Centring, counted as a sublattice

Centring is usually introduced as a way of drawing a cell: take a rectangle with a point in the middle, and notice that the same points can be described by a smaller rhombus. That is true and it makes centring sound like a matter of taste in draughtsmanship.

It is not. Adding the midpoint of every cell to a lattice produces another lattice, one that contains the original as a sublattice of index two, and the operation can be applied to any lattice whatever. Asking what it does to each of the five in turn settles several questions the drawing-a-cell account leaves open — including why the classification stops at five rather than growing to ten.

Index two, and what it means geometrically #

The relation between a lattice and its centred version has a name in group theory and a picture in geometry, and they are the same fact.

Every lattice is a group under addition. The original lattice sits inside the centred one, and the quotient has two elements: a point of the centred lattice is either in the original or is one of the added midpoints, and adding two midpoints gives an original point. That is what index two means.

The geometric version is the one a figure can show: the centred lattice’s cell has half the area of the original’s. Half the cell area means twice the density of points, which means the original is half of what is there. Every row of the figure above reports that ratio, and the ratio being exactly two is asserted rather than assumed — a centring construction that produced a ratio of anything else would be a construction with an arithmetic error in it.

The five are closed under it #

The first thing the computation says is a closure statement: centring one of the five gives one of the five.

That could have failed. Nothing in the definition of the operation says the result has to be a lattice of a type already on the list, and if some centring had produced a sixth type the classification would have been wrong. It does not, and the reason it cannot is the crystallographic restriction — a lattice of any kind has one of five possible holohedries, so a lattice of any kind is one of the five types.

The closure is what makes the table finite and it is why crystallography does not carry entries for “centred oblique” or “centred square”. Those lattices exist; they are simply oblique and square lattices, already described.

Two come back unchanged #

Centring an oblique lattice gives an oblique lattice. Centring a square lattice gives a square lattice — rotated by an eighth of a turn and smaller by a factor of √2, and square.

For the square case the conclusion is immediate. A centred square cell describes a square lattice, so it is a bigger cell for a lattice that already has a smaller one of the same shape, and the description buys nothing. Crystallography’s convention — use the smallest cell that displays the full symmetry — settles it without a second thought.

The oblique case is the same conclusion reached from the other end. An oblique lattice has the minimum symmetry available, its centred version has the same, and there is nothing for a larger cell to display.

The pair that exchanges #

Two of the five swap places, and the swap is the whole content of the word “centred” in this subject.

Centring a rectangular lattice gives a rhombic one. Centring a rhombic lattice gives a rectangular one. So the two are not two lattices with a relation between them; they are one family, and which of the two names is used depends on which cell has been chosen. That is exactly what the essay on cm and pm argues from the operations, arriving here from the geometry instead.

The convention picks one of the two descriptions and it does not pick the smaller. A rhombic lattice’s mirrors run along the diagonals of its primitive cell, so a reader working in that cell has axes that are not aligned with the symmetry. Take the centred rectangular description instead, and the mirrors run along the cell edges at the cost of doubling the area. Crystallography takes the deal every time, which is the trade the whole cell-choice essay is about.

The hexagonal case, which loses #

The most interesting row of the table is the one that goes downwards.

Centring a hexagonal lattice produces a rectangular one. Its point symmetries fall from twelve to four: the sixfold rotation is gone, and so are four of the six mirrors. Adding points to a lattice has destroyed symmetry, which sounds impossible until the mechanism is clear — the added points are not placed symmetrically with respect to the sixfold axis, so no arrangement containing them can have one.

The consequence for the classification is a small piece of practical advice. A crystallographer who describes a structure on a centred hexagonal cell has almost certainly made an error, and the error is not one of taste: the cell describes a lattice with a quarter of the symmetry the hexagonal label claims. In three dimensions the same trap has a well-known form — the rhombohedral and hexagonal settings of the same lattice, which are related by a centring, and which are a standing source of confusion in the literature for exactly this reason.

Why the list is not five doubled #

The natural expectation, before any of this is computed, is that centring should double the classification: five lattices, each available primitive or centred, giving ten. It gives five, and the table says exactly where each of the missing five went.

