Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Symmetry is decidable — page 5

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.
May be piezoelectric, against may be optically active. Two questions asked of all thirty-two classes, and the classes where the answers part company. 14 classes are in both lists, 6 in only the first, 1 in only the second and 11 in neither. Both lists are computed from the same character sum with a different tensor, so a class appearing in one and not the other is a statement about which representation survives rather than about anything measured. Every entry is a permission: a class in a column is a class whose symmetry fails to forbid the effect, which is a weaker statement than it is usually read as. What symmetry decides

Each permits what the other forbids

432 and 4̅3m are both cubic, both of order twenty-four, both without a centre. One of them can be piezoelectric and the other can be optically active, and it is not the same one — which is as clean a demonstration as the subject offers that "amount of symmetry" is not a quantity.

 The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell. Lattices

The cell nobody chose

Every unit cell on this site is a convention, and one construction escapes the warning entirely: the region of the plane closer to one lattice point than to any other. It needs no basis, no axes and no rule — and its combinatorics are decided in integers, with the square roots confined to drawing it.

 Subgroups of index 3, across the seventeen. Every subgroup of index 3 with cyclic quotient in each of the seventeen plane groups, sorted into the two kinds: 4 keep all the translations and lose operations, 22 keep all the operations and lose translations, and the total is 26. The split is decided by whether the homomorphism onto ℤ3 kills the two lattice translations, which is a property of the kernel and not a judgement. Every one of them is found by enumeration inside the finite quotient by 3Λ, and the count for the whole classification is a measurement. The classification

Three colours, and why most patterns cannot have them

Seventy-four of the seventeen plane groups' subgroups have index two, and every group but one has at least one. At index three there are twenty-six, and ten of the seventeen have none at all — because a symmetry of order two cannot survive being asked to permute three colours.

 13 of one on 12 of the other. Two rows of atoms whose spacings are in the ratio 1.042. Every 12 cells of the substrate come to within 3.97 per cent of 13 cells of the film, so the two are nearly in register at those points and out of register between them. There is no exact coincidence anywhere, and there cannot be: exact coincidence needs the ratio to be rational, and no measured ratio is. Symmetry at work

Two different lattices never coincide, and the question becomes how nearly

Grow one crystal on another and their spacings are in a ratio that no measurement ever makes rational, so exact coincidence is unavailable in principle. What is left is the best rational approximation inside a tolerable repeat — a quantity that jumps rather than drifts as the ratio changes, and whose acceptability is decided by elasticity rather than by arithmetic.

 The seventeen signatures, and the seventeen groups. Every combination of features costing exactly two, beside the plane group each one names. The left column is produced by an accounting identity that has never heard of a lattice; the right by reading seventeen groups' own operations — their rotation centres and orders, which of those lie on mirrors, and how many closed curves the mirror lines make once equivalent lines are identified. The map between the two lists is a bijection, and the figure does not appear unless it is one — in both directions. A signature with no group and a group whose signature is not on the list are both refused, and so is the failure that actually happens: two groups deriving one signature, which costs exactly two and passes every check but injectivity. The classification

Seventeen dollars

Conway's magic theorem prices the features a folded-up pattern can have — a handle costs two, a mirror boundary one, a cone point of order n almost one — and requires the total to come to exactly two. There are seventeen ways to pay, and the classification falls out of an accounting identity that never mentions a lattice.

 Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, piezoelectric moduli from 18 down to 0, dielectric tensor from 6 down to 1, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as. What symmetry decides

A filter of great precision and no predictive power

The whole table in one place — thirty-two classes, six properties, 192 exact integers. What it settles, what it merely permits, and why knowing which of the two is happening at any moment is the entire skill of using it.

