Centring, and why cm is not pm

Of the five plane lattices, four are described on a cell with one lattice point and the fifth is not. The centred rectangular lattice gets a cell with two — a rectangle with a point in the middle — and the choice is deliberate, standard, and the source of a persistent confusion about what a systematic absence means.

The confusion is worth clearing up early. Centring is a property of a description, and lattice type is a property of a set of points. The letter at the front of a Hermann–Mauguin symbol reports the first, which is why cm and pm can be different groups while c and p are not different kinds of thing.

What centring is #

Take a rectangular lattice: points at the corners of a grid of rectangles. Now add a point at the centre of every rectangle. The result is a new lattice — the points still form a group under addition, so it qualifies — and it is not a rectangular lattice, because its shortest vectors run diagonally.

That new lattice can be described in two ways.

Primitively, on the rhombus whose edges are two of the shortest vectors. One point per cell, minimum area, and edges at an angle to the mirror lines the lattice carries.

Centred, on the original rectangle with its extra point. Two points per cell, twice the area, and edges along the mirror lines.

Both describe the identical set of points. Neither is more correct, and crystallography takes the second every time, because coordinates aligned with the symmetry make every subsequent formula shorter.

Why it earns a name #

A reasonable objection: if the centred lattice is just a rhombic lattice in disguise, why is it one of the five rather than a special case of the oblique one?

The answer is about symmetry rather than shape. A general rhombic lattice — equal edge lengths, arbitrary angle — carries four point symmetries: the identity, a half turn, and two mirrors, one along each diagonal of the rhombus. A general oblique lattice carries only two. The rhombic lattice’s mirrors are a genuine extra, they are present for every angle, and they are what the classification is counting.

The count of five comes out of that classification and not out of shapes. The enumeration on this site produces the holohedry — the full point symmetry — of each type and compares it against the value the classification claims: two for oblique, four for rectangular, four for rhombic, eight for square, twelve for hexagonal. A sixth type would need a holohedry no lattice can carry, which is the restriction doing its work.

So the centred rectangular lattice is on the list because it has four point symmetries and a rectangular one does too, while having a different arrangement of them: its mirrors run along the diagonals of its primitive cell rather than along its edges. Two lattices with the same number of symmetries, differently placed, which is the same distinction that separates p3m1 from p31m one level up.

cm against pm, as groups #

The two groups make the case concrete, and the difference between them is not a difference of description.

pm has a rectangular lattice and a mirror. Its operations per cell are the identity and one reflection: two in all.

cm has a rhombic lattice and a mirror. Its operations per cell are the identity and one reflection: two in all.

Same count, same kinds. They are nevertheless different groups, and the reason is what the translations do to the mirrors.

In pm the reflection lines are spaced one cell apart and there is nothing between them. In cm the centring translation carries each mirror onto a line halfway between two of them — and that composite, a reflection followed by a slide along its own axis, is a glide. So a cm pattern has reflection lines and glide lines alternating at half-cell spacing, while a pm pattern has reflection lines only.

Alternating mirrors and glides is what a centred group looks like on the page, and it is the most reliable way to spot one by eye. Reflection lines spaced at half the repeat mean centring; reflection lines matching the repeat mean none.

The glide that is and is not there #

The previous section needs a qualification, and the qualification is the sharpest illustration this essay has of its own thesis.

Everything on this site works modulo the lattice translations, because that is what makes a wallpaper group finite and its symmetries comparable. In the primitive rhombic basis — the one the machinery uses for cm — the centring translation is a lattice vector. So the glide, which is the mirror composed with that translation, sits in the same coset as the mirror, and the operation list modulo translations contains one reflection and not two. The detector, asked what symmetries a cm pattern has, reports the identity and a mirror.

In the conventional centred rectangular basis the same composite is a reflection plus a half-cell slide, which is not a lattice vector of that cell, so it appears as a separate operation and the group has four entries rather than two.

Both accounts are right and they are counting different things. The glide lines are in the pattern either way: draw a cm pattern and there are axes at quarter-cell positions across which the pattern reflects with a slide, and no choice of description removes them. What changes with the description is whether the glide counts as a distinct operation once translations have been divided out.

That is worth sitting with, because it is the same phenomenon as the vanishing reflections and it is not a coincidence. Describing a lattice on a cell coarser than its own repeat splits one operation into several and makes half the reflections disappear, and both effects are bookkeeping. The pattern does not notice.

What the round trip checked, and how #

Both groups are generated from their standard generators and handed to the detector, which enumerates every operation the lattice permits and keeps those that map the point set to itself. Two facts come back.

