How many polytypes there are

Two stackings, one density establishes the free choice. A close-packed layer of spheres has two sets of hollows; the layer above takes one set or the other; and once two layers are down, the third faces a genuine decision — over the first layer, or over the hollows the second did not use. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours.

Repeat that choice indefinitely and there is not one structure, nor two, but a family — and every member of it is a genuine crystal with a space group of its own. Every sequence of letters A, B and C with no two adjacent letters alike is a close packing, and the ones that repeat are crystals. Counting them is a well-posed question, and it turns a familiar remark — silicon carbide has hundreds of forms — into an enumeration.

What counts as the same structure #

The counting question needs its equivalence stated before the number means anything, and three moves have to be quotiented out.

Rotation of the sequence. ABCABC and BCABCA describe the same crystal seen from a different starting layer. The sequence is a cycle, not a word.

Reversal. ABAC and CABA are the same structure viewed from the other end of the stack — the crystal turned upside down, which is a rotation of the whole thing rather than a different arrangement.

Relabelling. A, B and C name the three registrations, and which is called A is arbitrary. Permuting the three letters consistently gives the same crystal.

So the count is the number of necklaces in three colours with no two adjacent beads alike, up to reversal and colour permutation. That is a standard combinatorial object, and the numbers it gives — 1, 1, 1, 1, 2, 3, 6, 7, 16 at periods two to ten — are the ones above.

The two simplest are the ones everybody knows. Period two is AB, hexagonal close packing. Period three is ABC, cubic close packing. Periods four and five each admit exactly one, and the family only becomes interesting at six.

The two at period six, and what tells them apart #

At period six there are two: ABABAC and ABACBC. They have the same composition, the same density, the same coordination, the same first and second neighbour shells — and different space groups, which is the only thing that distinguishes them.

This is where the round trip does the work. The atom positions are generated from the sequence, handed to the detector in three dimensions which enumerates every operation the lattice permits and keeps those that map the set onto itself, and the group comes back without the sequence being consulted. So the claim “these two are different structures” is a computation rather than an assertion about the letters.

The rule of thumb that comes out is worth having. A sequence that is symmetric under reversal about some layer has mirror symmetry and lands in a hexagonal group; one that is not is rhombohedral. ABC is the extreme case: it is invariant under a cyclic shift of the labels, which is a rhombohedral operation, and the resulting group is the cubic Fm3̅m — cubic close packing being a rhombohedral stacking whose extra symmetry only appears when the cell is chosen properly.

Counting the words before counting the crystals #

The census reports two numbers at each period and the gap between them is the whole content of the equivalence.

At period six there are sixty-six admissible words — cyclic sequences of six letters over {A, B, C} with no two adjacent alike, the wrap-around included. Quotienting by all three moves at once leaves four classes, and two of those four are not period-six structures at all: ABABAB is hexagonal close packing written three times over and ABCABC is cubic close packing written twice. Discard those and two polytypes remain. At period eight the same three steps run 258 words, eight classes, six polytypes.

The step that removes the impostors is the one to watch, because it is not one of the three equivalences. Rotation, reversal and relabelling say when two words describe the same crystal; they have nothing to say about a word whose true period is shorter than the length it was written at. That is a separate filter and it is the reason the sequence of counts is not monotone in any obvious way — period four has two classes and one polytype, period five has one class and one polytype, period six has four and two.

The words are easy to count and mean nothing; the polytypes are what a diffractometer would distinguish.

Getting the quotient right is the delicate part, and there is a check available that costs nothing: the class sizes must divide the group order. The group acting is rotation by up to n, reversal, and the six colour permutations, and the orbit of every word must have a length dividing that order. A bug in the equivalence shows up immediately as an orbit of the wrong size.

There is a second check in the small cases, which is that the answer is known independently. At period two there is one packing and it is hexagonal; at period three one and it is cubic. If the quotient were too coarse the count would fall below one somewhere; too fine and period three would report two structures where every textbook says one.

Notation, and why there are two of them #

A polytype is named by its period and the shape of its cell, in Ramsdell notation: 2H is the two-layer hexagonal one, 3C the three-layer cubic one, 4H, 6H, 15R and so on, with the letter naming the lattice type rather than the crystal system. That is the notation the mineralogical and semiconductor literature uses, and it is a label rather than a description: 6H says the period is six and the lattice hexagonal, and does not say which of the two period-six sequences it is.

