A layer is not a wallpaper

A sheet of graphene repeats in two directions and has two sides. A wallpaper pattern repeats in two directions and has none — it is a pattern of the plane, and the plane has no thickness for anything to be on either side of.

That difference is not a quibble. It changes the classification: there are seventeen plane groups and eighty layer groups, and a monolayer, a surface, a slab cut from a crystal, a woven cloth or a cell membrane is described by one of the eighty. The extra structure is small and completely specific. Every operation of a layer group either leaves the two faces where they are or exchanges them, and that single bit per operation is the whole of the difference.

The word for these is subperiodic: a group of motions of d-dimensional space whose translations span fewer than d directions. Two families are crystallographically useful.

A rod group repeats along a line and lives in space: a polymer chain, a helix, a nanotube, a column of a crystal structure. There are seventy-five.

A layer group repeats across a plane and lives in space: a monolayer, an interface, a slab, one sheet of a layered mineral. There are eighty.

Between them and the familiar cases there is a neat pattern of counts — 7 friezes, 17 plane groups, 75 rod groups, 80 layer groups, 230 space groups — and the two new numbers are not obtainable from the old ones by any short argument. That is the first thing to notice: eighty is not seventeen times anything.

It is also not a coincidence that both new numbers are of the same size as their two-dimensional relatives multiplied by four or five rather than by twenty. The room a third dimension adds to a layer is small: the normal is a single direction and the operations can do only two things to it. The room it adds to a periodic structure is enormous, because the third direction acquires its own translations, its own screws and its own glides — which is why seventeen becomes two hundred and thirty and eighty stops well short.

The classes that keep a direction #

The first restriction is on the point group, and it is exactly computable.

A layer group’s operations must map the layer onto itself, so each one maps the layer’s normal to plus or minus itself. A rod group’s operations must map its axis to plus or minus itself. Both questions therefore reduce to the same test: which crystal classes keep a direction at all?

So a layer group and a rod group both draw their point groups from the same twenty-seven axial classes, which is a strong constraint: no layer and no rod can have cubic symmetry, whatever it is made of. A cube of crystal has cubic symmetry; a slab cut from that cube does not, because the slab has a normal and the cubic operations do not keep it.

One sign per operation #

Given a plane group, how many layer groups sit over it? The question has a precise answer and the answer is the arithmetic of a two-coloured pattern.

Write each operation of a layer group as a plane operation A together with a sign ε = ±1 recording what it does to the normal. The signs cannot be arbitrary: composing two operations composes their signs, so the assignment is a homomorphism onto {±1}. That leaves three possibilities.

• Every operation takes ε = +1. The layer has two distinct faces, like a sheet of paper printed on one side, and nothing turns it over.
• Every operation occurs with both signs. The layer has a horizontal mirror, and its two faces are the same.
• The signs are given by a non-trivial homomorphism: some operations turn the layer over as they act, and those form the complement of an index-two subgroup.

The third case is the interesting one and it is the same computation as a two-colouring, with the other side of the sheet in place of the other colour. The enumeration was written for the colour question and is reused here unchanged, which is the point rather than an economy: turning a layer over and swapping two colours are one piece of group theory in two costumes.

Sixty-three arrangements over the seventeen #

Counting those possibilities across all seventeen plane groups gives sixty-three, of which twenty-nine involve turning the layer over.

Sixty-three is not eighty, and the gap is instructive. What has been counted is the point-group half of the question: which linear parts a layer group can have over a given plane group. What has not been counted is the translation half — a plane across the layer can be a glide as well as a mirror, and the glide direction is a further choice. Adding those brings the total to eighty, and doing so completely needs the extension arithmetic in full, which is the same work the space groups need and is not attempted here.

That is a deliberate line and it is the same one this site draws at 230. Numbers this site derives are derived; numbers it takes from the literature are named as taken. Twenty-seven and sixty-three are computed above. Eighty and seventy-five are quoted.

The one-sided case is not a curiosity #

The layer with two distinct faces is worth its own paragraph, because it is what almost every real surface is.

