Crystallography & Symmetry
patterns derived from their groups
Symmetry at work

A net is a choice of what counts as a bond

A list of atomic positions does not contain a net. It contains distances, and somebody has to decide which of them are bonds — so the net is a fact about the cutoff as much as about the crystal, and moving the cutoff past a shell of neighbours changes the answer.

Assumes A structure with the distances thrown away and The placement nobody chose.

16 min read 9 figures The graph remembersSymmetry is decidableNo origin removes it

The three essays before this one have taken a net as given and asked what can be got out of it. This one asks where the net came from, and the answer is less comfortable than the rest of the ladder.

A crystal structure, as an experiment delivers it, is a list of atomic positions in a cell. There is no net in that list. There are distances — every pair of atoms has one — and a net appears only when somebody says which of those distances count as bonds. That is a decision, it is made with a number, and the number is not in the data.

two sites in a hexagonal cell: 3 cutoffs, degree 3 to 12. Two atoms per hexagonal cell, at the positions graphite's carbons occupy, read as a net at a ladder of bonding cutoffs. Each row takes the cutoff just past a shell of neighbours and reports the net that results: how many edges it has, the degree of its vertices, whether its cycles generate the whole translation lattice, and the group of its own barycentric placement. The net is not in the coordinates. There is no bond in a list of positions; there is a cutoff, and moving it past a shell gives a different net from the same atoms. A row marked as a supercell is a net whose own translations turn out finer than the cell it was described in — the description was on too large a cell and the machinery says so.
Fig. 1 Two atoms per hexagonal cell, at the positions graphite’s carbons occupy, read as a net at a ladder of cutoffs. The first row gives the honeycomb. The second gives a net of degree nine. Nothing about the atoms changed between the rows.

The shells

Start by listing what there is to decide about. For a periodic set of points, the distances between pairs are not spread evenly: they come in shells, sharply separated, because the points sit on a lattice and a lattice has a discrete set of vector lengths.

So the distance list is a staircase. Between one shell and the next there is a gap, and any cutoff placed in that gap gives the same net. The net is therefore not a continuous function of the cutoff — it is a step function, and the steps are where the decision actually lies.

That is a piece of good news and a piece of bad. The good news: a cutoff does not have to be chosen precisely, only placed in the right gap. The bad news: choosing the gap is choosing the net, and different gaps give genuinely different objects.

three sites at the half-lattice points: 3 cutoffs, degree 4 to 14. Three atoms per hexagonal cell, at the midpoints of its edges, read as a net at a ladder of bonding cutoffs. Each row takes the cutoff just past a shell of neighbours and reports the net that results: how many edges it has, the degree of its vertices, whether its cycles generate the whole translation lattice, and the group of its own barycentric placement. The net is not in the coordinates. There is no bond in a list of positions; there is a cutoff, and moving it past a shell gives a different net from the same atoms. A row marked as a supercell is a net whose own translations turn out finer than the cell it was described in — the description was on too large a cell and the machinery says so.
Fig. 2 Three atoms per hexagonal cell, at the midpoints of its edges. Past the first shell the net is the kagome net, of degree four. Past the second it is of degree eight and is a different net, with the same symmetry and nothing else in common.

What moves as the cutoff crosses a shell

Four things change when the cutoff steps past a shell, and it is worth separating them because they do not change together.

The degree changes, obviously, and by however many neighbours the shell contained.

The connectivity can change. Below the first shell the net has no edges at all; on some structures the first shell joins atoms into isolated clusters and the graph is disconnected. A disconnected quotient graph is not a net — it fails the voltage-lattice test at once, because a graph in pieces has no cycles reaching the whole of ℤ² — and the machinery reports it rather than proceeding.

The symmetry can change, and it can go up. This is the one that surprises. Adding a shell of longer bonds cannot remove a symmetry, because every symmetry of the point set permutes every shell; but it can leave the net’s symmetry unchanged while the degree doubles, and it can turn a disconnected graph into a connected one whose group is larger than any piece’s.

