tile.

One consequence of the linearity deserves emphasis because it is what makes the construction more than a trick. Every property of the ordinary cell that followed from its faces being bisector planes now follows from its faces being some planes, and almost none of those properties used the halfway position. Convexity, the tiling, the finite face count, the fact that a face is shared by exactly two cells — all of them survive verbatim. What does not survive is symmetry about the site, and that is the only casualty.

There is one property of the ordinary cell that is genuinely lost, and it should be named rather than glossed over. The Voronoi cell of a lattice point is symmetric about that point, because the lattice is: if v is a lattice vector then so is −v, so the bisector planes come in parallel pairs. A power cell is not centrosymmetric in general, because the displacement of each plane depends on the neighbour’s weight and the neighbour at +v need not be the same kind of atom as the one at −v. In a structure where they are — caesium chloride, rock salt — the symmetry survives; in one where they are not, it does not.

The rock salt row of that table is worth a second look. Both cells come out with six faces and equal volumes at equal weights, which says the ordinary Voronoi cell of the rock salt arrangement — the two interpenetrating face-centred lattices taken as one set of sites — is a cube. That is right and it is a little surprising: the sites form a simple cubic array, so of course the cell is a cube, and the two-element structure only differs from a simple cubic element in which atom sits where. Every geometric consequence of the arrangement is carried by the weights and by nothing else.

Three things that could not happen before #

The construction is not a refinement of the ordinary cell — it is a larger object with behaviour the ordinary one has no room for, and three of those behaviours are worth naming.

A site need not lie inside its own cell. The plane moves towards the smaller site and can pass beyond it. Nothing forbids that, and it is not a defect: the region of space closest-by-power to a small atom wedged between large ones may genuinely be somewhere else.

A site can have no cell at all.

The threshold is exactly 1/8 for this arrangement and it is not fitted — the plane between the two sites sits at a distance (d² − w_big + w_small)/2d from the small one, and setting that to zero with d = √3/2 and the two weights gives 2t − 1/4 = 0. So the disappearance is a threshold rather than a fade, and the ratio at which it happens is arithmetic in the structure.

Faces are lost at thresholds, not gradually.

A face disappears when a neighbour stops being one, and that is a discrete event. It is the same phenomenon a reduction with one rule meets when a Selling parameter reaches zero and a cell changes its combinatorial type: the shape varies continuously and the topology jumps.

A fourth behaviour is worth mentioning although it does not arise in the structures here: the cells need not be connected to the sites in the way intuition expects, and two sites of very different weight can produce a cell whose shape has no resemblance to a scaled version of the equal-weight one. That is what makes the construction useful for irregular structures and it is also what makes it hard to sanity-check by eye, which is why the tiling identity is the check that matters.

The threshold’s arithmetic generalises and it is worth writing out once. For two sites at distance d with weights w₁ and w₂, the plane sits at distance (d² − w₂ + w₁)/(2d) from the first, so the first site’s own cell survives only while that is positive: w₂ − w₁ < d². In words, a site loses its cell when a neighbour’s radius exceeds it by more than the distance between them — in squared terms. That is a local condition on one pair, so a site with many neighbours loses its cell as soon as any one of them satisfies it, and the threshold for a structure is the first of those to be reached.

The volume share is not what anybody guesses #

The natural expectation is that an atom twice the radius gets eight times the volume. It does not.

The reason is worth extracting because it is the whole character of the construction. A power boundary is decided by r₁² − r₂², a difference, while the cube law is about r₁³/r₂³, a ratio. Differences and ratios behave differently under scaling: multiply both radii by the same factor and the ratio is unchanged while the difference of squares is not, so the power diagram of a structure depends on the absolute radii and not only on their ratio. That is a real feature and it is the reason the weights are usually quoted as squared radii rather than as radii.

It is also why the volume share here should not be read as an atomic volume in any chemical sense. What counts as a bond makes the general point: a geometric partition of space is decidable and a chemical one is a convention, and the two answer different questions. What this construction gives is the partition that is convex, tiles exactly, and reduces to the ordinary cell when the sites agree — which is a strong set of properties and is not the same thing as being right about atoms.