The Zhdanov symbol does say. It records the sequence as a run-length encoding of the layer-to-layer moves: whether each step goes “up” or “down” the cycle A → B → C, written as counts of consecutive steps in the same sense. ABABAC becomes (2 2 1 1) and ABACBC becomes (3 3), and the two are told apart at a glance.

The pair is a good example of a general habit in this subject: one notation for indexing a thing in a catalogue, another for saying what it is. One class, two names is the same situation among the point groups, and the resolution there is the same — the two are answers to different questions and both survive.

Where the choice comes from, and why it barely matters #

The free choice exists because the two candidate positions for the next layer are related by a symmetry of the layer beneath. Nothing local prefers one, and that is not an approximation: the first and second neighbour shells of any atom are identical in every polytype, so any energy that depends only on those is exactly equal across the whole family.

The differences begin at the third shell and they are tiny — a few millielectronvolts per atom in silicon carbide, which is why so many polytypes are observed in the same material and why which one grows depends on temperature, impurities and the growth method rather than on any large energetic preference.

This site does not compute those energies and will not. The applied field’s standing rule is that a symmetry argument states a permission and predicts nothing, and the whole of this essay is a count of possibilities with the space group of each. Which of the possibilities a crystal takes is materials science.

The layers themselves, and what a cut through one leaves #

A polytype is a stack, so the natural companion question is what a cut through it presents — and the answer connects this anchor to the subperiodic one.

Cleaving hexagonal close packing on its basal plane leaves a layer whose symmetry is p6m as a plane group; cleaving between two layers leaves p3m1. The two faces are half a layer spacing apart and have different symmetries, so an adsorbed atom sees a different set of sites on each — which is what a cleave leaves computing exactly this, on exactly this group.

For a longer polytype the answer varies down the stack. In a period-six structure the six layers are not equivalent, so there are up to six inequivalent cleavage planes with their own sectional layer groups, and the surface of a 6H crystal depends on where in the repeat it was cut. That is a real and awkward fact in silicon carbide device work, where the polarity of a face — silicon-terminated or carbon-terminated — changes its chemistry entirely.

Silicon carbide, and what “hundreds” means #

Silicon carbide is the standard example and the numbers are worth stating carefully, because the familiar claim mixes two different things.

More than two hundred and fifty polytypes of SiC have been reported. That is a count of observed structures, with periods running from two up to several hundred layers — 6H and 4H are the technologically important ones, 3C is the cubic form, and the long-period ones include a 594R.

The enumeration in this essay says how many are possible at each period, and the two numbers diverge fast: sixteen at period ten, forty-three at period twelve, and the count grows like $2^n/n$ — the words are $2^n$ up to a sign and the group they are quotiented by has only about $12n$ elements, so the exponential survives the quotient and only its coefficient does not. By period twenty there are thousands, and by period a hundred the number is astronomical. So the observed polytypes are a vanishingly small sample of the possible ones, and the interesting question is not why there are so many but why so few of the possible ones occur.

The answers offered are growth-kinetic — screw dislocations at the growing face imposing a period, or stacking-fault energies favouring particular repeats — and none of them is symmetry. This essay’s contribution is the denominator.

A polytype is a stacking fault made periodic #

There is a way of seeing the whole family that connects it to the rest of the applied field.

Take cubic close packing, ABCABCABC…, and insert one wrong layer: ABCACBACB…. That is a stacking fault, a two-dimensional defect, and its cost is small for exactly the reason above — the local environment is barely disturbed. A crystal with a few faults is a cubic crystal with defects.

Now make the faults periodic. A fault every third layer is a new polytype with period six; a fault every seventh gives period fourteen; and the whole family is the set of periodic fault arrangements. A polytype is a stacking fault put on a lattice, which is the same move antiphase domains describes for ordering: a defect made periodic becomes a superstructure.

The connection runs both ways in practice. A polytype with a long period is hard to distinguish from a shorter one with occasional faults, and telling them apart needs the sharpness of the superstructure reflections rather than their positions — a periodic arrangement gives sharp extra spots, a random one gives streaks. That is the same distinction the freedom a crystal has not draws between a uniform phason strain and frozen disorder, and it is measured the same way.

