How far one lattice is from another

“Nearly hexagonal” is a phrase that does real work in crystallography. A lattice that is nearly hexagonal twins by pseudo-merohedry where an exactly hexagonal one would twin by merohedry and a plainly oblique one would not twin at all; a structure whose cell is nearly cubic is a headache for anybody assigning it a space group; a phase transition that lowers a lattice slightly away from a special shape leaves twins behind for exactly that reason.

The phrase deserves a number, and giving it one is harder than it looks. The obvious measures are all measures of a description: the difference between two cell angles, or the ratio of two cell edges. Both depend on which cell was written down, and the cell is a choice while the lattice is not.

The space this is measured in #

Every lattice shape is a point of one region, which the earlier essay built: take the reduced form of the lattice’s metric, read off the complex number τ whose real part is the ratio of the cell’s projections and whose imaginary part is its height, and the point lands inside a region bounded by two vertical lines and an arc.

The square lattice sits at i, at the top of the arc. The hexagonal lattice sits at the corner, where the arc meets the right-hand wall. Everything else is a lattice of no particular type, and the special types are the boundary and the corners rather than the interior.

A distance between two lattice shapes is therefore a distance between two points of that region — and the region has a geometry, which is the part worth being careful about.

Why the geometry is hyperbolic #

A distance between lattice shapes must not change when a lattice is described differently. That is the whole requirement, and it is enough to force the answer.

Changing the basis of a lattice acts on τ by a modular transformation — the maps τ ↦ τ + 1 and τ ↦ −1/τ and everything they generate, which is exactly the two moves that reach every basis. A distance that is a fact about the lattice has to be unchanged by all of them.

The hyperbolic metric on the upper half-plane,

$$d(z, w) = \operatorname{arccosh}\!\left(1 + \frac{|z-w|^2}{2\,\mathrm{Im}\,z\;\mathrm{Im}\,w}\right)$$

is defined by being invariant under exactly that group. A Euclidean distance is not: two points near the top of the region and two points near its bottom can be the same Euclidean distance apart and represent lattices that are related by a change of basis in one case and not in the other.

So the hyperbolic geometry is not decoration on the picture. It is the only geometry in which the question has an answer, and it arrives because of what the region is rather than because anybody chose it for elegance.

Reduce first, always #

The distance is computed after reduction, and skipping that step gives a number about a basis.

An unreduced form has a τ outside the region — a different representative of the same shape, possibly very far up or very far to one side — and its distance to i is then a fact about how badly the basis was chosen. Reducing moves it to the unique representative inside the region, and the distances measured from there are properties of the lattice.

The check is direct and it is run: take a form, apply three different unimodular changes of basis, reduce each, and require the distances to agree. They agree to the last bit, because reduction produces the same reduced form in every case.

What the region’s shape already says #

Before any distance is measured, the shape of the region carries information that is worth reading off, because it explains why “nearly special” is a natural idea at all.

The five plane lattice types are not five points. Two of them are: the square lattice at i and the hexagonal lattice at the corner. Two more are edges: the rectangular lattices along the vertical wall, the rhombic ones along the arc. And the fifth — oblique — is the whole interior.

So the special types are exactly the boundary and corners of a two-dimensional region, which is why a randomly chosen lattice is oblique with probability one and why the interesting lattices are the ones close to an edge or a corner. Being nearly rectangular means being near a line, which is a codimension-one condition; being nearly hexagonal means being near a point, which is codimension two and correspondingly rarer.

That accounting is also why a nearly hexagonal lattice is a more remarkable thing to find than a nearly rectangular one, and why a crystallographer’s suspicion is aroused more by the first. A distance gives the suspicion a number; the region’s shape says why it was aroused.

What the number does #

The behaviour of the distance along a family is the reason it is worth having, and it is not the behaviour of an angle.

Walk a lattice from square to hexagonal through a one-parameter family. A difference of cell angles falls linearly along that walk. The hyperbolic distance does not: it is flat near each special point and steep between them, so a lattice that is somewhat near hexagonal is measured as much nearer than a naive angle difference would suggest.