The cell can turn out to be too big. This is the interesting failure and it gets its own section.

three sites in a square cell: 4 cutoffs, degree 2.67 to 14.67. Three atoms per square cell, at the corner and two edge midpoints, read as a net at a ladder of bonding cutoffs. Each row takes the cutoff just past a shell of neighbours and reports the net that results: how many edges it has, the degree of its vertices, whether its cycles generate the whole translation lattice, and the group of its own barycentric placement. The net is not in the coordinates. There is no bond in a list of positions; there is a cutoff, and moving it past a shell gives a different net from the same atoms. A row marked as a supercell is a net whose own translations turn out finer than the cell it was described in — the description was on too large a cell and the machinery says so.
Fig. 3 Three atoms per square cell, at a corner and two edge midpoints. Four cutoffs, four nets, and a degree that runs from under three to nearly fifteen. Each row is a perfectly good net; only one of them is likely to be what a chemist means.

The description that was written on too large a cell

Take two atoms in a square cell, one at the corner and one at the centre. Join each to its nearest neighbours and the result is a net of degree four — and the machinery reports that its symmetry group contains a translation of half a cell along each diagonal, which the description does not have.

That is not an error. It is the arithmetic saying that the net’s own translation group is finer than the ℤ² the description was written against: the two atoms are related by a translation of the net, so one of them would have done, and the cell is a supercell.

two sites in a square cell: 2 cutoffs, degree 4 to 8. Two atoms per square cell, one at the corner and one at the centre, read as a net at a ladder of bonding cutoffs. Each row takes the cutoff just past a shell of neighbours and reports the net that results: how many edges it has, the degree of its vertices, whether its cycles generate the whole translation lattice, and the group of its own barycentric placement. The net is not in the coordinates. There is no bond in a list of positions; there is a cutoff, and moving it past a shell gives a different net from the same atoms. A row marked as a supercell is a net whose own translations turn out finer than the cell it was described in — the description was on too large a cell and the machinery says so.
Fig. 4 Two atoms per square cell, one at the corner and one at the centre. The net that results is the square net drawn on a doubled cell, and the extra translation shows up as a symmetry group twice the size it should be. The machinery reports the factor rather than quietly halving it.

This matters more than it looks. A structure deposited with a doubled cell — which happens, for perfectly good experimental reasons, when a weak superstructure reflection is missed — will produce a net with twice as many vertices as it needs, whose coordination sequence is right, whose group order is wrong by a factor of two, and which will match nothing in a database. Reporting the factor turns a mysterious non-match into a diagnosable one.

The general principle is the one the cell is a choice states for lattices, arriving here by a different route: the description carries conventions the object does not, and the only defence is to compute the object’s own and compare.

Why the cutoff cannot be got rid of

It would be pleasant to say that the right cutoff is the first gap and be done. It is not, and the reasons are chemical rather than mathematical.

A framework silicate has silicon-oxygen bonds at about 1.6 Å and silicon-silicon distances at about 3.1 Å, and the net a chemist wants is usually the one on the silicon atoms with the oxygens suppressed — which is not the first shell of anything, it is a shell of a derived point set. A layered material has strong bonds in the plane and weak ones across it, at distances that overlap; a cutoff that catches the interlayer contacts gives a three-dimensional net, and one that misses them gives a stack of two-dimensional ones, and both are true descriptions of the same crystal at different questions.

So the cutoff encodes a question, not a fact. What this collection can do is make the encoding visible: run the ladder, print what each rung gives, and let the choice be a choice rather than a default.

the honeycomb net at p6m, order 12. the honeycomb net drawn at the placement in which every vertex sits at the average of its neighbours in the cells its voltages name. The placement is the solution of one linear system per coordinate and is exact in the lattice basis, so its symmetry is detected by the same round trip every pattern here goes through: 12 operations, group p6m. Each detected operation is then required to carry every edge of the quotient graph to an edge, which is what makes it a symmetry of the net rather than of the point set.
Fig. 5 The net the first row of the first table gives, drawn at its own equilibrium with the group that placement turns out to have — p6m, of order twelve. It is the honeycomb, and it is the net anybody would have said graphite has, which is the case where the default is right and which is why the default survives. Nothing about the cutoff reaches this drawing: the ladder handed over a quotient graph, and the coordinates here were solved for from that graph alone.

Set the centred square arrangement’s ladder beside the honeycomb’s and the contrast is in the number of rows as much as in what they say. The honeycomb’s ladder has three rungs and the first of them is the answer everybody wants; the centred one has two, and both of them report a supercell, because the two atoms in that cell are related by a translation of every net the ladder produces. So a reader who took the first row as the answer in both cases would be right once and would have a doubled description the other time, with nothing in the row itself to say which had happened. That is the argument for printing the factor rather than the net.

The measurement that decides, when one exists

There is a case where the choice can be made by the structure rather than by the person, and it is worth stating because it is the honest version of “use the first gap”.