A crystal surface has vacuum on one side and crystal on the other. Nothing about it is symmetric under exchanging those, so its group is one of the one-sided ones: no horizontal mirror, no inversion, no two-fold axis lying in the surface plane. That single observation removes more than half of the eighty groups from consideration before any structure is measured, and it is why surface crystallography’s tables are shorter than the layer-group tables.

Where the layer groups actually turn up #

Four places, and the fourth is the one that surprises.

Two-dimensional materials. Graphene, boron nitride, the transition-metal dichalcogenides: single sheets whose symmetry is a layer group and whose properties — piezoelectricity in particular — depend on which. A monolayer of MoS₂ lacks the inversion centre its bilayer has, which is a difference of layer groups and shows up in a second-harmonic measurement.

Surfaces and interfaces. Reconstructions, adsorbed layers, and the sublattices they sit on.

Layered structures. Micas, clays and graphite are stacks of layers whose individual sheets have layer symmetry; the stacking then decides the space group, and polytypism is the same freedom this site meets in close packing.

And sections of ordinary crystals. Any plane through a three-dimensional crystal has a layer group — the subgroup of the space group that maps that plane to itself. That is how the subperiodic groups earn their place in the International Tables: not as exotica, but as the answer to what symmetry does this slice have, which is a question every electron microscopist asks.

The friezes, one dimension up #

The rod groups deserve the parallel because the site has already done the smaller case in full.

The seven friezes are the groups periodic in one direction inside two, and this site enumerates them: sixteen candidate combinations of a horizontal mirror, a vertical mirror, a half-turn and a glide, closed and renormalised, collapsing to seven. Rod groups are the same question inside three dimensions, where the extra room admits screw axes of every allowed order and rotations about the axis rather than only half-turns. Seven becomes seventy-five.

Reading a layer group’s symbol #

The symbols look like space-group symbols with one position that means something else, and knowing which position is the whole of learning to read them.

A layer group is written with a lower-case p or c for its two-dimensional centring, then three positions for the three symmetry directions — but the non-periodic direction is written first for a rod group and third for a layer group, by convention, and a numeral 1 holds any position with nothing on it. So p2₁/b11 is a layer group whose only symmetry beyond the lattice is a screw along a and a b-glide across it, while p112/m puts a two-fold along the normal.

The convention that a 1 must be written rather than dropped matters for exactly the reason it matters for p3m1 and p31m: the position a symbol’s entry sits in is what says which family of directions it belongs to, and dropping a trailing 1 merges two groups that are not the same. Layer-group notation keeps every position for that reason, which makes its symbols longer than a plane group’s and unambiguous in the same way.

The other reading worth knowing is the negative one. A layer group symbol with an m in the normal position has a horizontal mirror and is therefore two-sided; one with a 2 in a position lying in the plane has an axis that turns the sheet over. A symbol with neither is one-sided, and a surface scientist can rule a group in or out by looking at where the letters are.

Who derived them, and how long they waited for a volume #

The layer and rod groups were derived in 1929, in the same year and in the same milieu as most of the rest of this subject’s classification work — by Weber, and independently in the work of Alexander and Herrmann, at the moment when the space groups had settled and the remaining classifications were being cleared up.

Then they waited. The International Tables gave the space groups a volume in 1935 and again in every revision since; the subperiodic groups did not get one until Volume E in 2002, seventy-odd years later, edited by Kopský and Litvin. For most of the twentieth century a crystallographer wanting a layer group had to derive it or find it in a paper.

The delay is a fact about what was being measured rather than about the mathematics. Bulk structures were what diffraction could see, and a monolayer scatters far too little for a laboratory source; surface crystallography needed electron diffraction, synchrotrons and scanning probes before it had structures to classify. The classification was finished decades before there was anything to apply it to — which is the same order of events as the crystal forms, where the mineralogists tabulated what a class permits a century before anybody could measure a lattice parameter.

Where the exactness stops #

Three limits, stated plainly.

