Crystallography & Symmetry
patterns derived from their groups
Concept

Polytype — where it appears

One of a family of structures differing only in the sequence of identical layers, and sharing a composition, a density and a coordination. What separates two polytypes is a word rather than a chemistry, which is why one substance can have hundreds of them.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours.

Two stackings, one density

Stack spheres as tightly as they will go and the third layer has a free choice. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours — and their space groups are Fm3̅m and P6₃/mmc, which is the only thing that tells them apart.

applied · Packing How many close packings there are of each period. Every cyclic sequence over three letters with no two adjacent alike is a close packing, and two sequences describe the same structure when one becomes the other by rotating the cycle, reversing it, or relabelling the three positions. Counting the classes that remain gives 1 of period 2, 1 of period 3, 1 of period 4, 1 of period 5, and 38 altogether up to period 10. Period two is hexagonal close packing and period three is cubic; everything above them is a polytype, equally dense and equally close packed, and silicon carbide has been found in more than two hundred of them. Nothing in the geometry chooses. What chooses is an energy difference of a few thousandths of an electron volt per atom, and no energies are computed here.

How many polytypes there are

One free choice per layer, repeated, gives a family of structures with the same composition, the same density and the same twelve neighbours — differing only in a sequence. Counting them up to rotation, reversal and relabelling turns "silicon carbide has hundreds of forms" into an enumeration.

applied · Packing One crystal, two rows: one streaks and one does not. Two rows of reflections from the same faulted crystal, at a fault rate of 0.05, each drawn against the same row from a perfect one. The upper row has h − k not divisible by three, so each layer contributes a different cube root of unity and the sequence of layers enters the sum: the sharp peaks collapse into a streak. The lower row has h − k divisible by three, the phase factor is one, every layer scatters in step, and the peaks are exactly as sharp as in the perfect crystal. The sorting is an integer condition — a reflection either can see the stacking or cannot, decided by h − k modulo three — which is the most direct evidence there is that the disorder is in the stacking and not in the layers. The profiles are averaged over 16 independently faulted crystals, because one crystal gives speckle rather than a diffuse profile.

The streaks a faulted stack makes

Close packing settles two directions and leaves the third to chance. A crystal that chooses wrongly now and then has a lattice in the plane of its layers and none across them — and its diffraction pattern says so, with some rows of spots as sharp as ever and others smeared into streaks, sorted by an integer condition.

diffraction · Diffraction One local configuration, and a number of structures that doubles. How many kinds of adjacent pair a close-packed stack has, how many kinds of triple, and how many stackings of each period there are. The first column never moves: every pair of layers is congruent to every other, at every period, which is what an order-disorder family is. The last two agree with 2ⁿ + 2(−1)ⁿ, which is the chromatic polynomial of a ring of n layers at three colours, because a stacking of period n is exactly a proper three-colouring of that ring. The gap between the first column and the last is the whole subject.

A stack with no space group

Every pair of layers in a close-packed stack is congruent to every other pair, and the number of stacks doubles with every layer added. A family whose local configuration is completely determined and whose global structure is not determined at all has no single symmetry group — what it has is a set of operations that compose only when their ends match, which is a groupoid.

applied · Packing One tile, two lattices. The smallest polyomino on exactly two lattices with no slide between them, of 5 cells. On the left its copies sit on one lattice and on the right on another, and neither tiling can be turned into the other by moving rows.

One tile, two lattices, and a row that slides

A crystal chooses its lattice, and it is natural to think a tile does too. Among all polyominoes to thirteen cells, more than five hundred that are not rectangles lie on more than one lattice, and some on seven. They fall into at most two families, and a family is a row of copies that can slide against the next row, which is exactly what makes polytypes possible in a real crystal. A tile with a prime number of cells has no such row unless it is a bar, and so it lies on at most two lattices.

classification · Isohedral

Named alongside it

The objects these essays reach for when they reach for this one.

Close packingSpace groupStacking faultStacking sequenceDisorderCoordination numberCosetDiffuse scatteringGroupIntegral breadthLattice tilingLayer group

All concepts