If the shells are widely separated, the choice is robust: any cutoff in a wide gap gives the same net, and the width of the gap is a measure of how little the answer depends on the number. If two shells nearly touch, the choice is fragile, and a structure refined to slightly different coordinates will give a different net. The gap width is computable and it is the thing to report — not the cutoff, which is arbitrary, but how far from the decision boundary the cutoff sits.

This collection’s tables print the squared distance each row is drawn past, so the gaps are visible. A row whose distance is barely larger than the one above it is a row nobody should lean on.

It is worth being blunt about how different the two answers are, because cutoff sounds like a tuning parameter and is not one. The honeycomb that the first row of the first table gives has a coordination sequence 3, 6, 9, 12; the degree-six net that a later row gives has 6, 12, 18, 24. They part company at the first term and the gap widens for ever — one has three neighbours per vertex and the other six, one has hexagonal faces and the other triangular, and the only thing they share is the set of atoms they were both read off. A change of cutoff is not a perturbation of a net. It is a different net, and any language that treats the cutoff as a knob to be tuned for agreement with something is language that has lost that.

The same decision, three fields over

The shape of this difficulty is not peculiar to nets, and recognising it elsewhere makes it easier to hold at the right level of alarm.

A twin law is decided by asking which lattice rows and planes are nearly perpendicular, and nearly is a threshold. The obliquity a twin misses by is a measured angle compared against a number somebody chose, and the choice decides which orientations are reported as twin laws and which are not. A coincidence site lattice is the same: almost nothing lines up exactly, so a grain boundary is described by the nearest exact coincidence and a measure of how far the real one sits from it.

In each case the pattern is identical. There is a decidable question — is this rotation an exact symmetry of the lattice, is this distance shorter than that one — and a real material answers it “not quite”, so an arbitrary number is introduced to turn the not-quite into a yes or a no. What separates a good treatment from a bad one is not avoiding the number, which is impossible, but stating it, varying it, and reporting how much the answer depends on it.

That is why this essay’s figures are ladders rather than single rows. A single row is an answer; a ladder is an answer with its sensitivity attached.

What Systre does with this

The identification of nets from structures is done in practice by a program called Systre, from Delgado-Friedrichs and O’Keeffe, and it is worth saying what it does and does not take responsibility for.

It takes a net as input — the bonds already decided — and computes a canonical form, so that two descriptions of the same net produce the same key however they were written down. That is the gauge and basis problem solved properly rather than by the bounded search this collection uses, and it is why the Reticular Chemistry Structure Resource can hand out stable three-letter names.

What it does not do is choose the bonds. That step is upstream and belongs to whoever prepared the input, and the fact that it is upstream is why two groups can publish different nets for the same compound without either of them being wrong about anything computable.

the kagome net: best at hexagonal, p6m. A net has no metric — it has no lengths and no angles — so asking for its symmetry means asking which metric it is allowed to be drawn at, and taking the best. The same barycentric placement is handed to the detector five times, once for each plane lattice type, and the number of operations that survive the edge check is recorded. the kagome net does best at the hexagonal metric, where its group is p6m of order 12. The smaller numbers are not wrong: they are the group of the same net drawn on a cell with less symmetry, which is a drawing anybody is free to make.
Fig. 6 And a reminder of a second place where a question has to be asked properly. Once the bonds are chosen the net is fixed, and its symmetry is still the maximum over the metrics its combinatorics permits — five detections rather than one, because the graph does not fix the shape of the cell.

The exactness that survives

For all the arbitrariness in the cutoff, everything downstream of it is exact, and the boundary between the two is worth drawing sharply.

The distances are compared as rationals. A squared distance in a lattice basis is a rational number — a quadratic form with rational coefficients evaluated at rational coordinates — so “closer than the cutoff” is a comparison of fractions and never of floating-point numbers. A distance that is exactly the cutoff is decided rather than rounded, and it is excluded, because a shell sitting exactly on the boundary is the case where the choice is most visible and a rule that silently swallows it hides that.

