A lattice described on somebody else's axes

Twenty-five candidate cells go into the Bravais enumeration and twenty-four of them get one of two clean verdicts: either the centring kept the system’s symmetry, or it did not. The twenty-fifth does both at once.

The verdict that is two verdicts #

The rest of the enumeration works by a rule with a clean statement: a centring survives when its set of centring vectors is closed under the system’s point group. Cubic C fails it, hexagonal C and hexagonal I-centring fail it, and everything else passes.

R passes and fails at the same time, and the reason is that it is closed under part of the hexagonal point group and not all of it. The threefold along c permutes the three vectors (0,0,0), (2/3,1/3,1/3) and (1/3,2/3,2/3) among themselves — which is why the lattice keeps its threefold. The sixfold does not: applying it to (2/3, 1/3, 1/3) gives a point the set does not contain, so the sixfold is destroyed and the twenty-four halves.

A centring that preserves half a point group is not a case the other twenty-four rows produce, and it is the only reason this row needs its own essay.

R-centring puts points at (2/3, 1/3, 1/3) and (1/3, 2/3, 2/3) inside a hexagonal cell. The resulting lattice has twelve automorphisms, not the twenty-four a hexagonal lattice has.

By the rule above, that is a failure: the centring has destroyed the sixfold axis and what is left belongs to a less symmetric system.

But twelve is the rhombohedral number, and the rhombohedral system is a row of its own in the enumeration, with its own primitive cell. So the lattice R produces is not merely “less symmetric” — it is a specific lattice, and that lattice is on the list.

So R is a duplicate, and the transformation says of what.

The three new vectors are worth writing out because they are memorable. In hexagonal coordinates the rhombohedral cell’s edges are

aR = (2/3, 1/3, 1/3), bR = (−1/3, 1/3, 1/3), cR = (−1/3, −2/3, 1/3)

and the first of them is one of the centring vectors. That is the whole trick: the centring points of the R description are the corners of the rhombohedral cell.

Check it: aR − bR = (1, 0, 0), which is the hexagonal a. bR − cR = (0, 1, 0), the hexagonal b. And aR + bR + cR = (0, 0, 1), the hexagonal c. So the three hexagonal axes are all in the rhombohedral lattice, as they must be, and the rhombohedral cell has a third the volume — index three, which is the number of centring vectors including the origin.

Why anybody uses the hexagonal setting at all #

If the rhombohedral cell is primitive and the hexagonal one has three lattice points in it, the primitive cell looks like the obvious choice. It is not the one most of the literature uses, and the reasons are worth having.

The rhombohedral cell has an awkward angle. Its three edges are equal and its three angles are equal, and the angle α is whatever the crystal makes it — 33° in one structure, 87° in another. Nothing about the number is memorable and two rhombohedral crystals with different α look completely unlike each other on paper.

The hexagonal setting has orthogonal-ish axes. Two equal axes at 120° and a third perpendicular to both is a cell most people can picture, and it is the same cell shape as every other trigonal and hexagonal structure — so a rhombohedral crystal and a hexagonal one can be compared directly.

Indices are easier. Miller indices on hexagonal axes have the familiar h, k, i, l form with i = −(h+k), and the reflection conditions read cleanly. On rhombohedral axes they do not.

What twelve looks like #

The rhombohedral holohedry is 3̅m and it has twelve operations, which is worth listing because it is the only holohedry of the seven that is neither obvious nor familiar.

Three rotations about the threefold axis: the identity, a third of a turn and two thirds. Three twofold axes perpendicular to it, at 120° to each other. That is six proper rotations, the point group 32. Compose each with the inversion and there are twelve.

The number sits awkwardly between the hexagonal twenty-four and the orthorhombic eight, and it does not divide either — 12 does not divide 24 evenly in the sense that matters here, because 3̅m is a subgroup of 6/mmm, index two. So a rhombohedral lattice is a hexagonal lattice with half its symmetry taken away, which is precisely what centring it did.

The “cube leaned on a corner” picture is worth pushing on, because it is exactly right in two special cases and those cases matter. At α = 90° the rhombohedral cell is a cube, and its automorphism count jumps from twelve to forty-eight. At α = 60° the rhombohedral cell is the primitive cell of a face-centred cubic lattice, and again the count is forty-eight. So the rhombohedral system contains two cubic lattices as coincidences, and the generic metric this site uses is chosen to avoid both — the assertion that rhombohedral gives twelve would fail immediately if it did not.

The obverse and the reverse #

There is a second choice hiding in the R description and it has no analogue anywhere else in the fourteen.

The centring vectors were given above as (2/3, 1/3, 1/3) and (1/3, 2/3, 2/3). The other possibility is (1/3, 2/3, 1/3) and (2/3, 1/3, 2/3) — the same construction with the thirds the other way round.