The detected group of a cm pattern is the identity and one reflection, on the rhombic lattice; the detected group of a pm pattern is the identity and one reflection, on the rectangular lattice. Neither detector is told what to look for and both match their generating sets exactly — and the two results are not the same result, because the lattice each was found on differs. The pair is the cleanest case on this site of two groups that a count of operations cannot separate and an enumeration of the lattice’s own symmetries can.

The check that earns its place is the one that runs on the diffraction. A centring point sits at $(1/2, 1/2)$ in the conventional cell, so it contributes $\exp(i\pi(h+k))$ to the structure factor, which is $-1$ whenever $h + k$ is odd. Every such reflection cancels exactly. The figure at the top of this page computes the structure factor for every reflection in a window and asserts, one by one, that it vanishes precisely when $h + k$ is odd — not approximately, and not for most of them.

The complementary assertion is the one that makes the point. The same lattice, described on its primitive cell with a single point in it, has no absent reflections at all, and that is computed rather than argued. Two descriptions of one set of points, one with half its reflections missing and one with none.

Where the exactness stops #

The absences above are exact and the interpretation of an absence is not, and the gap between those two sentences is where structure determination does its hardest work.

An absence in a measured pattern is a spot whose intensity is below the noise. That is a threshold judgement, and it is made harder by three effects that all put counts where none should be: multiple scattering, which lets a forbidden reflection borrow intensity from two allowed ones; overlap from a neighbouring spot; and detector background. The literature has a standing warning about it, since a weak reflection wrongly declared absent removes a whole family of candidate groups from consideration and a weak reflection wrongly declared present adds them back.

There is a second and cleaner limit. The absences from centring are indistinguishable, reflection by reflection, from absences that a glide would produce if the glide happened to remove the same set. Telling them apart requires knowing which cell the indices were assigned on, and that is information about the description rather than about the crystal. Nothing in the data supplies it.

So the practical rule is the one this essay opened with, and it is worth stating as a rule: an absence is a fact about a description until it is shown to be one about the structure. The showing is done by re-indexing on the primitive cell and seeing whether the absence survives.

Telling a centred pattern from a primitive one #

The practical question is how a reader confronted with an unlabelled pattern decides whether it is centred, and there are three tests of increasing reliability.

By the reflection lines. Reflection axes at half the repeat spacing mean centring, and axes matching the repeat mean none. This is the quickest test and the least reliable, because it needs the repeat to have been identified correctly first, and identifying the repeat is exactly what a centred pattern makes hard.

By the shortest vectors. Find the two shortest translations that map the pattern onto itself. If they are equal in length and the angle between them is not $90°$ or $120°$, the lattice is rhombic and any rectangular description of it is centred. This is a reduction calculation and it is exact.

By the diffraction. Index the pattern on the rectangular cell and look for reflections with $h + k$ odd. Their systematic absence is the centring signature, and it is the test an experiment actually performs, since a crystallographer never sees the pattern in real space at all.

The third test is worth dwelling on because it is the one that scales. In three dimensions there is no prospect of finding the shortest translations by looking, and the centring type of every structure ever solved was read off a pattern of missing spots before anything else about the structure was known.

The generalisation #

In three dimensions centring multiplies. The fourteen Bravais lattices include primitive, body-centred, face-centred and base-centred varieties, and each centring type removes its own family of reflections:

• Body-centred, with a point at $(1/2, 1/2, 1/2)$: everything with $h + k + l$ odd vanishes, so half the reflections go.
• Face-centred, with points at the centre of each face: everything whose indices are not all odd or all even vanishes, so three quarters go.
• Base-centred, with a point at the centre of one pair of faces: half go, and which half depends on which pair.

Those rules are the first thing read off a diffraction pattern, before any structure is contemplated, because they narrow the two hundred and thirty space groups to a manageable handful. A crystallographer who sees only reflections with all-even or all-odd indices has learnt that the lattice is face-centred, and has learnt it from where the spots are not.

The counting also explains why nobody centres everything. Adding a centring point to a lattice that does not need one produces a lattice already on the list under a different name: face-centred tetragonal is body-centred tetragonal in different axes, which is exactly the redundancy that made Frankenheim’s 1842 count come out at fifteen. Bravais found the duplicate in 1848 and the list has been fourteen since.

The surprising part #

The centred lattice is the one place in this subject where the standard description is deliberately redundant, and the redundancy is a considered trade rather than an oversight.