Two of the candidates are the lattice they started from, so they are not new descriptions but the same description at a different scale. Two more — centred rectangular and centred rhombic — are each other, so they are one new entry between them rather than two. And centred hexagonal is a rectangular lattice, which is already on the list under its own name.

Five candidates, and the five collapse onto zero new types. The classification counts lattices, and the only thing the centred description adds is a cell — a way of writing one of the five so that its symmetry is visible along the axes. That distinction, between a lattice and a description of it, is the single most useful thing to hold onto in this part of the subject, and it is the reason the count of Bravais types in the plane is five and not ten.

The same arithmetic in three dimensions is why there are fourteen Bravais lattices rather than the twenty-eight a naive doubling of the seven crystal systems would give — the centrings collapse onto one another there too, in a more elaborate pattern, and the fourteen are what survives.

The sublattice seen from the other side #

A relation between two lattices in real space is a relation between two lattices in reciprocal space as well, and it runs the other way.

If the centred lattice contains the original with index two, then the reciprocal of the centred lattice is contained in the reciprocal of the original, also with index two. Denser in one space is sparser in the other, exactly.

That is the reciprocal-space statement of the systematic absences a centred description produces. Half the reflections predicted on the larger cell are missing — not weakened, missing exactly — and they are missing because they were never there: the reflections that survive are precisely the reciprocal lattice of the true, primitive lattice. An absence caused by centring is not a cancellation happening in the crystal but a description generating reflection indices that the lattice has no points for.

That is why an absence means nothing until the cell is stated, and why the glide’s absences — which really are cancellations, between atoms that exist — have to be separated from the centring’s before either can be interpreted.

What the computation actually did #

Nothing in the table is a lookup, and the way each row is decided matters because it is the one place on this site where a tolerance appears.

For each type, a Cartesian basis is built, the midpoint of the cell is added, and the resulting pair of vectors is reduced by Gauss’s algorithm to the shortest basis of the new lattice. Then the type is read off the reduced basis by comparing two lengths and an angle — equal lengths and a right angle is square, equal lengths and 120° is hexagonal, a right angle alone is rectangular, equal lengths alone is rhombic, and anything else is oblique.

That comparison is not exact and cannot be. A hexagonal lattice in Cartesian coordinates carries √3, so asking whether an angle is 120° is a floating-point question with a threshold in it. Everywhere else on this site the corresponding question is asked in the lattice basis, where the operations are integer matrices and the answer is decided rather than measured — and the difference between those two situations is itself an essay. Here the threshold is declared, it is a part in a billion, and every lattice in the table is either exactly of a type or a long way from it.

The error this found #

Two of the five lattices on this site were being drawn with the wrong metric, and computing the table above is what exposed it.

The drawing code turns fractional coordinates into Cartesian ones through a basis chosen per lattice type, and its oblique case returned two vectors of the same length at an angle — which is a rhombus. A rhombic lattice has two mirrors that an oblique lattice does not have, so every drawing of the generic lattice, and every p1 and p2 pattern on it, was drawn in the one shape the type is specifically not allowed to assume. The caption under the oblique panel of the five-lattice plate read “no constraint on lengths or angle”, over a picture with both constraints satisfied.

Its rhombic case had the opposite fault: two vectors of different lengths at 60°, which is an oblique lattice with no mirror at all. So cm and cmm — the two groups the rhombic lattice exists for — were drawn on a lattice without the mirrors those groups are about, and their mirror operations came out as shears rather than reflections.

Neither could be caught by any gate this site has, and it is worth being precise about why. Symmetry here is decided in fractional coordinates against a stated holohedry, so both errors were invisible to the round trip: the claims were right the whole time. What was wrong was the picture. The fix is in the basis function, both cases now have the metric their type requires, and the effect is a change to forty-four built pages — a large blast radius for a two-line correction, and the reason it was worth measuring before making it.

The tolerance is avoidable, and it is worth saying how #

The table’s one threshold comes from comparing a reduced Cartesian basis against a lattice type, and Cartesian coordinates of a hexagonal lattice carry a square root. The question itself is decidable, and it is worth recording how, because the distinction between a tolerance in the question and a tolerance in the implementation runs through this whole collection.