Both are perfectly good R-centred lattices. They are related by turning the cell through 60° about c, or equivalently by reversing the sign of a and b. The first is called the obverse setting and the second the reverse, and the Tables use obverse.

That would be a footnote except that it is a real source of published error. Two crystals of the same substance, indexed by different people, can end up described on obverse and reverse settings, and comparing their atomic coordinates directly gives nonsense. The reflection conditions differ too: obverse R gives −h + k + l ≡ 0 (mod 3) and reverse gives h − k + l ≡ 0 (mod 3), so the diffraction pattern distinguishes them, which is the practical way the ambiguity gets resolved.

The condition under each cell is derived rather than remembered, and the derivation is one line: a reflection survives a centring when every centring translation puts a whole number of turns into its phase. For the obverse vector (2/3, 1/3, 1/3) that means (2h + k + l)/3 must be an integer, which is −h + k + l ≡ 0 modulo three because 2 and −1 differ by three. For the reverse vector (1/3, 2/3, 1/3) the same arithmetic gives h − k + l ≡ 0. Two conditions of the same shape, disagreeing on which pair of indices carries the minus sign, and a diffraction pattern answers the question that no amount of staring at the cell can.

The obverse/reverse distinction is not a symmetry question — both settings describe the same lattice type, and neither is more correct. It is a labelling question, and it belongs with the other things in this collection that turn out to be about description rather than about the object: which cell, where the origin sits, which axis is called a. The subject has an unusual number of these and it is worth noticing the pattern.

Reading a symbol that starts with R #

Because the two settings both exist, a rhombohedral space group has two symbols in circulation and the Tables print both.

R3 on hexagonal axes has nine general positions and its cell is three times the primitive volume. The same group on rhombohedral axes has three, and the Tables label the two R3 (hexagonal axes) and R3 (rhombohedral axes). They are not two groups and the number 146 refers to both.

The practical consequence is that atomic coordinates in an R group cannot be compared without knowing the setting. A structure reported on hexagonal axes and one reported on rhombohedral axes will have completely different coordinate lists for the same arrangement, and the transformation between them is the one at the top of this essay.

That is a specific, common and entirely avoidable source of confusion, and the Tables’ solution is a convention: hexagonal axes unless stated otherwise. It works about as well as such conventions do.

Trigonal is not a lattice system #

The word that causes the most trouble here is trigonal, and it causes trouble because it is a name for a set of point groups rather than for a lattice.

There are seven crystal systems by point group — triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, cubic — and the trigonal one contains the point groups whose highest rotation is threefold: 3, 3̅, 32, 3m and 3̅m.

There are also seven lattice systems, and they are not the same seven. The lattice systems are triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal and cubic. Trigonal is not among them and rhombohedral is.

The mismatch is entirely in this one place. A trigonal point group can sit on either a rhombohedral lattice or a hexagonal one — 3 gives P3 on hexagonal and R3 on rhombohedral, and both are trigonal crystals. So “trigonal” does not determine the lattice, and “rhombohedral” is not a point-group class.

Getting this wrong is common enough that the Tables spell it out, and the confusion has a specific shape: people say “trigonal” meaning “rhombohedral”, find a trigonal structure on a hexagonal lattice, and conclude something has gone wrong. Nothing has. Five of the twenty-five trigonal space groups are R and the other twenty are P.

What is actually rhombohedral #

The lattice type is real and the structures on it are worth naming, because “rhombohedral” can otherwise sound like a bookkeeping category.

Calcite — calcium carbonate — is the standard example, in R3̅c. Its structure is a rock-salt arrangement squashed along a body diagonal, with the carbonate groups lying in planes perpendicular to the threefold axis. The squashing is what takes it from cubic to rhombohedral: the cubic lattice’s four threefolds become one.

Corundum, which is aluminium oxide and therefore ruby and sapphire, is R3̅c as well.

Bismuth, antimony and arsenic are all rhombohedral elements, in R3̅m, and their structures are distorted simple-cubic arrangements — again a cubic lattice with the symmetry along one body diagonal kept and the other three lost.

The pattern across all of them is the same and it is the geometric reading of the twelve-out-of-forty-eight above. A rhombohedral lattice is what a cubic lattice becomes when it is stretched or squashed along one body diagonal. Three of the four threefolds go, one survives, and the forty-eight drops to twelve. That is a common thing for a real structure to do, which is why the lattice type is well populated despite looking like an oddity in the table.

The index three, which is unique here #

One last thing sets R apart from every other centring in the fourteen, and it is a number.

C-centring has index two. I-centring has index two. F-centring has index four. R-centring has index three, and it is the only odd index anywhere in the enumeration.