Here is the connection worth carrying away. Centring is a choice to describe a lattice as a sublattice plus cosets: the rectangular sublattice, plus one translated copy of it. That is exactly the structure of a quotient group, and the vanishing reflections are the characters of the quotient that fail to be trivial on the coset. The same arithmetic appears wherever a periodic structure is described on a coarser period than its own — in the folded Brillouin zones of a superlattice, in the extra spots that appear when an alloy orders, in the pattern of a doubled unit cell.

Which gives the general rule in its most portable form. Describe something on a cell larger than its true repeat, and the excess shows up as missing reflections rather than as an error. The description stays correct; it just carries a hole where the finer period would have put something. Learning to read that hole is most of what learning to read a diffraction pattern amounts to.

Who settled the conventions #

Bravais’s 1848 correction of Frankenheim gave the fourteen, and the naming that survives — P, I, F, C for primitive, body-, face- and base-centred — comes from the German Innenzentriert and Flächenzentriert by way of the International Tables.

The plane case is the smaller and older story. The centred rectangular lattice appears in every derivation of the seventeen from Fedorov’s 1891 onward, and it is the one type that some accounts of the classification get wrong by listing “rhombic” and “centred rectangular” as separate entries — the two-dimensional version of exactly the error Bravais corrected.

Pólya’s 1924 paper, which did more than any other to make the seventeen widely known, uses the centred description throughout, and the plates that accompany it are among the first published drawings in which the cell chosen is the symmetric one rather than the small one.

Where the ladder goes next #

The classification this type belongs to is the five plane lattices, and the freedom it exploits is the cell being a choice.

The experimental consequence is systematic absences, where the same calculation is run for a glide rather than for a centring point, and what a powder pattern loses, where the absences survive a further collapse.

The canonical description that removes the ambiguity altogether is the reduced basis, which is what a database uses when it has to decide whether two reported cells are the same lattice.

What the pictures here cannot show. One figure above does draw both cells over a single point set, so the reader is at least not asked to compare two separate drawings — but what it demonstrates is the counting, not the identity. That the rhombus and the rectangle describe the same lattice is a statement about the code that placed the dots, and a reader looking at the picture is taking the placement on trust. The same holds for the reflection panels: that a spot is absent rather than merely faint is a claim about an exact cancellation, and a drawing has no way of distinguishing a zero from a small number.

Why the plane has exactly one centring #

Three dimensions offers a menu — one face, all faces, the body — and the plane offers a single option. That is not because the plane is small but because the menu is a list of subgroups, and in two dimensions the list is short.

A centring adds points at halves of the cell vectors. Counted modulo the cell, the candidates are $\tfrac12\mathbf{a}$, $\tfrac12\mathbf{b}$ and $\tfrac12(\mathbf{a} + \mathbf{b})$ — the three non-zero elements of the group of two binary digits.

Two of them are not centrings at all. Adding $\tfrac12\mathbf{a}$ halves one edge and gives a lattice of the same kind with a shorter cell; the same for $\tfrac12\mathbf{b}$. Nothing has been produced that was not already on the list under a different description.

Which leaves one. The half-diagonal is the only candidate that produces a genuinely different lattice, so there is exactly one kind of centring in the plane, and the four-entry menu of space — where the same argument runs one dimension up — collapses to two entries here.

Body-centring and face-centring coincide. A two-dimensional cell has one interior point and its “faces” are its edges, so the distinction that separates body-centring from face-centring in space has nothing to separate here. The letter is c and there is no alternative to it.

Recognising one from the numbers #

A structure arrives as cell parameters and a coordinate list, and deciding whether the lattice is centred is a question about those numbers.

On the rhombic description the tell is an equality. A primitive cell of a centred rectangular lattice has its two edges the same length, with an angle that is not sixty, ninety or a hundred and twenty degrees. Two equal edges at a general angle is a centred rectangular lattice described primitively, every time.

On the rectangular description the tell is a coordinate. Every atom appears twice, at $(x, y)$ and at $(x + \tfrac12, y + \tfrac12)$, and a coordinate list showing that doubling throughout is a centred cell whether or not it says so.

And the diffraction pattern says it loudest. Index on the rectangular cell and half the reflections are systematically absent, in the pattern $h + k$ odd. That absence is a property of the description rather than of the crystal — the arithmetic is the same as the extinction any centring produces — and it is the first thing an indexing program tests.

The three tests agree or something is wrong. A cell with two equal edges, a coordinate list without the doubling, and no absences describe a lattice that is not centred and a cell that was chosen badly; the reduction of the previous rung is what settles it.