Where the symmetry actually differs #

If every polytype has the same density and the same neighbours, it is fair to ask what a symmetry argument can possibly distinguish — and the answer is a good deal, because a space group is a statement about the infinite structure rather than about a neighbourhood.

The lattice type differs. Sequences whose colour-permutation orbit closes after a cyclic shift are rhombohedral; the rest are hexagonal. That is the R or H in a Ramsdell symbol, and it decides which reflections are systematically absent — so two polytypes of the same period are told apart on a diffraction pattern by extinctions rather than by peak positions.

The site symmetries differ. In 2H every atom sits at a site of the same symmetry; in longer polytypes the layers are inequivalent, and a structure with period six has three or six crystallographically distinct atomic sites where 2H has one. That is directly measurable by any local probe — nuclear magnetic resonance sees a distinct signal per site, and the count of lines in a spectrum is one of the standard ways of identifying a polytype.

The band structures differ, and by more than a little: the band gap of silicon carbide runs from 2.4 eV in the cubic form to 3.3 eV in 2H, a spread of nearly a volt across structures with identical densities and identical neighbour shells. The mechanism is the zone folding the cell nobody chose describes — a longer repeat folds the Brillouin zone more times — and it is the reason polytypism matters commercially rather than only mineralogically.

None of those three is computed here. What is computed is the group, and each of them is a consequence of the group that some other discipline measures.

What the enumeration does and does not establish #

Four claims with different warrants.

The counts are exact. They come from generating every word of the given length over three letters with no two adjacent alike, and quotienting by the group generated by rotation, reversal and colour permutation. Nothing is sampled and no formula is quoted; the census reports the number of raw words and the number of classes at each period, so the quotient can be checked by hand at small periods.

The space groups are detected. The groups of the two close packings are rediscovered from their atom positions by the three-dimensional detector, and the round trip requires the detected set to equal the generating set exactly — in both directions, so a structure that turned out to be more symmetric than the sequence suggests would fail rather than pass quietly.

The density is measured. Both close packings fill π/√18 of space, computed on the structures rather than quoted, and the two agree to twelve figures — which is the arithmetic behind “the same density” in the first paragraph.

The observed counts are quoted. Two hundred and fifty-odd polytypes of SiC, the periods reported, the energy differences: all from the literature, none computed here, and named as such.

Where the ladder goes next #

The packing anchor now has two rungs: the choice and the count of what it produces. Three directions are open.

The space group of every polytype, rather than of the two simplest. The detector can do it; what is missing is a compact way to present a census in which each entry has a sequence, a Zhdanov symbol, a Ramsdell name and a group.

Packings that are not close. Body-centred cubic is denser than a simple cubic packing and less dense than close packing, and the sphere-packing question in three dimensions — settled by Hales in 1998, that no packing beats π/√18 — is a genuinely deep result that this site can state and cannot prove.

Layers that are not spheres. Every layered structure with two registrations has the same combinatorics: the transition-metal dichalcogenides, the clays, graphite and its rhombohedral polytype. The counting in this essay is about the sequence rather than about spheres, so it applies unchanged wherever the choice is binary — which is a considerably larger class of materials than the close packings.

How a polytype is identified #

The enumeration says which stackings exist. Telling which one a specimen has is a measurement, and it is one the arithmetic of this collection makes unusually direct.

Every polytype shares the same close-packed layer, so its in-plane lattice is the same and its reflections with indices making a whole number of the in-plane repeat sit at the same places whatever the stacking. The stacking shows up only in the third direction, and it shows up in the rows of reciprocal space that the layers’ registrations can distinguish — the same rows a faulted stack streaks.

For a periodic polytype those rows are not streaked; they are sharp, and the spots along them are spaced by the reciprocal of the period. So a stacking of period six gives six spots between the positions the cubic stacking would use, and counting them reads the period off directly.

That makes the identification two steps and no modelling. Count the spots along a streaking row to get the period. Then compare the intensities along the row against the candidates the enumeration lists at that period — two at period six, four at period eight — since the intensities depend on the sequence and the positions do not.

And the two steps have different characters, which is worth noticing. The period is a count and is unambiguous. The sequence is a comparison of intensities against a short list, so it is a fit — and where two candidates give similar intensity distributions, the identification is as good as the data and no better.

That is why the reported polytypes of a material are a list of periods with sequences attached, and why the long-period ones are reported by period more confidently than by sequence.