What a cleave leaves

A layer is not a wallpaper builds a layer group as a plane group with a sign attached to each operation, and leaves open the question of where such groups come from. In practice they come from cutting.

Cleave a crystal, grow one on another, or stop a calculation at a slab: whatever the reason, some plane through the structure has been singled out, and the operations that survive are those that map that plane to itself. The rest are still symmetries of the infinite crystal and are no longer symmetries of the thing in front of anybody. What is left is a sectional layer group, and it is a subgroup of the space group rather than a new object — which is the whole point, because it means the answer is computable and not merely nameable.

Two conditions, and the second is where the work is #

An operation (M, t) of a space group keeps the plane z = c where it is when two things hold.

The plane must not tilt. The linear part must send the z axis to ±z, which is the condition that the third row of M is (0, 0, ±1). Any operation with a component carrying z into the plane sends the plane somewhere else entirely, and there is nothing to discuss.

The plane must come back to its own height. With the sign ε = M₃₃ and the translation’s third component t_z, the plane goes to z = εc + t_z, and that must equal c modulo the lattice. For ε = +1 this reads t_z ≡ 0: the operation survives at every height or at none. For ε = −1 it reads 2c ≡ t_z: the operation survives at exactly two heights per cell, and at no others.

The second condition is the interesting one because it makes the answer depend on c. A twofold axis lying in the plane, a mirror perpendicular to the normal, an inversion centre — all of them are ε = −1 operations, and each is available only at two heights. A screw axis along the normal has ε = +1 with t_z = ½, so it satisfies neither: a screw axis survives no cut at all.

That last statement has a consequence for anyone computing on surfaces. A crystal whose bulk symmetry is largely made of screws presents a surface with almost none of it, and a slab calculation that inherits the bulk space group has assumed the answer to this computation rather than made it.

The sign each operation carries is the layer group’s own bit #

The operations that survive come with the sign ε already attached, and that sign is exactly the homomorphism onto ±1 that a layer is not a wallpaper builds layer groups from — arriving here from the other direction. An operation with ε = +1 leaves the two faces of the layer where they are; one with ε = −1 turns the layer over.

So a section carries three pieces of information rather than one: the plane group its operations project onto, the number of operations that survive, and how many of those exchange the faces. Two sections can project onto the same plane group and be different layer groups, because one turns the layer over and the other does not — which is the difference between a sheet printed on one side and a sheet printed on both.

The profile of a cut #

Sweeping the height through a whole cell gives a profile, and its shape is the same for every group: flat, with spikes.

The flat part is the general section, and it is what a cut through a generic plane leaves. The spikes are the special heights, where an ε = −1 operation comes back and the surviving group doubles or quadruples. In Pnma the general section keeps two operations and the special ones keep four; in P6₃/mmc the general section keeps six and the special heights keep twelve.

Two things are worth reading off the comparison. The first is that the ratio is never large but is always more than one for a centrosymmetric group — an inversion centre is an ε = −1 operation, so it is available at two heights and nowhere else, and those two heights are where the profile spikes.

The second is that the position of the spikes is not always at nice fractions. Fddd’s best sections are at eighths, which is the same eighth that makes two origins for one group a problem for that group: its inversion centres sit at (⅛, ⅛, ⅛) from the point of highest site symmetry, so the planes on which they lie are at eighths rather than at halves. A survey sampling only halves and quarters would report Fddd as having no special sections at all.

A section is a stabiliser #

The construction has a name in the language the rest of this site uses, and giving it that name says what kind of object it is.

The space group acts on the set of planes parallel to (001). That action has orbits — the planes at heights c, c + 1, c + 2 and so on are one orbit, and so are c and −c when the group contains an ε = −1 operation — and the stabiliser of a plane in that action is precisely its sectional layer group. Everything true of stabilisers is therefore true here: the orbit of a plane is as long as the group divided by its stabiliser, planes in one orbit have conjugate stabilisers, and the planes with large stabilisers are isolated while the general ones are not.

That is the same arithmetic as Wyckoff positions, applied to planes instead of points. The profile above is a Wyckoff table for the action on planes, the spikes are the special positions, and the flat part is the general one — with the orbit–stabiliser theorem accounting for the shape: a section of twice the symmetry has an orbit half as long, so it occurs at half as many heights.