That is the behaviour that matches what crystals do. Twinning by pseudo-merohedry happens when a lattice is near enough to a higher-symmetry shape for the mismatch to be absorbed at a boundary, and the threshold is sharper in reality than a linear measure would predict. The obliquity that crystallographers use for this is a different measure of the same thing, defined as an angle a twin operation misses by; the two agree about which lattices are near and disagree about how near, which is what one expects of two measures of one idea.

The three-parameter check #

The distance is claimed to be a property of the lattice, and there are three separate ways that claim could fail. Each is tested.

The basis. Three different unimodular changes of basis are applied to the same form; the reduced forms come back identical and the distances agree exactly. That is the test that catches a computation done on τ before reduction.

The scale. A lattice and the same lattice at twice the size are the same shape, and the reduced form of the doubled metric is the doubled reduced form, whose τ is unchanged. So the distance is scale-invariant, which it must be: shape is what is being measured.

The handedness. A lattice and its mirror image have forms differing by the sign of the cross term, and the reduction sends both to the same representative — the collection’s standing convention that a lattice and its mirror are one lattice. So a chiral distinction is invisible here, which is correct for lattices and would not be for structures.

Three invariances, three tests, and each of them is the kind that would pass silently if it were wrong: a distance that quietly depended on the basis would still produce plausible small numbers for nearly-special lattices.

The distance as a coordinate, not just a measure #

Two distances locate a point in a two-dimensional region up to a reflection, so reporting a lattice by its distances to the two special points is close to reporting the lattice’s shape entirely.

That has a practical use this collection can point at rather than perform. Cell parameters are six numbers in three dimensions and three in the plane, of which one is a scale; the shape is what is left, and it is two-dimensional here. So a plane lattice’s shape is exactly two numbers, and any two independent functions of it will do as coordinates.

The conventional choice — two lengths and an angle, less the scale — is a poor coordinate system for comparison, because the same shape has many such triples and choosing between them is the reduction this essay performs first. The pair of distances is a coordinate system with no such freedom.

Nothing here proposes changing how cells are reported. What is worth noticing is that the reporting convention and the comparison problem are different problems, and that solving the second in the first’s coordinates is what makes “nearly hexagonal” hard to pin down.

What “near” cannot mean #

Two limits, and both are the same limit this collection has drawn before.

Near is not a symmetry. A lattice at distance 0.02 from the hexagonal point has the group of an oblique lattice, exactly, with no six-fold rotation and no mirror. It is near-symmetry, which this collection treats as a measurement rather than as a property: the decidable question — does this lattice have a six-fold rotation — has the answer no, and the number 0.02 is a different kind of statement about it.

Near depends on a scale that is not supplied. Whether 0.02 is near enough for a crystal to twin depends on an energy, which nothing here computes. The distance orders lattices by how nearly special they are; converting that order into a prediction needs physics.

Both limits are worth stating because a number invites a threshold, and this number does not come with one.

The distance between two ordinary lattices #

Everything so far has measured distance to a special point, which is the useful case. The metric measures between any two lattices equally well, and that has its own uses.

Two crystals of different substances whose lattices are near in this sense are candidates for epitaxy — growing one on the other — because the interface’s misfit is small in every direction at once, which is a stronger condition than matching one axis. And two determinations of the same structure that produce lattices at a measurable distance from each other are two determinations that disagree, in a way that a comparison of six cell parameters does not make legible.

Neither is a claim this collection tests. Both are what a distance on the space of shapes is for, and they are the reason the moduli picture is more than a way of drawing the five lattice types.

Where the metric comes from, historically #

The upper half-plane with this metric is Poincaré’s model of hyperbolic geometry, and its appearance in the theory of lattices predates any application to crystals by a long way.

Gauss reduced binary quadratic forms in the Disquisitiones of 1801, and the region drawn here is his; the modular group acting on it is the object number theorists spent the nineteenth century on, because a lattice shape and a binary quadratic form are the same thing and forms are what represent integers. The hyperbolic metric arrived with Poincaré at the end of that century, and the recognition that it was the natural metric for the reduction theory came with it.

