Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: The same arithmetic, renamed

A crystal form is an orbit. A twin law is a coset. The domain states left by a phase transition are the cosets of the low-symmetry group in the high-symmetry one. Four subjects that grew up in different centuries and different departments, doing one piece of arithmetic under four names.
The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it. Symmetry at work

Why a crystal face carries small whole numbers

A crystal face is flat because it lies on a plane of lattice points, and its orientation is therefore named by three integers rather than by two angles. That the integers exist is a theorem about lattices; that they are usually smaller than four is a separate claim about growth, and the two are routinely run together.

 Two habits of {101̅}, one set of angles. The same 12 faces of class 6/mmm, grown to different distances from the centre. The outline changes completely and not one interfacial angle moves, because a face's orientation is set by the lattice and its extent by how fast it grew. That is Steno's law, and it is the reason a goniometer measures something about the substance rather than about the specimen. Symmetry at work

The angles belong to the substance, the shape to the specimen

Two crystals of the same mineral can look nothing like each other and still have exactly the same angles between corresponding faces. That is the oldest quantitative law in the subject, and what it measures turns out to be the shape of the unit cell — which means a brass instrument from 1809 was reading lattice parameters a century before anyone knew there were any.

 {111} in class m3̅m. The form {111} of crystal class m3̅m: 8 faces, being the orbit of one face under the 48 operations of the class, with a stabiliser of order 6. 4 poles lie in the upper hemisphere or in the plane of the page and are drawn filled; the other 4 lie below and are drawn open at the same positions, which is the stereographic convention and the reason only the upper ones carry their indices. The form is closed: the faces enclose a volume, so a crystal can be bounded by this form alone. Symmetry at work

A form is an orbit, and whether it closes is an integer question

Name one face of a crystal and its class names the rest. That set is a form, it is an orbit in exactly the sense this site has used since its first essay, and whether it encloses a volume — whether a crystal could be bounded by it alone — is decided without any lengths or angles entering the calculation anywhere.

 {100} offered to 5 classes: one form between them — the shape names none of the 5. The same face, {100}, handed to 5 crystal classes — m3̅m, m3̅, 432, 4̅3m, 23 — with the orbit each one returns drawn as a stereogram. Filled marks are poles in the upper hemisphere and open ones their partners below. The face counts are 6, 6, 6, 6, 6, taking 1 distinct value; the sets of faces take 1, which is the number that matters, since two classes can return the same count and different faces. Here every class returns the identical set, so a crystal bounded by this form alone has said nothing about which of them grew it. Symmetry at work

Five classes grow the same cube

A crystal's shape is the most obvious thing about it and the least informative. Five of the thirty-two classes produce an identical cube, diffraction cannot see an inversion centre and so collapses the thirty-two to eleven, and the measurements that finally separate them are etch pits, optical rotation and a heated crystal attracting ash.

 p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one. Symmetry at work

A twin is a symmetry the lattice has and the crystal does not

Two orientations of one structure, grown together across a boundary the lattice runs straight through. The operation relating them cannot be a symmetry of the crystal, or there would be nothing to see, and it must be a symmetry of the lattice, or the boundary would be a crack — which leaves exactly a coset, and a short computable list.

 Which classes can twin by merohedry, and how many ways. Every crystal class, with the number of twin laws its own lattice offers it. The index of the class in the point group of its lattice is the number of orientations available; 25 of the thirty-two have more than one, and the 7 holohedral classes have exactly one — their crystal already has every symmetry their lattice has, so there is nothing left over to twin by. The names along the right are the old mineralogical ones: hemihedral for half, tetartohedral for a quarter. Symmetry at work

Twenty-five of the thirty-two can twin, and seven cannot

The number of twin laws available to a crystal is the index of its class in the point group of its lattice, minus one. Doing that arithmetic for all thirty-two classes takes a moment and produces a census with a sharp edge on it — the seven classes that cannot twin this way are exactly the seven that already use everything their lattice has.

 The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law. Symmetry at work

Quartz has exactly three twin laws, and its lattice is why

Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

 p3, single and twinned. Left, the diffraction pattern of a single crystal of p3. Right, the same crystal twinned, with 50 per cent of it in one orientation. Not one spot has moved — the twin law is a symmetry of the lattice, so the two reciprocal lattices lie exactly on top of one another — and 72 of the 81 reflections drawn have changed intensity. At a fifty-fifty twin the pattern acquires the full symmetry of the lattice's point group and is indistinguishable from a crystal that genuinely has it. Symmetry at work

A merohedral twin moves no spot at all

The twin law is a symmetry of the lattice, so the two individuals have reciprocal lattices lying exactly on top of one another. Nothing splits, nothing appears in a new place, and the only thing that changes is that pairs of intensities which were different have been averaged — which produces a diffraction pattern with a symmetry the crystal does not have and no sign that anything is wrong.

 m3̅m → 4mm: 6 domain states. The transition from class m3̅m to class 4mm loses 40 of the parent's 48 operations, so the child has index 6 and the crystal comes apart into 6 domain states. Each colour is one state — one coset of 4mm in m3̅m — and each holds the same 8 poles. The lost operations are what carries one state onto another, and they survive in the crystal as the relation between its domains rather than as symmetries of any part of it. The descent changes the crystal system, so the states differ in shape as well as in orientation and the transition is ferroelastic. Symmetry at work

How many domains a transition makes is an index

Cool a crystal through a symmetry-lowering transition and it has to choose one of several equally good low-symmetry arrangements. Different parts of it choose differently, and the number of available choices is the index of the new group in the old one — a number available before any crystal is grown, and one of the few predictions in this field that is a count rather than a bound.