The eighty and the seventy-five are quoted. What is computed here is the axial-class list and the point-group arrangements over each plane group. Reaching the full classification means enumerating the translation extensions and quotienting by the changes of basis and origin that describe one group twice — the work the arithmetic classes show the shape of, at a scale this site has done for named classes and not for a whole family.

The round trip does not run here. Every pattern figure on this site hands its point set to a detector that rediscovers the group; there is no detector for layer groups, so the layers drawn above are generated and not verified. They are drawn from a plane group’s operations lifted by an explicit sign assignment, which is exact — but it is a construction rather than a round trip, and the difference is worth naming rather than eliding.

And a layer group is not a description of a real layer’s thickness. It says which motions are symmetries; it says nothing about how thick the slab is, and two slabs of different thickness cut on the same plane have the same layer group. That is a feature — the classification is about symmetry — but it means a layer group cannot distinguish a monolayer from a hundred-layer film, which physically are very different objects.

What the extra sign decides about a material #

The whole difference between a plane group and a layer group is one sign per operation, and it is worth following that sign into a measurement, because it decides properties an ordinary wallpaper classification has no way to discuss.

The sign records whether an operation turns the sheet over. An operation with ε = −1 exchanges the two faces of the layer, and among such operations the important one is the inversion, which a layer group may have and a plane group cannot even express. Whether a layer has one decides, by Neumann’s principle applied to the layer group, whether it may be piezoelectric — since an odd-rank polar property cannot survive a centre.

That gives a prediction about stacking that is checkable and has been checked. Take a layer whose own group has no inversion — a sheet with two inequivalent atoms per cell, arranged so that the two sublattices are distinguishable, which is the honeycomb with unlike vertices. One such sheet is not centrosymmetric and may be piezoelectric. Stack two of them in the arrangement that puts one sublattice over the other, and the pair acquires an inversion centre between them: the bilayer is centrosymmetric and the effect is forbidden outright.

Add a third layer and the centre is gone again. So the property alternates with the number of layers, present for odd counts and absent for even ones, and it does so for a reason that is entirely a matter of counting signs. Nothing about the chemistry changes between two layers and three; a symmetry element appears and disappears, and a rank-three tensor is killed and restored.

This is the clearest thing the subperiodic classification buys. A wallpaper group has no way to say that a sheet has two sides, so it has no way to distinguish the sheet from the bilayer, and a property that alternates with layer count is invisible to it. The eighty exist because the seventeen cannot express a sign, and every consequence of that sign is a consequence the seventeen cannot state.

The same argument runs for the other operations. A layer with a mirror in its own plane is one whose two faces are identical, so no property distinguishing up from down may occur — no polarisation perpendicular to the sheet, and no bending response to a uniform field. A layer with no such mirror has two chemically different faces and may have all of them, which is what a surface reconstruction or an adsorbed monolayer is. The one-sided case is not a degenerate corner of the classification; it is the case every real interface sits in.

There is a general moral in that alternation, and it is about what a classification is for rather than about layers. A property table indexed by the seventeen would report one answer for a sheet, its bilayer and its trilayer, because all three project to the same plane group and the projection is what the seventeen see. The subperiodic classification does not merely add detail: it adds the one distinction the measurement is sensitive to, and everything else it carries is the same. That is the usual reason a finer classification earns its place — not that it is finer, but that the thing it distinguishes turns out to be the thing that varies.

Where the ladder goes next #

The obvious next rung is the rod groups themselves: the helices, where a screw axis of order that no lattice permits — a 10₃ in a polymer, a 7₂ in a protein fibre — is perfectly legal, because there is no two-dimensional lattice in the way. That is the one place in crystallography where the restriction quietly does not apply, and the reason is worth an essay.

One caution belongs with that. The alternation described above is a statement about the ideal, freely suspended stack, and a real sample sits on a substrate. A substrate is on one side and vacuum on the other, so it removes any operation exchanging the two faces whatever the stack itself would have had — which restores the forbidden property in every layer count, weakly, and for a reason nothing about the layer decides. That is the same boundary the whole of this collection draws: the group says what an arrangement permits, and what is measured is an arrangement together with everything around it.