Work with the Gram matrix rather than with the basis. A lattice’s Gram matrix is the array of dot products of its basis vectors, and in the lattice’s own basis every entry is rational — the hexagonal lattice’s is a scale times [[1, −½], [−½, 1]], with no square root anywhere.

Centring is then a rational change of basis: the new basis is the old one times a matrix of halves, so the new Gram matrix is a rational matrix conjugated by a rational one, and every quantity stays rational throughout. Reducing it is Gauss’s algorithm on integers after clearing denominators, and identifying the type is asking which equalities hold among the reduced entries — a = c, b = 0, 2b = ±a — each of which is an exact comparison of rationals.

So the whole table can be computed with no threshold, and the one that appears is a consequence of having chosen to draw the lattices before classifying them. That is the ordinary shape of a tolerance in this subject: it enters when a metric quantity is computed in a basis that does not respect it, and it leaves when the computation is moved into the lattice basis — which is the same move the character essay makes for a different quantity.

The same exchange, one dimension up #

The pair that swaps places under centring has a three-dimensional counterpart, and it is the best-known instance of the reciprocal-space reading above.

The reciprocal of a face-centred cubic lattice is body-centred cubic, and the reciprocal of a body-centred cubic lattice is face-centred. The two exchange, exactly as the rectangular and rhombic lattices do in the plane, and for the same reason: adding points in one space removes them in the other, so a centring of index n in real space is a thinning of index n in reciprocal space.

That is why a physicist’s picture of a face-centred cubic metal’s Brillouin zone is the Wigner–Seitz cell of a body-centred cubic lattice — a truncated octahedron — and why the two shapes are so often met together without the relation being stated.

And it is the same statement as the absences. The reflections a face-centred lattice permits are those with indices all even or all odd, which are the points of a body-centred reciprocal lattice; the ones it forbids are the points that thinning removed. A centring in real space and a systematic absence in reciprocal space are one fact, and the index is the same integer in both.

Index three, and why it is a different subject #

Halving is the only case computed here, and it is worth saying what the next case looks like, because it is where the plane’s tidiness ends.

A sublattice of index three is a lattice containing a third of the points, and there are four of them in any lattice rather than one. In a hexagonal lattice one of those four is again hexagonal, rotated by thirty degrees and larger by a factor of √3 — a relation with a name in its own right, because it is what a √3 × √3 superstructure means in surface science and what the three-colouring of a triangular lattice is built on.

Index three therefore does not have the clean structure this essay found for index two. There is no single “centring” operation, the results depend on which sublattice is chosen, and the classification of what happens is a piece of number theory about the ideals of the ring of Eisenstein integers rather than a table of five rows.

Nothing here computes any of that, and the point of mentioning it is to be clear about what “centring is a sublattice” buys. Index two is special: the sublattice is unique given the cell, the arithmetic closes on the five types, and the answer fits in a table. One step further and neither is true.

Where the exactness stops #

These are lattices, not patterns. Everything above is about point sets with no motif on them. A pattern built on a centred lattice has its own group, and that group is not determined by the lattice — five lattices carry seventeen groups between them.

Index two is the only case computed. Sublattices of index three, four and higher exist and are the subject of a much larger theory — sublattice enumeration is how superstructures and commensurate modulations are described. Nothing here bears on them, and the halving-of-the-area assertion would be the first thing to generalise.

The plane only. In space the same operation has three inequivalent forms — base-centred, body-centred and face-centred — and applying them to the seven crystal systems is exactly the computation that produces fourteen Bravais lattices rather than twenty-eight. That is the next field this site opens and none of it is derived here.

Where the ladder goes next #

The group-theoretic version of the same argument, where cm and pm are told apart by their operations rather than by their cells, is centring, and why cm is not pm.

The reason crystallography keeps the larger cell even though a smaller one exists is why the bigger cell wins, which is this essay’s other half.

The classification these five come from is five lattices and no others, and the canonical description that removes the ambiguity between two reported cells is the shortest basis.

What the pictures here cannot show. The table is a table of computations, and the claim in each row — that a centred lattice is a lattice of some type — is a claim about a reduced basis that the figure does not draw. A reader who wanted to check a row would have to add the midpoints, find the two shortest vectors, and measure the angle between them, which is what the code does and what no picture of a lattice displays.