Five copies, and the gap they leave

A crystal may not have a five-fold axis. That is the restriction, it is one line of arithmetic, and nothing in this collection contradicts it.

A crystal particle may look very much as though it has one. Gold and silver grown from solution, copper deposited from vapour, silicon nanoparticles, and the fivelings that mineralogists have described in native gold for two centuries: five wedges of perfectly ordinary metal arranged about a common edge, with a five-fold axis running down the middle of the assembly.

Nothing is violated, because no lattice is being asked for a five-fold rotation. What is being asked for is that five pieces meet, and the arithmetic of whether they do is the subject here.

The angle, from the lattice rather than from a table #

The twin operation here is a mirror in a {111} plane, which is the commonest twin law in cubic metals and the one a stacking fault produces. Two such planes sharing a ⟨110⟩ direction meet at an angle that is a fact about the cubic lattice.

In a cubic lattice the plane (hkl) has normal (h, k, l) in Cartesian coordinates, so the angle between (111) and (11−1) is the arccosine of

$$\frac{1 + 1 - 1}{3} = \frac{1}{3},$$

which is 70.5288°. It is the tetrahedral angle — the angle at the centre of a regular tetrahedron subtended by two vertices is its supplement — and its irrationality in degrees is the whole problem.

Five of them come to 352.644°. The circle needs 360. The gap is 7.356°, or 2.04 per cent of a full turn.

A symmetry that does not close #

The pretty way to say it is that a five-fold twin is a symmetry operation that fails to close, and that statement is checkable rather than rhetorical.

Two mirrors half a dihedral angle apart compose to a rotation through the dihedral angle, which is the operation carrying one unit onto the next. Compose five of those and the result is a rotation of 352.644° about the axis — not the identity. A group would have closed. This does not, and the residual is the same seven and a third degrees.

The parity is worth keeping too. Five mirrors is an odd number, so the fifth unit is the mirror image of the first rather than a copy of it. That is consistent — a mirror image of a face-centred cubic lattice is a face-centred cubic lattice — and it is the reason the arrangement is a twin rather than a strained single crystal.

What the lattice says about the five-fold axis itself #

It is worth asking the direct question, since the particle looks so much as though it answers it.

If the assembly really had a five-fold axis, the rotation by 72° about it would map the lattice onto itself, and would therefore be an integer matrix in the lattice basis. The trace of an integer matrix is an integer, and the trace of a rotation by 72° in the plane across the axis is 2cos 72° = 0.618 — the reciprocal of the golden ratio, and not an integer.

So the assembly’s five-fold axis is not a symmetry of any of its five pieces, and the pieces are what the crystal is. The axis is a property of the arrangement, in exactly the sense that a virus’s sixty-fold symmetry is a property of the molecule and not of the crystal it sits in.

Why five and not four or six #

The particle grows five units rather than some other number, and the reason is in the same table.

Four units leave a wedge of 78°, which is not a small gap in any sense — it is more than a fifth of the circle, and a particle with it would be a fan rather than a particle. Six units overrun by 63°, which is worse: the material would have to be compressed by that much, and the sixth unit has nowhere to go.

Five is the only count whose deficit is small enough to be absorbed, and the reason is arithmetic rather than anything about metals: 360 divided by 70.5288 is 5.104, so five is the nearest whole number and the fractional part is what is left over. The particle is a rounding error made physical.

That also explains why the same shape appears in every face-centred cubic metal and in silicon, whose structure is cubic with the same {111} twin plane. The angle depends on the lattice type and not on the substance, so the deficit is 7.356° in all of them — a number that belongs to the arithmetic of the cubic lattice rather than to any material.

Where the seven degrees go #

Three things can happen and all three are observed. The arithmetic does not choose between them; it says only that one of them must.

A gap. The five units meet along the axis and separate as they go outwards, leaving a wedge of empty space. This is seen in particles grown at low temperature, where atoms cannot move to close it.

Strain. The units deform elastically to fill the wedge, which costs energy and stores it. The strain is of order the deficit divided by a full turn — about two per cent — and it is largest at the axis. This is the usual case, and it is why decahedral particles stop being favourable above a certain size: the stored energy grows with the volume while the surface energy that made the shape favourable grows more slowly.

A defect along the axis. The closure failure is carried by a disclination: a line defect whose Burgers-like content is a rotation rather than a translation. That is the same object a circuit round a defect measures, and a five-fold twin is the standard physical example of one — a positive wedge disclination of 7.36°.

The third is the one that connects this essay to the rest of the collection, because the deficit angle of a disclination in a lattice is normally required to be a rotation the lattice permits. A wedge of 7.36° is not one. So the disclination in a five-fold twin is not a lattice disclination at all — it is one whose closure failure the lattice cannot supply, and the strain field is what makes up the difference.

Twenty of them, and the icosahedral particle #

The same accounting run on the other cluster geometry gives the shape that dominates at small sizes.

