Why the bigger cell wins

Two descriptions of one lattice sit side by side in every crystallography textbook. One uses a rhombus with a point at each corner and nothing inside. The other uses a rectangle of twice the area with a point at each corner and one in the middle. The points are identical; only the accounting differs.

Crystallography chooses the larger one, consistently, and teaches the choice as a convention. It is a convention, and it is not arbitrary: the larger cell buys something specific, pays for it in a specific currency, and both sides of the transaction are visible in a diffraction pattern.

What the bigger cell buys #

The purchase is a set of coordinate axes that agree with the symmetry.

A rhombic lattice’s mirror lines run along the diagonals of its primitive cell. Write the operations in that cell’s basis and the mirror is the matrix that exchanges the two coordinates — perfectly correct, and not the form anybody wants to read. In particular the two axes are not perpendicular, so a distance computed from coordinates needs the angle between them, and a mirror “along the x axis” is a phrase without a referent.

Now take the centred rectangular cell. Its edges are along the mirror lines. The mirror across the first axis is the matrix that negates one coordinate and leaves the other, the axes are perpendicular, and every subsequent statement about direction can be made in the ordinary way.

This is worth stating in its general form because it is the reason for conventions all over the subject. A description is chosen so that the symmetry is visible in it, and “visible” has an exact meaning: the point group acts on the axes by permuting and negating them, so its matrices have one non-zero entry per row. Everything a crystallographer wants to do — index a reflection, name a direction, state a Wyckoff position, look a group up in the tables — is easier in a basis where that is true.

What it costs #

The cost arrives in reciprocal space, and it is not small.

A centred cell has a lattice point at its middle, so every reflection index is computed as though the cell contained two lattice points rather than one. The contribution of the centring point differs in phase from the corner’s by half a turn whenever the index sum is odd, and those reflections cancel — exactly, and for every structure described on that cell.

Half the reflections predicted by the description do not exist. They are not weak, they are absent, and they are absent because the description invented indices for which the lattice has no points. The essay on centring and cm computes this from the structure factor; the sublattice view says the same thing more briefly — the reciprocal of a denser lattice is a sparser one, and the surviving reflections are precisely the reciprocal lattice of the primitive cell that was set aside.

Why the absences are worth paying #

An experimenter looking at a diffraction pattern sees missing reflections and has to work out why they are missing. There are two possible answers and they are not equally interesting.

A glide plane removes reflections because atoms cancel. Two halves of a structure related by a glide scatter out of phase along a particular row, and the cancellation is a fact about where the atoms are. It identifies the glide, which is otherwise invisible.

A centred cell removes reflections because the indices were never real. No cancellation happens; the reflections are absent in the same sense that reflection number 2.5 is absent.

So the price of the convention is a permanent ambiguity that has to be resolved by declaring the cell. Crystallography accepts it because the alternative — primitive cells everywhere — would make the directional information in a symbol unreadable, and directional information is precisely what an experiment measures. A notation organised around directions is what Hermann–Mauguin is for, and it needs axes that mean something.

The rule the convention actually follows #

Stated as an algorithm, the choice of a conventional cell is short:

Take the smallest cell whose edges lie along the symmetry directions. Not the smallest cell — the smallest cell of the right shape. For the rhombic lattice that rule produces the centred rectangle, at twice the minimum area, and for every other plane lattice it produces the primitive cell, because there the smallest cell already has the right shape.

That is why exactly two of the seventeen groups carry a c in their symbol. cm and cmm are the groups living on the rhombic lattice; the other fifteen are on lattices whose primitive cell already displays their symmetry, so no centring is needed and none appears.

The primitive description, written out #

It is worth being concrete about what is actually harder in the smaller cell, because “the operations are awkward” is the kind of claim that is repeated without ever being checked.

In the primitive rhombic basis the lattice’s four point symmetries are the identity, the half turn, and two reflections — and the reflections are the matrices that exchange the coordinates, with and without a sign. They are integer matrices, they are exactly correct, and they have their non-zero entries off the diagonal.

In the centred rectangular basis the same four operations are the identity, the half turn, and the two matrices that negate one coordinate each. Diagonal, obvious, and immediately readable as “a mirror perpendicular to the first axis”.