There is a small pleasure in the special case. A group with no ε = −1 operation at all has a constant profile, every plane has the same stabiliser, and the action on planes is free — which is the same word, and the same property, that makes p1 and pg safe to draw a motif in. Freeness is what “no landmarks” means, whether the object being acted on is a point or a plane.

A worked cut: the close-packed layer #

The clearest example is the one every metallurgist already has in mind, and it is worth following because the answer is a number rather than a description.

P6₃/mmc is the space group of hexagonal close packing, and it is a group with a screw axis: the 6₃ carries a sixth-turn with a half-translation along c, which is precisely what makes the AB stacking repeat every two layers rather than every one.

Its profile has a general section of order six and special sections of order twelve, at z = 0 and z = ½ — the planes of the atoms themselves — with another kind of special section at the quarter heights, which are the planes between the layers. The screw axis appears in none of them, exactly as the second condition requires: the 6₃ has ε = +1 with t_z = ½, so it fixes no plane whatever.

So a hexagonal close-packed metal cleaved along its basal plane presents a layer whose symmetry is p6m as a plane group, and a cut between the layers presents p3m1 — a different group, from the same crystal, half a layer spacing away. The two faces have different site symmetries for an adsorbate, and two stackings, one density is where the two packings’ bulk groups are told apart in the first place.

A section that arrives centred, and gets its name back #

The (001) section of an F-centred group contains the pure translation (½, ½, 0). In the parent’s basis that is an operation with a fractional translation, and no plane group has one — so the naming routine returns nothing and the section looks unnameable.

It is not: on the lattice it actually generates, that translation is simply a lattice vector, and the group is p4m. So the pure translations are collected, a basis of the finer lattice is taken by a two-dimensional Hermite normal form, and the operations are re-expressed on it. This is why the bigger cell wins in reverse: a centred description hides the group’s name and the primitive one shows it.

Before that step, twenty of the space groups here produced at least one section with no name. After it, every section of every group is named — which is the check that the reduction is doing what it claims rather than relabelling something arbitrarily.

What a surface scientist gets from this #

Three consequences, and the first is the one that changes an experiment rather than a description.

The surface’s periodicity is the section’s, not the bulk’s. Low-energy electron diffraction from a clean surface produces a pattern with the two-dimensional symmetry of the sectional layer group, and the extra spots of a reconstructed surface are indexed against that. A pattern indexed against the bulk projection instead will disagree about which reflections should be present.

The surface’s site symmetries are the section’s. An adsorbed atom sits at a site whose symmetry is the site symmetry within the layer group, and that is generally lower than the site symmetry the same position has in the bulk — some operations that fixed it have gone. Wyckoff positions is the plane version of this arithmetic, and the sectional layer group is what it must be applied to.

Which faces a crystal shows, and which cleave. A cleavage plane is one that costs little energy to make, and low energy correlates with dense packing and with the plane being a mirror or lying between weakly bound layers. This computation does not decide cleavage — it is a symmetry argument and predicts nothing about energy — but it says what symmetry the cleaved face has if it cleaves there, and that is what a micrograph is interpreted against.

The census over forty-five groups #

Running the profile for every space group this site defines turns the individual observations into a table with three columns worth reading.

Twenty-one of the forty-five have a flat profile: no height leaves more than any other, so every cut through them is alike. Those are the groups whose ε = −1 operations are absent or are screws and glides that fix no plane — and a flat profile is a statement about the group rather than about the sampling, since a grid of twenty-fourths carries every special height the site’s groups can have.

None of the forty-five has a section larger than the group it came from, which sounds too obvious to check and is exactly the sort of thing worth checking: a section is a subgroup, so its order divides the parent’s, and a bug in the plane-part reduction that duplicated an operation would break it silently.

The best-to-general ratio is one or two, and never anything else. That is forced rather than observed. The operations with ε = +1 either survive at every height or at none, so they contribute the same amount everywhere; the ones with ε = −1 form a single coset of that subgroup, and a special height admits either all of them or none. So a section is the general one or exactly twice it, over all forty-four groups. A crystal’s faces come in two kinds of symmetry, not in a graded series of them.

Where the exactness stops #

Four limits, and they are all about scope rather than precision.

Only (001) sections are computed. The same two conditions work for any plane, with the normal replaced by a general reciprocal-lattice vector, and the arithmetic is no harder. What stops it here is presentation: a section on (111) needs a two-dimensional lattice that is not the parent’s a and b, so the plane group has to be found on a basis the figure would also have to explain.