Twenty tetrahedral units meeting at a point, each sharing faces with three neighbours, would make an icosahedron. The solid-angle deficit is larger in proportion than the decahedron’s plane-angle deficit, and the strain is correspondingly greater — but for a very small particle it is worth it, because the icosahedron has only {111} faces and those are the lowest-energy surfaces of a cubic metal.

The result is a size sequence observed in gold and silver: icosahedral particles up to a few hundred atoms, decahedral in a middle range, and ordinary single crystals above a few thousand. The crossover sizes depend on the metal and on how the particle was made; what is not in doubt is the direction, and it follows from strain growing faster with size than surface does.

What separates this from a quasicrystal #

The distinction matters and is easy to lose, because both objects show five-fold features in a diffraction experiment.

A five-fold twinned particle is made of ordinary crystalline pieces. Each piece has a lattice, each piece diffracts to sharp spots at that lattice’s reciprocal positions, and the five-fold appearance of the whole comes from five sets of spots superposed at 72° to one another. Rotate the particle and the sets move together; take the particle apart and each piece is a crystal.

A quasicrystal has no lattice anywhere in it. Its diffraction is sharp and its ten-fold symmetry belongs to a single set of spots rather than to five superposed ones, and no piece of it is periodic.

Telling them apart is a real experimental question and was one of the first raised against Shechtman’s measurement: the pattern he reported could in principle have come from a five-fold twin, and ruling that out took dark-field imaging showing that single grains — not assemblies — gave the pattern.

What the picture cannot show #

The strain field is not drawn anywhere here. The figures show wedges and gaps; a real particle has neither, because the material has redistributed itself. The deficit is a statement about undistorted pieces, and the distortion is what makes the particle exist.

Nor is the size dependence. Whether a given metal grows decahedral particles at a given size is a question about energies, and no energy appears in this collection. What is computed is the geometric deficit, which is the same 7.356° for gold, for silver and for silicon.

And the axis is drawn as though exact. In a strained particle the five units are not related by exactly the tetrahedral angle any more; they are related by 72°, with the difference taken up by elastic distortion. The figures draw the unstrained arrangement, which is the one whose angles are computable.

The measurement that separates the cases #

There is a laboratory test distinguishing the three fates of the seven degrees, and it is worth stating because it is the point at which this arithmetic becomes checkable.

Diffraction from a single particle. If the units are strained, their lattice parameters differ slightly from the bulk value and differ across each unit — so the spots are broadened and shifted in a way that maps onto the strain field. If a gap has been left instead, the units are unstrained and the spots are sharp, with five sets at exactly 72°.

Direct imaging. At atomic resolution the boundary regions can be seen, and a gap is a gap. This is how the modified decahedra were identified: the particle is not five perfect wedges but five wedges with re-entrant surfaces, which is a shape that reduces the strain by giving the material somewhere to go.

And the size distribution. Strain energy grows with volume, surface energy with area, so the crossover between shapes is at a size that can be predicted from measured energies and compared with what is observed. That prediction is physics rather than symmetry and is not made here; what the symmetry supplies is the 2.04 per cent that everything else is a response to.

Who found it #

Five-fold twinned crystals of gold were described in the nineteenth century as a mineralogical curiosity. Segall identified the decahedral structure in electrodeposited copper in 1957, and Ino and Ogawa worked out the geometry of both the decahedral and the icosahedral particle in the 1960s, including the strain accounting. Marks developed the modified shapes — the truncated decahedron that minimises the total energy — in the 1980s.

The deficit angle itself has a much older life. It is the same 7.36° that makes five regular tetrahedra fail to fill the space around an edge, a fact known to Aristotle’s commentators and stated wrongly by Aristotle himself, who believed the tetrahedron tiled space. The correction is due to Regiomontanus in the fifteenth century, and it is one of the earliest cases in this collection’s subject of an angle being computed rather than assumed.

The same shortfall in one dimension less #

A cleaner version of the arithmetic is worth having, because the three-dimensional picture hides how ordinary it is.

Take a regular pentagon’s interior angle, 108°, and ask how many meet at a vertex of a tiling. Three come to 324° and leave 36°; four come to 432° and overrun. Neither works, which is why regular pentagons do not tile the plane — and the 36° left over is the plane’s version of the seven degrees here.

The difference between the two cases is what can be done about the shortfall. A tiling has no way to absorb 36°: the tiles are rigid and the gap stays. A particle has three ways, because a metal is not rigid, its pieces are finite, and a defect line can carry a rotation. A shortfall that is fatal in a tiling is merely expensive in a solid, and that is the whole reason five-fold particles exist while five-fold tilings by regular pentagons do not.

The pentagon’s case is worth finishing, because the obvious conclusion from it is wrong. Regular pentagons do not tile the plane; convex pentagons do, in fifteen distinct ways, and every one of them is a pentagon whose angles were chosen to close rather than made equal. So the shortfall is a fact about the regular shape and not about five-sidedness, and the parallel with the particle is exact in that respect too: nothing forbids five pieces from meeting, and what forbids it is five pieces of this shape, whose angle the lattice fixed.