The difference is not one of difficulty for a computer — both are two-by-two integer matrices and no arithmetic is harder — but of what a symbol can say. Hermann–Mauguin names an operation by the direction it is perpendicular to, and that naming only works when the directions in question are the cell’s own axes. In the primitive basis the mirrors are perpendicular to the diagonals of the cell, and the notation would have to name them by something other than a position in the symbol.

The reduction figure above makes the complementary point. Reduction is the algorithm that finds the smallest cell, it is completely blind to symmetry, and applied to this lattice it returns the rhombus every time. Two different conventions cannot both be produced by one algorithm because they are answers to different questions.

Telling which cell is in use #

Since an absence means nothing until the cell is stated, a reader who has been handed indices and not told the convention has a small forensic problem. Three things settle it.

Count the lattice points per cell. One is primitive, two is centred, and the count is done by giving each corner a quarter, each edge point a half and each interior point a whole. It is elementary, it is the definition of primitive, and it is the check nobody does.

Look at the absence pattern. Centring removes a systematic half of the reflections with a rule that involves only the index sum. A glide removes a systematic half of one row. The two patterns look different at a glance once the difference is known: one is a chequerboard across the whole pattern, the other is a rule along a line.

Check the cell against the point group. If the operations do not lie along the axes, the cell is not conventional, whatever it is. That is the criterion the convention is built to satisfy, so it is also the test for whether the convention has been followed.

Where the same trade appears again #

The pattern — accept a larger description to make the symmetry legible — recurs, and recognising it makes several conventions stop looking arbitrary.

In three dimensions. Seven crystal systems and four centring types would give twenty-eight lattices; the collapses leave fourteen. The surviving centred entries exist for the same reason cm does: a primitive cell of the right shape does not exist for those lattices, so a larger cell of the right shape is used instead, and the systematic absences that come with it are the standard first evidence of the lattice type in an experiment.

In the choice of origin. The tables fix the origin at a point of high site symmetry, even where a different origin would give simpler coordinates, so that the operations have no translation parts where they need not. Two origin choices are listed for many space groups precisely because the rule does not always pick uniquely — and a structure reported on the wrong one is a standing source of irreproducible coordinates.

In the reduced cell. The opposite convention exists as well and is used for a different job. When the question is whether two reported lattices are the same, the answer must not depend on which conventional cell each author picked, so a canonical smallest description is computed instead — which is the reduced basis, and it is deliberately blind to symmetry. Two conventions, two jobs: display the symmetry, or identify the lattice.

What the round trip checked, and how #

The figures here rest on two computations that are checked separately.

The first is the pattern: cmm’s operations are generated from its generators, the orbit of an asymmetric motif is built, and the resulting point set is handed to a detector that finds every symmetry it has and must return exactly the four operations the group was built with. That is the site’s standard round trip and it is what keeps the drawn mirror lines honest.

The second is the absence calculation: the structure factor is computed from the atom positions alone at every reflection in a window, and the claim tested is a pair rather than a fraction. Every reflection with an odd index sum vanishes, and the primitive description of the same lattice has no absences at all. An earlier version of this site’s centring figure asserted that half the reflections vanish, which is false on an odd-sized window — forty absent against forty-one present — and the pair above is both stronger and true.

The two computations share only the atom positions. One is integer arithmetic on matrices; the other is a sum of complex exponentials.

What the halved reflection list costs an experiment #

The absences are usually described as an inconvenience of bookkeeping. They are also a real cost in data collection, and putting a number on it makes the size of the trade clear.

An experiment measures reflections one at a time, or in batches, and its total time is roughly proportional to how many it collects. A centred description names twice as many reflections as the lattice has, so half of any complete sweep is spent measuring reflections that are systematically zero unless the collection strategy is told to skip them.

Modern software is told, and the saving is real. But the same fact has an experimental use that outweighs the cost: the pattern of which reflections are absent is how the centring is identified in the first place. An experimenter who did not measure any of them would not know which lattice type they had.

So the standard practice is to measure a sample of them — enough to establish the absence rule with confidence — and skip the rest. That is a decision about how much evidence a systematic claim needs, made routinely and rarely written down, and it is the same kind of decision as choosing a tolerance: the answer is not a number that can be derived, it is a judgement about how much checking a claim deserves.

