The cell is a choice, the lattice is not

A lattice is a set of points, and it is completely determined by the pattern it came from. A unit cell is a parallelogram drawn on that set, and it is not determined by anything at all. There are infinitely many, they are all equally valid, and crystallography picks one by convention.

That distinction sounds pedantic until a symbol is misread because of it, which happens constantly. Nearly every apparent disagreement between two accounts of the same structure turns out to be a disagreement about conventions that neither account stated.

Two bases, one lattice #

Take a lattice with basis vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. Now use $\mathbf{a}_1$ and $\mathbf{a}_1 + \mathbf{a}_2$ instead. Every point reachable by whole-number combinations of the first pair is reachable by whole-number combinations of the second, and conversely. The lattice has not changed. The description has, and since the lattice is what constrains every other symmetry, nothing about the pattern’s permitted operations has changed either.

The general rule is exact. Two bases generate the same lattice precisely when one is obtained from the other by a matrix of whole numbers with determinant $\pm 1$ — a unimodular transformation. There are infinitely many such matrices, so every lattice has infinitely many bases, and no arithmetic can pick a preferred one out of the set.

The determinant condition is doing real work. Determinant $\pm 1$ means area-preserving with a whole-number inverse, and the whole-number inverse is what guarantees that the change of basis can be undone without leaving the lattice. A matrix with determinant $2$ generates a sublattice of index two — half the points — which is a different lattice with a legitimate use of its own, but not the same one.

What actually is determined #

Not everything about a cell is arbitrary, and it is useful to know exactly what survives.

The area is determined. Every primitive cell of a given lattice has the same area, because the unimodular transformation that relates any two bases has determinant of absolute value one. Skew the cell as violently as convention allows and its area does not budge.

The number of lattice points is determined, once the counting rule is agreed: a primitive cell contains exactly one, counting corners as a quarter each.

The lattice type is determined, since the holohedry is a property of the point set and not of the description.

The shortest vector is determined, and so is the second-shortest that is not parallel to it. That pair is nearly canonical, and the qualification is what the next few sections are about.

Primitive is not always preferred #

The obvious way to fix the convention is to demand a primitive cell — one lattice point per cell, the smallest possible area. Crystallography frequently declines.

The reason is the centred rectangular lattice. Its primitive cell is a rhombus, and the rhombus’s axes sit at an awkward angle to the lattice’s mirror lines. Describe the lattice in those axes and every formula involving the mirrors acquires the angle as a nuisance parameter, and the symmetry that motivated the classification becomes invisible in the coordinates.

The alternative is a rectangular cell with a lattice point at its centre: twice the area, two lattice points, and axes along the mirrors. Crystallography takes that trade every time, and the letter at the front of a group symbol records it. The c in cm and cmm means the cell is centred; the p in the other fifteen means it is primitive.

So the choice between p and c is not a statement about the lattice. It is a statement about which cell was drawn on it.

Counting points, carefully #

The bookkeeping that decides whether a cell is primitive is worth doing once, slowly, because it is the source of a common miscount.

Draw a parallelogram with its corners on lattice points. Each corner is shared between the four cells that meet there, so it contributes a quarter. Four corners at a quarter each is one point, and a cell with nothing else in it is primitive. Put an extra point at the centre — shared with nothing, so contributing one whole — and the cell contains two lattice points and is centred.

The rule generalises without surprises: a point on an edge is shared between two cells and counts a half. In three dimensions a corner is shared between eight cells and counts an eighth, a point on a face between two, and a point in the body counts one. Face-centred cubic contains four lattice points per conventional cell, body-centred cubic two, and both are routinely miscounted by readers meeting them for the first time.

The count matters because it multiplies through everything downstream. The density of a crystal is the cell contents divided by the cell volume, and a factor of two in the cell contents is a factor of two in the density — an error large enough to be caught, which is fortunate, since it is an error easily made.

The reduced cell, and what it is for #

There is a way to make the choice canonical, and it exists because databases need one.

A reduced cell is defined by a list of conditions — take the shortest vector, then the shortest independent of it, break ties by prescribed rules about angles and signs — and the conditions are arranged so that exactly one basis satisfies them all. Paul Niggli set out the three-dimensional version in the 1920s and it is now the standard way to decide whether two reported structures are the same structure.

That question arises more often than it should. Two laboratories measure the same compound, choose different cells, and report numbers that share not one digit. Reducing both to the Niggli cell settles it in seconds, where comparing the raw parameters settles nothing.