So the picture is: one arbitrary number at the front, and after it a chain of integer arithmetic with no tolerance anywhere. That is a better position than it sounds. An arbitrary number that is stated and varied is a parameter; an arbitrary number buried in a subroutine is a bug waiting for somebody to disagree with it.

the kagome net: 3 vertices, 6 edges, quotient graph. The quotient graph of the kagome net: 3 vertices, 6 edges, and on each edge the pair of integers saying which cell its far end sits in. That is the whole net — an infinite graph written as a finite one. three vertices, six edges, degree four — the midpoints of the triangular net's edges. Its cycles generate the translations at index 1, which is what makes this description honest rather than one written on too large a cell.
Fig. 7 The kagome net’s quotient graph, which is what the first row of the second table produces. Three vertices, six edges, twelve integers — and from here on nothing in the ladder depends on a distance again.

The count that does not depend on the cutoff

One quantity is worth singling out because it survives the arbitrariness, and it is a quantity the second rung already introduced.

Whatever cutoff is chosen, the resulting net has a coordination sequence, and the first term of it is the degree — which is exactly the number of neighbours inside the cutoff, so it moves with the cutoff and settles nothing. But the sequence’s later behaviour, its period and its slope, are properties of the net that a small change of cutoff does not touch, because a small change of cutoff inside a gap changes nothing at all.

So the reporting discipline is: state the cutoff, then report invariants that do not depend on where in the gap it was put. A figure that quotes a degree has quoted the cutoff; a figure that quotes a period has quoted the net.

the kagome net at p6m, order 12. the kagome net drawn at the placement in which every vertex sits at the average of its neighbours in the cells its voltages name. The placement is the solution of one linear system per coordinate and is exact in the lattice basis, so its symmetry is detected by the same round trip every pattern here goes through: 12 operations, group p6m. Each detected operation is then required to carry every edge of the quotient graph to an edge, which is what makes it a symmetry of the net rather than of the point set.
Fig. 8 And the net the second table’s first row settles on, at its own equilibrium: the kagome net, whose placement also comes out at p6m and whose three vertices land on the half-lattice points they were read off. Every number in the essays either side of this one — the shell counts, the group, the mechanism count — is downstream of the decision that table’s first row makes, which is why the decision is worth a ladder of its own rather than a default.

Two things this does not license

It does not license reading a net off a picture. A drawing of a structure has lines in it that somebody drew, and those lines are a cutoff already applied, usually silently. The lines in a published figure are evidence about the author’s intentions and not about the compound.

It does not license calling the result “the” net of a substance. A substance has as many nets as it has sensible questions, and the definite article belongs to the pair (structure, cutoff) rather than to the structure. Where these essays name a net — the honeycomb of graphite, the kagome net of the mineral it is named after — the cutoff is the first shell and the gap is wide, and both of those are stated rather than assumed.

12 nets, 10 with a group. Every net this collection draws, with the size of its quotient graph, the degrees of its vertices, the index at which its cycles generate the translations — one for every honest description — and the plane group of its own barycentric placement, detected rather than declared. The last row is a net whose placement puts two vertices at one point, so it has no drawing and is refused a group.
Fig. 9 The nets this collection works with, all of which arrived as quotient graphs rather than as structures. That is the honest position for a collection that cannot run an experiment: the nets here are the objects, and the structures are what they would be for.

A worked case, in four rows

The table that opens this essay is worth reading a row at a time, because every step in it is a decision somebody makes without noticing.

Two carbon atoms sit in a hexagonal cell at the positions graphite’s layers put them. The nearest-neighbour distance is the carbon–carbon bond; the next shell is the distance across a hexagon; the shell after that is the second ring of neighbours. Placing the cutoff past the first shell gives a net of degree three, which is the honeycomb, which is what everybody means by the structure of a graphite layer.

Placing it past the second gives a net of degree nine. That net is not wrong — those atoms really are that far apart, and there really are that many of them — and it is not the structure of anything. Its coordination sequence is different, its quotient graph is larger, and it would match nothing in any database. It is what happens when a program is handed a cutoff a little too large and reports the result without comment.

The third row is worse in a more interesting way, because the degree keeps climbing and the symmetry does not change at all. A net can grow denser without becoming more symmetric, and a reader who was using the group as a check on the bonding would notice nothing. The group is a property of the net, and every net in the column is a perfectly good net.

The criterion with no length in it

A cutoff is a length, and a length is exactly the thing a net was supposed to have thrown away. There is a family of criteria that avoid choosing one, and they are worth knowing because they replace the arbitrariness rather than hiding it.

Build the Voronoi cell of every atom — the region nearer to it than to any other — as this collection builds the cell nobody chose for a lattice. Two atoms whose cells share a face are neighbours in a sense that mentions no distance at all: the face exists or it does not, and whether it does is decided by the whole arrangement rather than by a number somebody picked.