The convention is not universal #

One more thing is worth saying, because a reader meeting the conventional cell everywhere could reasonably conclude it is the only description in use.

Several fields work in primitive cells by preference. Electronic structure calculations use the primitive cell because the cost scales with the number of atoms and the conventional cell can carry twice as many for no benefit — the symmetry is applied to the calculation directly rather than being read off the axes. Reciprocal-space work on the Brillouin zone uses the primitive cell for the same reason.

So the choice is not between a right description and a wrong one. It is between a description that makes the symmetry legible to a person reading a symbol, and one that makes a computation cheaper. Crystallography’s tables serve the first, and a structure moving between the two communities gets converted at the boundary — which is a routine operation and a routine source of factor-of-two errors in reported cell contents.

Where the exactness stops #

The convention is a convention. Nothing computed here says the centred cell is correct. It says what each description costs and buys, and the choice between them is a decision about legibility that a person makes.

Absences are exact here and approximate in a laboratory. A reflection that vanishes in this arithmetic is exactly zero. A measured one is small, and deciding whether it is absent or merely weak is a threshold question of the kind this site otherwise avoids — with the added complication that multiple scattering can give a genuinely absent reflection a measurable intensity.

Two dimensions. The three-dimensional statements above are described rather than derived; this site’s machinery is planar and periodic, and the fourteen Bravais lattices are not computed anywhere on it.

Where the ladder goes next #

The construction underneath all of this, and what it does to each of the five lattices, is centring counted as a sublattice.

The essay that separates cm from pm by their operations rather than by their cells is centring, and why cm is not pm, and the general form of the argument is the cell is a choice, the lattice is not.

The experimental consequence, where absences of the two kinds have to be told apart, is systematic absences.

What the pictures here cannot show. The claim that a description is easier — that operations with one non-zero entry per row are better to work with — is not a claim a figure can establish. It is an argument about notation and habit, and the pictures on this page can only show that the mirrors are parallel to the cell edges in one description and not in the other. Whether that is worth doubling the cell is a judgement, and crystallography’s is a century old.

Where the convention is dropped #

The preference is not universal, and the place it is abandoned says most clearly what it was for.

A calculation in reciprocal space uses the primitive cell. The zone belonging to a cell is the reciprocal cell, so a description with four times the volume has a zone with a quarter of the volume — and every band that would have run across the larger zone is folded into the smaller one, four times over.

Folding costs nothing in principle and everything in practice. The states are the same states, relabelled, exactly as a bigger cell folds a zone describes. But a calculation now solves a problem four times larger at each of four times as many wavevectors as it needs, and the band structure it prints has four branches where the material has one — indistinguishable, on the plot, from four genuinely different bands.

So band-structure work uses primitive vectors and crystallography uses conventional ones, and the same crystal is routinely described both ways in the same paper. The translation between them is a matrix with fractional entries, and it has to be applied to the wavevectors as well as to the positions.

Which is the general shape of the trade. The conventional cell buys axes aligned with the symmetry, which is what makes an operation legible and a symbol short. The primitive cell buys the smallest problem, which is what makes a computation cheap. Neither is a property of the crystal, and using each where it pays is the whole of the convention.

Two arithmetic slips the convention invites #

The cost is not only in reciprocal space. Two ordinary calculations go wrong in a way that leaves no trace, and both are consequences of the cell having more lattice points than it needs.

The first is density. A crystal’s density is the mass in a cell divided by the cell’s volume, and both quantities have to belong to the same cell. Take the conventional volume and the primitive count of formula units and the answer is wrong by the centring factor — two, or four — which is a large enough error to be caught and a plausible enough number to be believed.

The second is multiplicity. A Wyckoff position’s multiplicity is quoted for the conventional cell, so a position in a centred group has a multiplicity already multiplied by the centring factor. A structure listing one atom at a general position of a C-centred group has eight atoms in the conventional cell, not four, and a composition computed from the coordinate list without that factor comes out at half the truth.

Both errors have the same cause and the same cure. Every count in a crystallographic description is a count per conventional cell, including the ones that look like properties of the structure rather than of the description. Stating which cell a number belongs to costs a word, and it is the word that makes the whole convention safe to use.