A section is a symmetry, not a surface. A real surface relaxes: the outermost atoms move, sometimes by a substantial fraction of a bond length, and a reconstruction can change the periodicity outright. The sectional layer group is the symmetry of the unrelaxed cut, which is the starting point of a surface calculation rather than its answer.

Nothing here decides which layer group of the eighty a section is. The plane group it projects onto is named, the count of operations is exact, and the ε signs are known — which determines the layer group — but this site does not carry the eighty standard symbols, and a layer is not a wallpaper quotes eighty rather than deriving it. The number is the literature’s here, and it is named as such.

The cut is ideal. Every operation is tested against an infinite plane through an infinite crystal. A real slab is finite in the third direction, so operations that map the plane to itself but move the two surfaces of the slab onto one another are symmetries only if the slab is symmetric about its middle — which is a statement about the slab, not the crystal.

Where the ladder goes next #

The subperiodic anchor now has two rungs: what a layer group is, and where sectional ones come from. Three directions are open.

Rod groups. The same argument with a line rather than a plane: the operations that map a given direction to itself, which are the subperiodic groups of a chain or a nanowire. There are seventy-five, and the counting is a genuine enumeration this site could make rather than quote.

Sections on a general plane, which is a matter of presentation rather than arithmetic and would give the layer groups of the low-index faces every surface scientist works with.

The eighty, derived. The layer groups are extensions of plane groups by a sign, and enumerating them is the same construction the ten ways of being mm2 makes for space groups, one dimension down. That would replace the last quoted count in this field with a computed one.

What the surface is then permitted #

A sectional group is a symmetry group, so everything this collection says about what a group permits applies to it — and applying Neumann’s principle to a section rather than to a bulk gives the standard explanation of a whole family of surface measurements.

A surface always loses the centre. Whatever the bulk group contains, an operation that inverts through a point on the plane sends the material below the plane to where the vacuum is, so it cannot be a symmetry of the terminated crystal. Every surface of every crystal is therefore non-centrosymmetric, including surfaces of crystals whose bulk is not.

And a property forbidden by a centre becomes permitted there. Second-harmonic generation — light in at one frequency, out at twice it — is forbidden in any centrosymmetric medium, because the effect is described by a tensor of odd rank and a centre sets every odd-rank tensor to zero. In a centrosymmetric crystal the bulk contributes nothing, and the surface, which has no centre, contributes everything.

That is why the technique is surface-sensitive at all. The signal is not surface-sensitive because the light stops at the surface; it penetrates perfectly well. It is surface-sensitive because symmetry silences the bulk, so the few atomic layers where the centre is broken are the only source. A measurement of the harmonic is then a measurement of the section this essay computes, and its polarisation dependence reports which operations survived.

The same argument runs for a polar surface. If the sectional group has no operation reversing the normal, the surface is permitted a dipole along it, and a slab terminated that way carries a net polarisation. The energy of that polarisation grows with the slab’s thickness, so such surfaces are unstable and reconstruct — adsorbing, faceting or rearranging until the dipole is cancelled. Which terminations do that is decided by the sign $\varepsilon$ attached to each surviving operation, which the profile already carries.

Which cut the crystal chooses #

The computation here answers what a cut at a given height leaves, and a reader is entitled to ask which height a real cleavage takes. The two questions are independent, and running them together is the commonest way this material is misapplied.

Symmetry does not decide where a crystal breaks. A crystal cleaves where the work of separation is least — across the planes held together most weakly, which are usually the widely spaced ones with few bonds crossing them. Mica splits along its basal plane because the layers are bound by weak interlayer forces; halite splits on the cube faces because those cuts break the fewest ionic bonds; graphite is the extreme case of the same statement.

Those planes are often the symmetric ones, which is why the two questions get conflated. Wide interplanar spacing tends to go with high symmetry, since a plane densely occupied by atoms is also one that many operations map to itself. The correlation is real and it is not a rule, and nothing in the profile above predicts a cleavage energy.

So the profile answers the second question, not the first. Given a cut, it says what symmetry the surface has, and therefore what the surface is permitted to do. Which cut occurs is a question about bonding, and the honest reading of the spikes is that they mark the terminations worth computing rather than the terminations that happen.

Which is still the useful direction for a calculation. A slab calculation has to choose a termination, and the choice determines the answer; knowing which heights leave which symmetry says which slabs are physically distinct and which are the same slab described twice.