 One hundred and twenty-two magnetic point groups. The three kinds, counted. Thirty-two ordinary groups, which contain no primed operation; thirty-two grey groups, which contain time reversal on its own and are the symmetry of anything magnetically disordered; and fifty-eight black-and-white groups, one for each way of splitting a class into a subgroup of index two and its complement. The last number is the one that has to be computed: the index-two subgroups are found by closure inside each class, reduced up to conjugacy, and reduced once more by an equivalence that needs a rotation no lattice may have. 32 + 32 + 58 = 122, and every term is a measurement. What symmetry decides

The operation that reverses time

A magnetic moment is a current loop, so running time backwards reverses it and moves nothing. Admitting that as a symmetry operation turns the thirty-two crystal classes into a hundred and twenty-two — and eight of the merges needed to reach that number require a rotation no lattice may have.

 The symmetry of a cut through Pnma. Every height in one cell, and the number of operations of Pnma that map the plane at that height to itself. The answer is 2 almost everywhere and rises to 4 at the special heights, where the plane group named above the spike is what a reader looking down at that surface would see. The rule is two conditions and no more: the operation must not tilt the plane, and the plane must come back to its own height — so a twofold axis lying in the plane survives at two heights per cell and a screw axis along the normal survives nowhere. The classification

What a cleave leaves

A surface is a crystal that has been cut, and the symmetry it presents is what the space group leaves of itself on that plane. Two conditions decide it — the plane must not tilt, and it must come back to its own height — and the answer changes with where the cut was made.

 The classes that permit a spontaneous magnetisation. Every magnetic point group permitting a spontaneous magnetisation — 31 of the 122 — with the number of independent components each allows. an axial vector, reversed by time reversal — a ferromagnet has one and nothing else does. The count comes from averaging the character over the group, with a primed operation's contribution multiplied by −1 because the property reverses when time does. That is Neumann's principle with one extra sign in it, and it reproduces the numbers the literature records without being given them. What symmetry decides

Which magnetism a class permits

Neumann's principle with one extra sign in it decides which of the hundred and twenty-two magnetic classes may have a spontaneous magnetisation and which may show the magnetoelectric effect. The answers are thirty-one and fifty-eight, and they come out of the same average that counts elastic constants.

 How many close packings there are of each period. Every cyclic sequence over three letters with no two adjacent alike is a close packing, and two sequences describe the same structure when one becomes the other by rotating the cycle, reversing it, or relabelling the three positions. Counting the classes that remain gives 1 of period 2, 1 of period 3, 1 of period 4, 1 of period 5, and 38 altogether up to period 10. Period two is hexagonal close packing and period three is cubic; everything above them is a polytype, equally dense and equally close packed, and silicon carbide has been found in more than two hundred of them. Nothing in the geometry chooses. What chooses is an energy difference of a few thousandths of an electron volt per atom, and this site computes no energies. Symmetry at work

How many polytypes there are

One free choice per layer, repeated, gives a family of structures with the same composition, the same density and the same twelve neighbours — differing only in a sequence. Counting them up to rotation, reversal and relabelling turns "silicon carbide has hundreds of forms" into an enumeration.

 The thirty-two, in both notations. Each class with the symbol crystallography uses and the symbol spectroscopy uses, both derived from the class's own matrices. The Hermann–Mauguin symbol is a report on three families of directions, read in an order the lattice system fixes. The Schoenflies symbol is a report on a construction: a principal axis of order n, whether there are n twofold axes across it, and which mirrors were added. Neither can be computed from the other without going back to the group, which is why the two lists are set beside each other rather than one derived from the other. What symmetry decides

One class, two names

Hermann–Mauguin names directions and Schoenflies names a construction, and the two are derived here from the same integer matrices by computations that share no step. Neither can be obtained from the other without going back to the group — which is why a molecule has one kind of symbol and a crystal has both.

 Two half-turns make a translation. The half-turn about (0.25, 0.25) followed by the half-turn about (0.75, 0.5) is the translation by (1, 0.5) — twice the vector between the two centres, and not the vector itself. The open lens is a third centre, and it is not the midpoint of the two drawn: it is where the half-turn about the first lands when it is composed with one repeat vector of the lattice, which is half a repeat along. That is the step that puts two-fold centres on the half lattice and gives a p2 cell four inequivalent ones. Both the translation and the forced centre are computed from the operations and compared with the construction in exact rational arithmetic. Operations

Where the product is

Composing two symmetries lands on a third — and the third one is somewhere. Two half-turns make a translation by twice the distance between their centres, and that single fact puts the lattice into a pattern before anybody chooses one.