That is the contact criterion, and it has a real advantage. It adapts automatically to a structure with atoms of different sizes and to one with a distorted environment, where a single cutoff either misses a long bond or admits a short contact. It also gives a natural strength for each neighbour — the solid angle the shared face subtends — so neighbours can be ranked rather than merely listed.

The arbitrariness does not disappear; it moves. A Voronoi cell in a real structure has many faces, most of them tiny slivers between atoms that nobody would call bonded, so a threshold on the face’s area or solid angle is still needed. What changes is what the threshold is a threshold on: a dimensionless fraction of the total solid angle rather than a length in ångström, which is at least a quantity that means the same thing in every structure.

The criterion chemistry supplies

There is a third route, it comes from outside the geometry entirely, and it is the one a structural chemist would reach for first.

Bond valence assigns each interatomic distance a number — a bond order — falling exponentially with the distance, with two parameters fitted for each pair of elements from many previous structures. The rule is then that the orders around an atom must sum to its oxidation state.

That turns the cutoff into a constraint rather than a choice. Every neighbour contributes something, distant ones contribute almost nothing, and the sum is checked against a number chemistry supplies independently. A structure whose valences do not sum correctly is a structure with a misassigned atom, a wrong occupancy or a genuine error, and the check is one of the standard tests applied to a new determination.

What it buys here is a bonded set with no threshold in it, since every neighbour is included with a weight. What it costs is that the answer is no longer a graph: a net needs bonds to be present or absent, and a valence calculation gives a continuum. Turning it back into a net means thresholding again — which is the essay’s point, arriving after two attempts to avoid it.

That is worth stating as the general conclusion rather than as a disappointment. A net is a discrete object and a structure is a set of positions; going from one to the other discards information, and every criterion is a different way of deciding what to discard. The honest response is the one this collection takes elsewhere with tolerances: state the criterion, print the ladder, and let the reader see what moved.

Where this ladder goes next

That closes the four rungs on nets as objects. The next two read the same nets as frameworks: replace each edge by a rigid bar and each vertex by a free joint, and ask whether the thing can move. The answer is decided by the rank of a matrix built out of the equilibrium placement, the count that everybody uses is wrong on some of them, and one of the motions goes all the way rather than stopping at first order.

The cutoff also has a cousin two fields away. Deciding which reflections are systematically absent rather than merely weak is the same shape of decision — a threshold applied to measured numbers, with everything downstream exact — and it is worth noticing that the crystallographer’s version has a hundred years of convention behind it and the net-builder’s has thirty.

Where this goes next

the honeycomb net: cmm against p6m. the honeycomb net drawn twice. On the left a placement chosen by hand, whose symmetry group is cmm of order 4; on the right the placement in which every vertex sits at the average of its neighbours, whose group is p6m of order 12. The graph is identical in the two — the same vertices joined the same way — so every symmetry of the left-hand drawing is a symmetry of the net and the right-hand drawing has them all. Each detected operation is then required to carry every edge of the quotient graph to an edge, which is what makes it a symmetry of the net rather than of the point set. The placement nobody chose A net has no coordinates, so drawing one means inventing them. There is exactly one way to invent them that involves no choice: put every vertex at the average of its neighbours. The drawing that results has the largest symmetry group the net admits, and this site's own detector finds it. The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry. How many vectors of each length Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven. Two cells of equal area on one rhombic lattice. One lattice — the rhombic lattice that cm sits on — with two parallelograms drawn on it: the conventional cell, and a sheared cell whose edges are the integer combinations (1, 0) and (1, 1) of it. The points are identical in both outlines; only the description changes. Each cell's contents were counted by writing every lattice point in that cell's own coordinates and sharing each one out between the cells that meet at it — a quarter at a corner, a half on an edge, one inside — and the totals come to 1 and 1, which are the determinants of the two matrices. The alternative has determinant one, so its inverse is integral and it generates exactly the same lattice; that is the whole condition, and it is why a lattice has infinitely many bases and no arithmetic can prefer one. The cell is a choice, the lattice is not Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.

What this makes readable

Essays that name this one as a prerequisite.

What links here

The 8 essays that link to this one and share the most of its objects, of 10 that link here.

The objects this essay names

Each one links to every other essay that touches it.

Bonding cutoffCoordination shellCrystal netCrystal net recognitionInteratomic distancePrimitive cellSupercell