The reduced cell is not, however, the cell anybody publishes. It is frequently ugly — an oblique-looking cell for a structure whose symmetry is high — and the conventional cell is chosen to display symmetry rather than to be canonical. Both are in use, for different purposes, which is exactly the situation in which stating which one is in force matters.

Cell parameters, and what they conceal #

A cell is usually reported not as vectors but as cell parameters: two lengths and an angle in the plane, three lengths and three angles in space. The compression is convenient and it loses something.

What it loses is orientation. Two crystals with identical parameters may sit in quite different orientations relative to a laboratory frame, and for anything involving a direction — a polarisation, an applied field, a cleavage plane — the orientation is the whole question. Parameters describe the shape of the cell and say nothing about where it points.

What it also loses, more subtly, is the distinction between an exact equality and a measured coincidence. A cell reported as $a = 5.431$, $b = 5.431$, angle $90.00°$ might be genuinely tetragonal, with the equality forced by symmetry, or it might be orthorhombic with two axes that happen to agree to the precision measured. The symmetry claim cannot be read off the numbers; it has to come from the diffraction pattern’s own symmetry and from which reflections are absent.

This is a general feature of the subject and worth carrying forward. Symmetry is not a numerical coincidence that becomes true below some tolerance. It is present or absent, and the way to find out is to test the operation rather than to compare lengths. A structure refined in a symmetry it does not have will refine perfectly well and give slightly wrong answers everywhere, which is among the more common failure modes in real structure determination.

What the centred cell costs #

Taking twice the area to get axes along the mirrors is a good trade, and it is not a free one. The cost is that the description becomes redundant, and the redundancy shows up in the experiment.

Index a centred lattice on its conventional rectangular cell and only half the reflections that cell predicts actually occur. Every reflection with $h + k$ odd cancels exactly, because the centring point sits at $(1/2, 1/2)$ and contributes a phase of $\pi$ to precisely those terms. The pattern of missing spots is a centring absence, and it is present for every structure described on that cell, whatever the structure is.

This is worth separating carefully from the absences a glide produces, because the two look identical on a film and mean quite different things. A glide absence says something about the crystal: an operation is present that translates by half a cell as it reflects. A centring absence says something about the paper: a cell was drawn with more points in it than the pattern needs. Describe the same lattice on its primitive rhombic cell and the centring absences do not appear at all — not because the crystal changed, but because the indices did.

So the conventional cell trades one nuisance for another. The primitive cell has awkward axes and no spurious absences; the centred cell has clean axes and a systematically half-empty reciprocal lattice that every reader must know to discount. Crystallography chose the second because clean axes help constantly while the absence rule has to be learnt once.

The general lesson is the one this whole page is about, in its most concrete form: a systematic absence is a fact about a description until it is shown to be a fact about a structure. Distinguishing the two requires knowing which cell was drawn, which is exactly the information a bare list of numbers omits.

The same statement, as a quadratic form #

There is a change of language that makes the whole of this page into one sentence, and it is the language the classification is actually done in.

Collect the dot products of the basis vectors into the metric tensor $G$, whose entries are $\mathbf{a}_i \cdot \mathbf{a}_j$. Every length and angle in the cell is read off it, so $G$ carries the cell parameters and nothing else. Change basis by an integer matrix $P$ of determinant $\pm 1$ and the metric becomes $P^{\mathsf{T}} G P$.

So a lattice is a positive-definite quadratic form, up to that equivalence, and the cell is a choice of representative. What is determined is exactly what the equivalence preserves: $\det G$, which is the squared area; the smallest values the form takes, which are the shortest vectors; and the form’s own reduced representative, which is the reduced cell.

That last identification is worth stating plainly, because it says where the reduction rules came from. Gauss reduced binary quadratic forms, and the conditions a reduced cell must satisfy are his conditions rewritten in the vocabulary of crystallography. The subject did not invent its canonical cell; it inherited one, from a theory that had been asking the same question about $ax^2 + bxy + cy^2$ for a century before anyone indexed a diffraction pattern.

The origin is a choice as well #

Fixing the axes leaves one more decision, and it is the one most often left silent.

Where is the origin? Nothing in the lattice marks a point as special; every lattice point looks like every other. Convention places the origin at a point of high site symmetry — a rotation centre if there is one, a mirror intersection otherwise — and where several candidates exist, the choice changes the numbers in every subsequent formula.