Crystallography inherited the picture and mostly does not use it. Cell parameters are reported as six numbers, comparisons are made parameter by parameter, and the fact that all of it is a point in a two-dimensional region with a natural geometry is not part of the standard training. That is a missed inheritance rather than a mistake, and this rung exists to point at it.

Two lattices at the same distance are not the same lattice #

A distance from one point does not locate anything, and it is worth saying so because a single number invites the mistake.

The set of lattices at a given distance from the hexagonal point is a circle in the region — a hyperbolic circle, which in this model looks like a Euclidean circle with a displaced centre — and every lattice on it is equally near hexagonal. They are not equally near each other, and they are not the same shape.

So the number answers “how near is this to hexagonal” and does not answer “what is this lattice”. Reporting a lattice by its distances to both special points comes closer, since two distances in a two-dimensional region usually locate a point up to a reflection — which is exactly what the figures here print.

The general lesson is the ordinary one about summary statistics: a projection of a two-dimensional object onto one number keeps what it was designed to keep, and a reader who wants the object should be given the point.

The distance written in the forms themselves #

The distance above is defined on the upper half-plane and computed after a reduction. There is an equivalent formula that never mentions τ and never needs the reduction, and it is the one that generalises to any dimension.

Two lattices are two Gram matrices, A and B. Form A⁻¹B and take its eigenvalues λ₁, λ₂. Then

$$d = \sqrt{\textstyle\sum_i (\log \lambda_i)^2}$$

is a distance between the two lattices, and every property claimed above falls out of it in a line. It is basis-independent, because a change of basis conjugates A⁻¹B and conjugation does not move eigenvalues. It is symmetric, because inverting the matrix negates the logarithms. And it is zero exactly when the two lattices are the same, since A⁻¹B is then the identity.

Scale appears as the one thing this version keeps and the τ version throws away: doubling B multiplies both eigenvalues by four and moves the distance by a fixed amount. Discarding it is one subtraction — measure log λ₁ − log λ₂ instead — which is where the two accounts meet.

The hyperbolic geometry is now a consequence rather than a choice. The space of positive quadratic forms of determinant one, with this metric, is the hyperbolic plane in two dimensions, and in n dimensions it is a symmetric space of dimension n(n+1)/2 − 1. So the awkwardness of the plane case — that the natural geometry is not the flat one — is not a peculiarity of lattices at all; it is what the space of positive definite matrices looks like, and crystallography meets it in its smallest instance.

Two ways of being nearly symmetric #

There is a second number in this collection measuring almost the same thing, and separating the two is worth doing because they are used interchangeably and are not the same.

The obliquity of a twin law is an angle: how far a particular operation is from carrying the lattice onto itself. The distance here is a length in the space of shapes: how far the lattice is from one that has some symmetry.

The difference is that obliquity is attached to an operation and the distance is not. A lattice sitting near the hexagonal corner is near a lattice with twelve extra operations, and every one of those twelve has its own obliquity. The distance is a single number summarising a family of angles, and it summarises them the way a norm summarises a vector — usefully, and with information lost.

Which one to use depends on the question. Will this crystal twin on this law? is about one operation and wants the obliquity. Is this lattice suspiciously close to a higher type? is about the lattice and wants the distance, because it does not require choosing an operation in advance — which is exactly the situation a crystallographer is in when a refinement is behaving oddly and the cause has not been identified.

Where this goes #

The obvious extension is three dimensions, where lattice shapes form a five-dimensional space with a group of changes of basis acting on it and the same question — how far apart are two lattices — has the same shape and much more machinery in it. That is where a real comparison of cell determinations would live.

The nearer neighbour is the similar sublattices of the previous rung, where the reduction used here as a tool is used to decide an exact question rather than to measure an approximate one — and where the same walk into the same region separates the sublattices that share the parent’s shape from the ones that merely share its symmetry.