 What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order. Operations

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

 The first 4 zones of the square lattice. Zones one to 4, each in its own shade. The n-th zone is the set of wavevectors with exactly n − 1 reciprocal lattice points nearer to them than the origin is, so the boundaries are the perpendicular bisectors and nothing else. The zones get further out and break into more pieces — 1, 4, 8, 12 fragments — and every one of them has the area of a single cell. Lattices

The zones above the first

The second Brillouin zone is a scattering of disconnected fragments in a different part of reciprocal space from the first, and it has exactly the same area. So does the third, and the seventh. The reason is that each of them is the first zone, cut up and moved.

 Six integers that do not depend on the description. The same monoclinic lattice written in 4 different bases, each obtained from the last by an integer matrix of determinant one, and each reduced by Niggli's algorithm. Every one of them gives the same six integers — the squared lengths and twice the dot products of the reduced basis. That is what makes the reduced form a fingerprint of the lattice: two cells with no number in common are the same lattice exactly when their reduced forms agree, and the comparison has no tolerance in it. Lattices

The cell that settles the argument

Two determinations of one compound can report cells that share no number and describe the same lattice. Reduction is the procedure that decides — six integers that depend on the lattice and not on anybody's choice of axes, and that agree exactly when the lattices do.

 How much a count of descriptions over-counts. For each plane group that has any two-colouring at all: how many colourings it has, how many designs those come to, and the ratio between them. Over the seventeen the ratio is 1.61, and group by group it runs from 1.00 — where nothing is identified — to 3.50 at p2, whose seven colourings fall into one class of six and one of one. The tick on each row is that row's largest single class, and it is at least the bar and usually more. The largest class anywhere is p2's 6, and that same group over-counts by only 3.50, because a factor is a mean over the group's classes and a mean reaches its largest term only when every term equals it. Reading the largest class as the over-count is therefore an over-statement, always. And the factor varies from group to group, which is why no single correction turns a count of descriptions into a count of designs after the fact. The classification

Seventy-four colourings, forty-six groups

This site counts the two-colourings of the seventeen and gets seventy-four. The literature says there are forty-six two-colour wallpaper groups. Both numbers are right, and the gap between them is a disagreement about when two coloured patterns are the same pattern.

 Thirteen ways to hold a lattice. Every finite group of integer matrices in two dimensions, up to a change of integer basis: 13 of them. Ten different abstract groups appear, and three of the ten hold a lattice in two inequivalent ways — a mirror along an axis or along a diagonal, and the same for 2mm and for 3m. The enumeration is a search: every subgroup of the two maximal holohedries, merged by conjugacy under integer matrices of determinant ±1, with the answer checked for not depending on how wide the search was. What a lattice forbids

Thirteen ways to hold a lattice

The crystallographic restriction is about one matrix. A crystal has a whole group of them acting on one lattice at once, and asking how many such groups there are gives thirteen — not the ten of the plane point groups, and not the seventeen of the plane groups.

 The most of an icosahedron a crystal can keep. Every subgroup of the sixty rotations of an icosahedron, found by closure, with the crystallographic ones marked — those whose rotation orders are all among the 1, 2, 3, 4 and 6 that a three-dimensional lattice admits. The largest is 23, of order 12, at index 5; everything containing a fivefold axis is refused. So a crystal containing an icosahedral molecule may fix a twelfth of the molecule's own symmetry and no more, and the remaining 5 orientations have to be related by something other than the site's symmetry. What a lattice forbids

The most of an icosahedron a crystal can keep

C₆₀ sits in crystals and virus capsids sit in crystals, and neither of them stops being icosahedral. What a lattice can fix is a subgroup — and the largest crystallographic subgroup of the sixty rotations has order twelve, at index five. The five are Kepler's five cubes.