That has consequences that show up in every calculation. A C-centred cell’s reflection condition is a parity condition — h + k even — and parity conditions are the ones everybody’s intuition handles. R’s condition is −h + k + l ≡ 0 modulo three, and a third of reflections survive rather than a half.

The volume arithmetic is the same story. A body-centred cell holds two lattice points and a face-centred one holds four, so both are comfortable powers of two; the hexagonal cell of a rhombohedral lattice holds three, and its volume is three times the primitive one.

And the d glide, which exists because centring makes a half-diagonal into a lattice vector and a quarter of it into a legal slide, has no R analogue at all — a third of a diagonal is not half of anything, so R-centring creates no new glide.

The odd index is a small fact with a long tail, and it is the reason rhombohedral calculations feel different from every other kind.

The choice that becomes a twin #

The obverse and reverse settings are described above as two descriptions of one lattice, and there is a situation in which a crystal contains both at once — which turns a convention into a defect with a name.

A rhombohedral crystal can grow with some regions built on the obverse arrangement and others on the reverse. The two are related by a rotation about the three-fold axis carrying one pair of centring vectors onto the other, and that rotation is an operation of the hexagonal lattice and not of the rhombohedral one — which is exactly what a twin law is: a symmetry the lattice has and the crystal does not.

So obverse–reverse twinning is twinning by merohedry on an R lattice, it is common in rhombohedral structures, and its index is one — the two orientations share every lattice point, so their reciprocal lattices coincide exactly and no reflection is split.

The tell is in the absences. A pure obverse crystal extinguishes the reflections failing its centring condition; a pure reverse one extinguishes a different set; a twinned crystal extinguishes neither completely, and the pattern shows weak reflections where an R-centred lattice permits none. A crystallographer meeting that has three candidate explanations — a supercell, a second phase, or this — and the third is settled by refining a twin fraction against the two orientations.

The consequence of missing it is a structure refined in the wrong lattice. Taking the extra reflections at face value means abandoning the R centring and describing the crystal on a primitive hexagonal cell three times too small in reciprocal terms, with a structure that fits and is not the one there.

The condition that looks like a hexagonal centring #

There is a reading of the same arithmetic that is worth stating because it is the shape a beginner’s error takes.

Written on hexagonal axes, the R lattice’s condition on the indices looks exactly like a centring condition on a hexagonal lattice: a linear congruence modulo three, extinguishing two reflections in every three, of the same form as the conditions an A, B, C, F or I-centring produces.

And that reading is not wrong so much as incomplete. The condition is a centring condition; what it does not say is that the lattice being centred is not the one the axes belong to. A hexagonal cell with those centring points is a rhombohedral lattice described on hexagonal axes, and its point symmetry is twelve rather than the twenty-four the axes suggest.

So the absence tells a determination that the lattice is centred and does not tell it which lattice system it is in. That has to come from the intensities — from the Laue class, which will be 3̄m or 3̄ rather than 6/mmm — and a determination that read the axes as hexagonal and the condition as a centring would arrive at a group that does not exist.

Where the argument stops #

Three limits.

The obverse/reverse choice is not modelled here. This site’s R centring is the obverse one, everywhere, and the reverse setting is described in prose rather than computed. The two are related by a change of basis this machinery could verify, and it is not one of the eight recorded transformations because the enumeration does not need it — obverse and reverse are the same lattice type and the count is unaffected.

The rhombohedral metric here is generic, as everywhere else. Its three equal edges and three equal angles are integers chosen so nothing accidental holds; a real rhombohedral crystal has a specific α and the classification does not depend on it. At α = 60° the lattice becomes face-centred cubic and at α = 90° it becomes simple cubic, which are genuine coincidences and would be caught by the automorphism count coming out at forty-eight instead of twelve.

The hexagonal setting of a rhombohedral group is not a hexagonal group. This is the confusion the whole essay exists to prevent and it is worth one more sentence. R3 written on hexagonal axes has a hexagonal-shaped cell and is not a group of the hexagonal lattice system — its lattice has twelve automorphisms, not twenty-four, and the cell shape is a description. Reading the cell shape as the lattice type is the mistake, and it is easy because for every other entry in the fourteen the cell shape and the lattice type do agree.

Twelve is the automorphism count of the lattice, not the crystal’s symmetry. A rhombohedral lattice permits 3̅m, order twelve. A crystal on it may have any subgroup of that — R3 has three operations before centring, R3̅ has six, R3m has six — and the lattice’s twelve is an upper bound that most structures do not reach.

The last of those is the general shape of everything in this anchor. A Bravais lattice is the scaffold and its symmetry is the most a crystal built on it can have; what the crystal actually has is decided by what sits at each point, which is the space group’s business rather than the lattice’s.