The absolute-scale dependence has a practical consequence worth stating. Two structures that are geometrically similar — the same arrangement with every length multiplied by a factor — have the same ordinary Voronoi cells up to that scaling, and they do not have the same power cells unless the weights are scaled by the square of the factor as well. So a power diagram is not a property of a shape; it is a property of a shape together with a set of lengths. That is the right behaviour for a physical model, in which a radius is a real length and not a proportion, and it is a trap for anybody carrying intuition over from the unweighted case.

There is a third reading of the sweep which is the most useful of the three. The curve is monotonic and continuous while the face counts jump, so two structures with slightly different radius ratios can have cells of the same volume and different shapes, or the same shape and different volumes. Volume and combinatorial type are independent measurements of the same object, and reporting one without the other describes half of it. The zones above the first makes the same distinction in reciprocal space, where the shape of a zone and the volume it encloses come apart for the same reason.

What it is for #

Two uses, and both are about making an existing tool work on a structure with a basis.

The first is coordination. The ordinary cell defines a neighbour as a site whose bisector plane contributes a face, and that definition is the reason a body-centred lattice has fourteen neighbours rather than eight — the cell nobody chose works that out. The same definition applied to a power diagram gives a neighbour count for a structure of two elements, and the counts change with the radius ratio at the thresholds above rather than being fixed by the space group. That is a statement about a structure that the space group alone does not contain.

The second is that a Delaunay triangulation has a weighted analogue with the same duality. The ordinary Voronoi diagram is dual to the Delaunay triangulation; the power diagram is dual to what is called the regular triangulation, and every property that follows from the duality follows there too. This collection does not build either, and it is worth recording that the duality is what makes both computable at scale.

The first of those is the strong one and it is the reason the others can be believed. Summing the cell volumes and getting the lattice’s own cell volume to fifteen digits is a statement that every point of space was assigned exactly once — no gap, no overlap, no rounding into a crack. A construction that got the geometry subtly wrong would fail it by a visible amount, and none of the pruning and half-space arithmetic that makes the computation fast can hide from it.

There is also a use this construction is not for, and it is the one most likely to be attempted. It does not decide which atom an arbitrary point of space “belongs to” in any sense a chemist would defend. The cells partition space and the partition is well defined; what it is not is an electron-density argument, and the boundary a density puts between two atoms is a surface where a gradient vanishes rather than a plane. Those two surfaces are close in a symmetric structure and are not the same object, and reporting a power-cell volume as an atomic volume without saying which construction produced it is how two papers come to disagree about a number they both computed correctly.

One more property of the thresholds is worth recording. The sequence of face counts — fourteen, then eight, then none — runs through the same combinatorial types the ordinary Voronoi cells of the cubic lattices have: the truncated octahedron, the octahedron and the cube. That is not a coincidence and it is not deep either. The candidate faces come from the same set of neighbours in the same cubic arrangement, and which of them survive is decided by one parameter; so the types available are the types the arrangement’s own symmetry allows, which five parallelohedra and no others enumerates for the translation case.

There is a small point about the computation itself that is worth recording, because it is where the first version was wrong. The vertex search intersects every triple of bounding planes and tests each candidate against all the others, which is cubic in the number of planes — so feeding it every neighbour within two cells, six thousand of them, is ten million triples per cell and does not finish. Sorting the planes by their distance from the site and keeping the nearest forty makes it instant, and it is safe for a reason worth stating rather than assumed: the tiling identity would fail loudly if a plane that mattered had been dropped. A pruning justified by an independent check is a different thing from a pruning justified by intuition.

A last connection worth drawing. The unweighted cell is what covering and packing want different lattices is about: the cell’s inradius and circumradius are the packing and covering radii, and the two are optimised by different lattices. The weighted version has the same pair of quantities per site and they are no longer properties of the lattice alone — they depend on the weights, so a structure can be a good packing for one of its two elements and a poor one for the other. That asymmetry has no counterpart in the equal-radius theory and it is the shape of most of what makes real structures interesting.

Where this stops #

The weights here are squared radii and the radii are chosen, not derived. That is the honest limit: a power diagram is exactly as good as the weights fed to it, and this collection has no way to compute an ionic radius from a structure. What it can say is what follows from a choice of weights, which is everything above.

Two extensions are not done. A cell for more than two kinds of site is the same code with a longer basis and no new idea; a cell for a structure whose sites are not at special positions is likewise. And the inverse problem — given a set of cells, what weights produce them — is a genuinely different question, and it is the one that would be needed to turn a measured structure into a set of radii rather than the other way about.