 N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does. How it is known

Whether there is a centre is a statistic

Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does.

 Two candidates, and the sign that chooses. The phase of reflection (2, 3), recovered from three measurements and no model. The circle is every complex number of the measured amplitude; the isomorphous difference fixes the cosine of the angle between the unknown phase and the heavy atom's, leaving the two candidates marked; the anomalous difference fixes the sine, which picks one. The recovered phase agrees with the true one to fifteen decimal places, and the true phase was never used in the calculation. How it is known

One experiment gives the cosine, the other gives the sine

Friedel's law holding exactly is what makes the phase unreachable. Its breaking is what hands it back: an isomorphous difference fixes the cosine of the phase and leaves two candidates, and the anomalous difference fixes the sine, which chooses.

 Aem2: one plane, two glides. The plane of Aem2 that carries two operations, drawn edge-on with each slide beside it. The two differ by the A-centring translation, which lies inside this plane — that is the whole condition for a plane to carry two, and it is why no primitive group has one. Both slides are axial, b and c, so neither letter has a claim on the symbol, and before 1992 the Tables simply chose. The letter e is the choice being refused. Into space

The plane that carries two glides

A plane can hold two glide operations at once, with slides that have no claim on each other. Every symbol printed before 1992 chose one of them, so the name recorded a convention rather than a group — and the International Tables invented a letter to stop it.

 Pnma has 6 names. The group Pnma with its three axes relabelled in each of the six possible ways. Every row is the same group, and each row is checked as it is drawn: the change of basis has determinant one, conjugating back gives the original operations exactly, and the census of screws, glides, mirrors and rotations is unchanged — and 6 different symbols come out. Each symbol is derived from the conjugated operations, not looked up, by the same routine that has to reproduce the symbol every group was entered under. Into space

Six ways to name one group

Pnma is also Pmnb, Pbnm, Pcmn, Pmcn and Pnam. Nothing about the crystal changes between those six; what changes is which axis was called a. In an orthorhombic group the axes are inequivalent and unlabelled, and naming them is a choice made six ways.

 Three shapes, and nothing else. The dielectric tensor of a crystal is an ellipsoid, and averaging a generic one over a point group leaves exactly three possibilities: a sphere, where all three principal values agree and the crystal is optically isotropic; a spheroid, where two agree and there is one optic axis; and a general ellipsoid, with two. The counts are 5, 19 and 8 of the thirty-two classes, and they were found by computing the eigenvalues rather than by sorting the classes by system. What symmetry decides

Three optical characters, and the arithmetic that assigns them

A cubic crystal cannot be birefringent, whatever it is made of. Between crossed polars it stays dark at every rotation, and the reason is that averaging any ellipsoid over a cubic point group leaves a sphere — a permission computed before anybody measures anything.

 Three lattices at 2 forms each: 6, 14, 12 faces. The shape each cubic lattice predicts, built as the solid bounded by its top 2 forms, with each face's distance from the centre inversely proportional to its interplanar spacing. The three lattices have the same metric and the same list of indices; every difference between these solids comes from which reflections are systematically absent. Pm-3m leads on {100} and comes out with 6 faces; Fm-3m leads on {111} and comes out with 14 faces; Im-3m leads on {110} and comes out with 12 faces. Taking more than the leading form matters only where the extinction correction has moved something: in a cubic metric a form's planes are placed at a distance proportional to the root of the sum of the squares of its indices, which is exactly where the corresponding corner of the cube already is, so an uncorrected second form arrives tangent and cuts nothing off. Symmetry at work

Which faces a crystal shows

Rock salt grows as cubes, fluorite as octahedra, garnet as dodecahedra. All three have cubic lattices and the same list of possible faces, and what separates them is which reflections are systematically absent — a rule about diffraction predicting a shape a mineralogist can hold.

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