The International Tables list two origin choices for many space groups, and structures published under different choices differ by a fixed shift in every atomic coordinate. Failing to notice is a routine source of confusion in structure comparison, and it is why coordinate lists are conventionally accompanied by the origin choice as well as the group symbol.

Settings, and why a symbol can be ambiguous #

There is a further layer, and it is the one that makes some group symbols non-unique.

The Hermann–Mauguin symbol records generators relative to the axes. Change which axis is called the first and the symbol can change with it, even though the group has not. In the plane this is mild: pmg and pgm describe the same group with the two axes exchanged, and only one of the two is standard. In three dimensions it is not mild at all — the monoclinic system alone has several conventional settings, and the same space group appears in the literature under more than one symbol.

The remedy adopted by the International Tables is to declare a standard setting for each group and to tabulate the alternatives explicitly. It works, provided everybody reports which setting they used, and the failures are all failures to report.

The pair that convention cannot separate #

One case deserves special mention, because it is the sharpest illustration that the convention question has real content.

The groups p3m1 and p31m have the same lattice, the same point group and the same number of operations. They differ in where the mirrors sit relative to the threefold centres: through them in one case, between them in the other. No relabelling of axes converts one into the other, and no change of origin does either — they are genuinely different groups, and the digits in their symbols are the notation’s way of saying which is which.

A reader who treats the symbols as arbitrary names will find the pair inexplicable. A reader who knows the symbols record generator positions relative to axes will find them obvious, which is a good argument for learning to read the notation rather than memorising the list.

What this site fixes, and why #

Every figure here needs a convention, and stating it is cheaper than having a reader guess.

The basis is the conventional one for each lattice type: axes along the mirror directions where there are mirrors, at $120°$ for the hexagonal lattice, and unconstrained for the oblique one. The cell is the conventional cell, centred where the convention centres it. The origin is at a point of highest site symmetry, which is what makes the symmetry-element diagrams comparable with those in the International Tables.

Internally the calculations are done in fractional coordinates along that basis, and that is the choice that makes everything exact: a lattice vector is a pair of whole numbers, a rotation is an integer matrix, and a glide’s translation is a fraction with a small denominator. The detector enumerates candidate operations in those coordinates, which is why it can answer yes or no rather than reporting a residual.

The same problem, three-dimensionally #

In space the conventions multiply, and so do the opportunities for misreading.

There are seven crystal systems and fourteen Bravais lattices, and the centring options include face-centred, body-centred and base-centred varieties. Several apparently distinct choices turn out to describe the same lattice — a face-centred tetragonal lattice is a body-centred tetragonal lattice in different axes — and the standard list is careful to include each type once. Moritz Frankenheim’s 1842 enumeration reported fifteen because it counted one such pair twice; Bravais corrected it in 1848.

That error is the three-dimensional version of counting “rhombic” and “centred rectangular” as two of the plane’s five. The hazard is identical and it is entirely about description.

Who standardised all this #

The conventions in force today come from the International Tables for X-ray Crystallography, first published in 1935 by a committee that had grown tired of the alternative.

Before them, every laboratory used its own axes, its own origin, and often its own notation. Comparing two published structures meant reconstructing both from scratch. The Tables fixed a standard setting for every space group, tabulated the alternatives, listed the general and special positions, and drew the symmetry-element diagrams that this site’s figures imitate.

They are one of the more successful acts of standardisation in the sciences, and their success is measurable: the reason a crystal structure published in 1950 can be read without correspondence today is that both author and reader were working from the same book.

Where the ladder goes next #

The notation the conventions serve is Hermann–Mauguin, and it repays half an hour: the symbols are instructions rather than names, and once that is seen the seventeen stop being a list to memorise.

The constraint that makes any of this finite is the crystallographic restriction, which limits lattice rotations to five orders by an argument one line long.

What the pictures here cannot show. Two of the figures above do make the comparison — one point set, two outlines, the contents of each counted rather than asserted — and that is as far as a drawing gets. What no drawing here shows is the origin: every symmetry-element plate on this page is drawn at one choice of it, and that a shift of a quarter cell would leave the same group with different coordinates is a relation between two coordinate lists rather than a feature of either. Nor can a figure establish that the conventions used here are the ones the International Tables use. That is a claim about how the code was written, and it is checked by comparing element diagrams with the published ones rather than by looking at a picture.