 Everything class m3̅m can descend to. The 25 crystal classes that are subgroups of m3̅m, arranged by order, with the 56 maximal steps between them drawn as edges. The order of each row is printed down the left, so the index of any step is the ratio of the two rows it joins. A symmetry-lowering transition can only be continuous when it goes down one of these edges, and the index on the edge is the number of domain states the transition produces. A descent of several steps is possible but has to happen discontinuously or through the intermediate classes. Symmetry at work

The descent of symmetry is a lattice, not a tree

Which classes a crystal can fall to when it loses symmetry, drawn as a graph with the index on every edge. It is routinely called a tree and it is not one — a class can be reached from its parent by several different routes of the same total index, and which route a material takes is a physical question the diagram deliberately leaves open.

 An antiphase boundary in p4. Where two antiphase states meet. Above the line the species alternate one way and below it the other, so at the boundary two cells of the same species sit next to one another and the ordering is out of step. This is a domain wall with no change of orientation across it: the crystal is not twinned, its lattice is undisturbed, and diffraction sees it only in the width of the superlattice reflections. Symmetry at work

The domains a lost translation makes, which nothing optical can see

An ordering transition can leave the crystal class untouched and take away translations instead. The domains that result have the same orientation, the same shape and the same optical properties as each other, and where two of them meet the ordering is simply out of step — a boundary with no change of direction across it and no way to find it except by looking at the ordering itself.

 The seventeen, arranged by what they can lose. Each group at the height of its own order, joined to every maximal subgroup that keeps all of its translations. Reading downwards is a crystal losing operations at a phase transition. The edges are the maximal ones only — every other containment is a path through these — and the whole graph is enumerated by closing every subset of each group's operations, so nothing is here because a table said so. Operations

The descent with no shortcut

A subgroup can give up operations, or it can give up translations. Hermann's theorem says that a *maximal* subgroup does one or the other and never both at once — which is why a crystal losing symmetry can be followed one clean step at a time, and why every route from p6m down to p1 has exactly three steps.

 Fddd has two published origins. The two conventions, computed from the operations. The International Tables place the origin at the point of highest site symmetry, and also at a centre of inversion, and for this group those are different points — so the group is printed twice, with every coordinate in the second table shifted by (-0.125, 0.125, -0.125) from the first. A structure published on one and read on the other has every atom in the wrong place by that vector, the refinement fails in a way that looks like bad data, and nothing in the symbol says which was used. Into space

Two origins for one group

The International Tables place the origin at the point of highest site symmetry, and also at a centre of inversion. For twenty-four of the two hundred and thirty those are different points, so the group is printed twice with every coordinate shifted — and nothing in the symbol says which table a structure was written against.

 The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique. Operations

The same pattern, described twice

Two coordinate lists for one structure can disagree in every number and describe exactly the same arrangement, because a group does not fix its own origin. The operations that may be applied to a description without changing what it describes are its normaliser, and they can be found by looking at pictures rather than at matrices.

 How many ways each class can lose symmetry. Every crystal class, with the number of distinct classes it can descend to — 247 parent-and-child pairs in all across the thirty-two, counted up to conjugacy in the parent, which is the equivalence that says two descents differing only by which axis was chosen are one transition. The count rises steeply with the order of the parent, which is why the cubic and hexagonal holohedries dominate the list of materials with rich domain structures. Symmetry at work

Two hundred and forty-seven descents, or two hundred and twelve

How many distinct ways can a crystal lose symmetry? Counting parent-and-child pairs up to conjugacy in the parent gives 247. The standard enumeration in the ferroics literature gives 212, and the operation that merges the extra thirty-five turns out to be a rotation through forty-five degrees — which no lattice may have, and which no integer matrix in a lattice basis can therefore express.

 Six classifications, and which are enumerated here. The families of symmetry groups by how many directions they repeat in and how many they live in. The thirty-two crystal classes, the seven friezes and the seventeen plane groups are each built from their own operations and counted. The seventy-five rod groups, the eighty layer groups and the two hundred and thirty space groups are numbers from the literature, marked as such wherever they appear: reaching them needs the translation extensions and their equivalences in full, which is the content of the classification rather than an application of it. The subperiodic cases sit exactly between the two halves, which is why they are so easy to assume are already known. The classification

A layer is not a wallpaper

A sheet repeats in two directions and lives in three, and its symmetry group is not one of the seventeen. There are eighty of them, the difference between one and another is a single sign per operation, and the arithmetic that supplies those signs is the arithmetic of a two-coloured pattern.

 Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence. Symmetry at work

Turn a lattice against itself and almost nothing lines up

Two copies of one lattice rotated about a shared point share that point and, at almost every angle, no other. At a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation costs a fraction of what a general one costs, because a fraction of the atoms are already where both sides want them.

 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.

 Two species on one lattice, ordered at index 2. Every position is a lattice point of the parent and none of them has moved. What has changed is which atom sits where: the larger marks are a sublattice of index 2, the smaller ones its other 1 coset, and the outlined cell is the new repeat. The lattice of positions is untouched and the repeat of the contents is 2 times as large, which is the whole of what an ordering transition does and the reason its signature is in reciprocal space rather than in the positions. Lattices

The reflections a superlattice adds

Centring a lattice makes reflections vanish. Ordering two kinds of atom onto a sublattice makes new ones appear, exactly n − 1 of them per parent cell, and their intensity is a difference rather than a sum — which is why an ordered alloy of two neighbouring elements can be invisible to X-rays and obvious to neutrons.

 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.

 One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours. Symmetry at work

Two stackings, one density

Stack spheres as tightly as they will go and the third layer has a free choice. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours — and their space groups are Fm3̅m and P6₃/mmc, which is the only thing that tells them apart.

 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 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.

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