Crystallography & Symmetry
patterns derived from their groups
Concept

Root of unity — where it appears

A complex number some power of which is one, so that the rotation it represents closes up after finitely many turns. Whether a given number is one decides whether a tiling's orientations are finite in number, and it is an exact question in a ring rather than a numerical one.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

The h0l layer of P2₁/c. The h0l reflections of P2₁/c out to 4 in each index, with each spot decided by summing the structure factor over the group's operations: 44 survive and 36 vanish identically, whatever the atoms are. The pattern of holes is the condition h0l: l even, read back off the spots rather than imposed on them.

The reflections that are not there

A screw axis and a glide plane leave no mark on the intensity of any reflection. What they do is delete some, exactly, for every possible arrangement of atoms — and the pattern of deletions is computed here from the sum a crystallographer writes down, rather than read from a table.

diffraction · Diffraction 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

aperiodic · Aperiodic Two turns and their undoing leave a slide. A turn g by 90° about the point c and a turn h by 60° about d. The marked point p is carried back 60° about d, back 90° about c, forward 60° about d and forward 90° about c, and does not return: it arrives displaced by a vector of length 2.371, which is 4·sin 45°·sin 30°·|c − d|. Two other points put through the same four motions move by the same vector, drawn beside them, because the commutator g h g⁻¹ h⁻¹ of two rotations of the plane is a translation — (I − A)(I − B)(c − d) exactly — whatever the angles and the centres.

What forces a lattice

Every enumeration here starts from a lattice of translations, and the lattice is usually taken as given. It need not be. A group of motions that is discrete, and leaves no point far from an orbit, has to contain one — in the plane by an argument four lines long, each line a picture, and in space by an inequality whose threshold turns out to be the six-fold rotation.

restriction · Finiteness When a three-species superstructure's spots can go dark. Every structure of three species on the square lattice with three to eight sites in its primitive cell, using all three species, and every superlattice spot of each, sorted by whether the spot is dark for every choice of scattering contrasts, dark at exactly one ratio of the two contrasts, or never dark; with the number of distinct ratios and their degree over the rationals. At the primes 3, 5 and 7 no spot is dark for all contrasts and no dark ratio is rational: none at three sites, quadratic ones at five and cubic at seven. At four and six sites every dark ratio is rational, at eight some are and some are not, and at six and eight some spots are dark for every contrast.

Three species hide a spot at the golden ratio

With two species and a prime number of sites in the cell, no superlattice spot can ever be missing, because a sum of distinct roots of unity of prime order vanishes only when it is all of them. With three species the structure factor weighs two sums against each other with two scattering contrasts, and the argument changes shape without breaking. At a prime size no spot is dark for every contrast and none at a rational ratio of them, but a spot can be dark at one irrational ratio: at five sites, a power of the golden ratio. It happens only in structures with a centre of symmetry.

operations · Counting Three-species arrangements whose sums are parallel at every spot, with and without a centre. Every arrangement of three species, all present, on a cyclic cell of four to nineteen sites. The third column counts those whose two phase sums are parallel at every superlattice spot, the fourth those among them with a centre of inversion, the fifth those without one, and the last how many kinds the exceptions make up to translation, inversion, relabelling the spots and renaming the species. Primes are shaded. At 5, 7 and 11 sites every parallel arrangement has a centre; at 13 sites 936 do not, and they are all one arrangement; at 17 there are 17 kinds and at 19 there are 48.

Thirteen sites obey Friedel's law without a centre

At five and seven sites the earlier census found that a three-species structure has a contrast that darkens a spot exactly when it has a centre, and asked whether that is a theorem. It is not. The two phase sums are parallel exactly when the A–B pairs are as common at every separation forwards as backwards, and exactly when no anomalous scatterer can produce a Bijvoet difference anywhere in the pattern. At 5, 7 and 11 sites only a centre achieves that. At 13 sites one arrangement does it without one.

operations · Counting

Named alongside it

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

Cyclotomic polynomialsStructure factorEnumerationSuperstructureAnomalous scatteringAperiodicityBijvoet differenceCentringCentrosymmetryChiralityCommutatorContinued fraction

All concepts