Twenty of them, and the deficit counted the same way #

The icosahedral case is stated above in proportions and it deserves the arithmetic, because the two deficits are computed from the same number and come out an order apart.

A tetrahedron bounded by four {111} planes of a face-centred cubic lattice is regular, and the solid angle it subtends at its own vertex is the spherical excess of a triangle with three angles of arccos(1/3): three times that angle, less π, which is 0.5513 steradians. Twenty of them come to 11.026, against the 12.566 steradians of a whole sphere. The deficit is 1.541 steradians, or 12.26 per cent of the solid angle round the point — against the decahedron’s 7.356° out of 360°, which is 2.04 per cent. The icosahedral shell is six times as frustrated as the decahedral wheel, exactly.

But twenty is not chosen the way five is, and that is the part the proportions hide. Five is the nearest whole number to 360° ÷ 70.5288°, so the decahedron is a rounding of a division. Four π divided by 0.5513 is 22.79, and twenty-two tetrahedra would leave only 3.5 per cent — so if the icosahedral count were a rounding it would be twenty-two. It is twenty because twenty is how many faces an icosahedron has: the units share faces with one another, three at every outer vertex, and that combinatorial requirement fixes the count before any angle is measured. The deficit is then whatever it is, and it is large.

That is why the strain in an icosahedral particle is quoted as a radial compression rather than as a wedge. Closing the shell means pulling every vertex in towards the centre until the circumradius is 0.951 of the edge, which is the icosahedron’s own ratio — a uniform five per cent, distributed through the volume rather than concentrated on five boundaries.

The two shapes therefore fail in different currencies, and the currency decides which one a small particle can afford. The decahedron’s shortfall is one angle on five interfaces, so it can be paid by shearing five thin slabs or by leaving one visible gap. The icosahedron’s is a solid angle spread over twenty units and thirty shared edges, so there is nowhere to put a gap and the only available payment is compression everywhere. A gap is cheap for a large particle and a uniform squeeze is cheap for a small one, which is the mechanism behind the size sequence rather than a restatement of it.

How much strain the gap costs #

The deficit is an angle and the three fates of it are qualitative. Turning the angle into a strain takes one line and it says why the second fate is available at all.

Seven and a third degrees, shared equally over the five boundaries, is a little under one and a half degrees at each. Closing a wedge of that angle by shearing the material on either side of it means a shear whose tangent is the tangent of that angle: about two and a half per cent.

That is a large strain for a metal and not an impossible one. Elastic strains of a per cent are routine in small particles, of a few per cent are reachable before anything yields, and of ten per cent are not. So the closure failure sits in the narrow band where elastic accommodation is expensive and possible, which is exactly what makes the three fates compete rather than one of them winning outright.

It also explains the size dependence without any energies being computed. Strain energy is stored throughout the volume, so the cost of closing the gap grows as the cube of the particle’s size, while the surface energy the shape is buying grows as the square. Small particles can afford the strain and large ones cannot, and somewhere between the two the ordinary single-crystal shape becomes cheaper. Where that crossover sits is a measurement; that there is one, and which way round it runs, is arithmetic.

Which wheels close exactly, and why the answer is familiar #

Five units leave a gap and six would not — for a different twin law — and it is worth asking in general which numbers of units can close exactly. The answer turns out to be a list this collection has met from three other directions.

Closing exactly means the twin operation’s angle is 360°/n for a whole number n. And the twin angle here was not chosen: its cosine came out as 1/3 from the lattice, and the cosine of any angle between lattice planes is rational, because it is a ratio of integer dot products.

So the question is which rational multiples of a full turn have rational cosines. Niven’s theorem answers it: the only ones are the angles whose cosine is 0, ±½ or ±1 — that is, multiples of a quarter turn, a third, or a sixth. So a wheel of crystallographic twins closes exactly only for n equal to 1, 2, 3, 4 or 6.

That is the crystallographic restriction again, and it has arrived here by a completely different route: not from an integer matrix having an integer trace, but from a number-theoretic fact about cosines. The five-fold particle is therefore not an exception squeezing past the restriction; it is the restriction, saying that five cannot close and leaving open how large the failure is. Whether the leftover is seven degrees or seventy is a question about the particular twin law, and that it is non-zero is a theorem.

Where the ladder goes next #

Sideways, to the general form of the defect. The circuit that does not close takes the closure failure as the object of study rather than as an obstruction, and the disclination in a five-fold twin becomes one entry in a classification of things a lattice permits to go wrong.

Upwards, to what a crystal keeps of a forbidden symmetry. The most of an icosahedron a crystal can keep asks the same question at the level of groups rather than of particles: given that the sixty rotations are unavailable, which subgroup survives, and the answer is order twelve at index five — the five being the five cubes inscribed in the icosahedron, which is the same five as the five units here.