Crystallography & Symmetry
patterns derived from their groups
Concept

Lattice type — where it appears

Which of the five plane or fourteen space lattices a given lattice is, decided by which integer matrices preserve its metric. It is a question about equalities among the entries of a reduced form, and an equality is a boundary of the space of all lattices.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

Centring, and why cm is not pm

A centred cell has a lattice point in the middle and twice the area it needs, and crystallography prefers it anyway. The preference has a price, and the price is paid in reflections that vanish for reasons that have nothing to do with the crystal.

lattices · Centring Centring the five lattices. Each of the five plane lattices with the midpoint of every cell added, and the type of lattice that results — read off the reduced basis of the new point set rather than looked up. Every centring halves the cell area, so the original lattice is a sublattice of index two in the centred one, and every centred lattice is again one of the five. Two of the five come back as themselves and are therefore no richer for being centred. The rectangular and rhombic lattices exchange, which is what makes them one family under two descriptions. And the hexagonal lattice centred is rectangular — its holohedry falls from 12 to 4, so centring destroys the symmetry it was meant to display.

Centring, counted as a sublattice

Adding the centre of every cell to a lattice produces another lattice, containing the first with index two. Doing it to each of the five in turn shows why the list is five rather than ten, and why only one of the five has a centred description worth keeping.

lattices · Centring The region every plane lattice lands in. The shape of a plane lattice is one complex number, τ, and every lattice can be brought by a change of basis into the region shaded here: the strip between 0 and a half, outside the unit circle. Its interior is the oblique lattices. Its left edge is the rectangular ones, its arc and its right edge the centred rectangular ones, and its two corners are the square lattice at i and the hexagonal lattice at ρ. Five kinds, and they are a region, three arcs and two points rather than five things of one sort. The region is unbounded upwards, where the cell gets longer and thinner without limit.

The space every lattice lives in

Five lattices in the plane is the number of *kinds*. The number of lattices is a continuum — and it has a shape: one two-dimensional region with two corners, three edges and an interior, where the five kinds turn out to be a region, three arcs and two points rather than five things of one sort.

lattices · Moduli Four groups whose description count a metric can raise. Every plane group, the number of ways of writing one arrangement down on the lattice the group requires, and the number on the most symmetric lattice it may sit on. Eight groups already occupy the most symmetric lattice available to them and have nowhere to go. Four have a metric that raises the count, by two and in one case by six. The starred rows belong to groups whose normaliser has a free direction, where the quantity is a count of grid points rather than an index and cannot be compared.

The normaliser is not a function of the group

How many ways there are of writing one structure down is computed from the group and printed in a table beside its name. It is not a property of the group. Draw a p2 pattern on a hexagonal cell rather than an oblique one and the number goes from four to twenty-four, with nothing done to the group at all.

operations · Normalisers Three rules for 'square' are three shapes, and the cell falls between them. A measured cell placed by how far it is from each of the two conditions a square cell needs — equal edges across, a right angle up — each in its own standard uncertainties, so the square cells are the origin. Three rules for accepting the cell as square at the five per cent level are three regions. The two-condition χ² accepts inside the larger circle, of radius 2.45; testing each condition separately accepts inside the square of half-side 1.96; reading the distance as if it were one condition accepts inside the smaller circle, of radius 1.96. The measured cell, 2.22 and 0.75 from the two conditions, is inside the first region and outside the other two.

Two conditions are not one condition twice

Whether a plane structure has a higher-symmetry supergroup is decided by its measured cell, and the obvious measure — how many standard uncertainties the cell sits from a right angle, or from equal edges — cannot rank the candidates on its own. A square needs two conditions and a rectangle one, so a single distance means different things on each; counted properly, a square can score above the equal-edged lattices that contain it, and walking up the diagram one condition at a time is the reading that keeps the ranking coherent, at a price that has to be paid in its error rate.

operations · Subgroups The centrings each holohedry allows, prime by prime. For each of the seven holohedries on its conventional lattice and each prime from 2 to 7, how many of the lines and planes of the lattice reduced modulo that prime the holohedry leaves invariant, and which of them give a lattice other than the one being centred. At 2 they give C for the monoclinic system, C, I and F for the orthorhombic, I for the tetragonal and I and F for the cubic; at 3 they give the rhombohedral lattice from the hexagonal one; at 5 and 7 nothing new.

Only two and three can centre a lattice

The centring letters P, A, B, C, I, F and R are usually a list brought to the Bravais lattices. They can be derived instead: a centring is a subspace of the lattice reduced modulo a prime, and a centring that keeps a crystal system's symmetry is a subspace its holohedry leaves invariant. Enumerating every one for the seven holohedries and the primes up to seven gives exactly the fourteen, with new lattices only at two and three. The reason is the crystallographic restriction, one step removed: every holohedry has an order made of twos and threes.

lattices · Centring The centrings six four-dimensional lattice groups allow, prime by prime. For each of the six four-dimensional lattices whose symmetry groups are maximal, with the group's order, and each prime from 2 to 7: the subspaces of the lattice modulo that prime the group keeps, sorted by what centring by each one makes. New Bravais types appear only at 2, three of them, all over the cube-and-line lattice, and at 5, where A₄ centred by fifths is A₄*. Both centrings of the hypercubic lattice by halves make D₄, which has three times the symmetry.

The body-centred hypercube is the checkerboard lattice

In three dimensions the centring letters come out as the subspaces a holohedry keeps modulo a prime, and only two and three make new lattices. In four dimensions the same census, run on the six lattices whose symmetry groups are maximal, finds a prime that the three-dimensional argument never could: five centres A₄ into its dual. It also finds that the most familiar centring does not survive the extra dimension. The body-centred hypercubic lattice is not a centred hypercube at all: its body centre is exactly as far away as an edge, and it is D₄, with three times the symmetry.

lattices · Centring Every stratum a primitive cubic cell lies on, and the steps between them. The ten lattice strata a primitive cubic cell lies on, from triclinic, with no condition, to cubic, with five, each with the number of linear conditions it imposes on the cell's six parameters, joined wherever one adds conditions to the other with nothing between. There are 14 steps and 10 paths from the bottom to the top: nine of four steps, through a monoclinic, the orthorhombic and a tetragonal stratum, and one of two, through the rhombohedral.

A cubic cell has fourteen chances to fail

In the plane, deciding whether a measured cell is square meant two steps up a short diagram, and dividing the error rate between them fixed the test. In space, a primitive cubic cell sits on ten strata joined by fourteen steps on ten different paths, and a walk that tests every step at five per cent calls more than a quarter of true cubic cells something less. Dividing the level by the steps of one path is not enough and dividing it by all fourteen is too much. Measured directly, the right level is two-thirds of a per cent a step, and calibrated that way the walk notices a tetragonal distortion more often than the single test it replaces.

operations · Subgroups

Named alongside it

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

HolohedryCentringBravais latticeMetric tensorBasis reductionCentred latticeCrystallographic restrictionPrimitive cellSubgroupSublatticeToleranceChange of basis

All concepts