Crystallography & Symmetry
patterns derived from their groups
Concept

Crystal class — where it appears

One of the thirty-two ways a crystal can be symmetric about a point, being a point group whose operations a lattice is able to carry. They are enumerated here as the subgroups of two holohedries, quotiented by conjugacy with every merge witnessed by a matrix.

Named by 24 essays across 5 fields — each of them below, with the objects they name alongside it.

The thirty-two crystal classes. Every crystallographic point group, as a stereogram. Each was found by enumerating the subgroups of m3̅m and of 6/mmm, and each diagram is the orbit of one general direction under the group, filled where the pole is in the upper hemisphere and open where it is in the lower — which is the only thing in the picture that tells a rotation from a rotoinversion.

Thirty-two, and no others

There are exactly thirty-two ways a crystal can be symmetric about a point. Not thirty-two that anybody has catalogued — thirty-two that a finite search produces, from two starting groups, with every step of the reduction counted separately so that no two of them can quietly compensate.

point-groups · Crystal classes The merge, and what witnesses it. The two holohedries are enumerated separately and their class lists merged where the element signatures agree. That returns the right total, which is not the same as being right: 33 merges are made and every one of them is checked by constructing an explicit change of basis carrying one group onto the other.

A fingerprint that gave the right answer

The thirty-two classes were merged on a fingerprint — the census of operation types — and the fingerprint returned thirty-two, which is correct. Returning the correct answer is not the same as being entitled to it, and the difference took three wrong constructions to close.

point-groups · Crystal classes Reading 4/mmm off its own directions. Each position of 4/m2/m2/m reports one symmetry direction of the tetragonal system: the highest-order axis lying along it, and whether a mirror is perpendicular to it. Nothing is looked up — every row is computed from the group's own matrices.

Reading a class off its own axes

A Hermann–Mauguin symbol is not a name that was assigned. It is a report on three directions, read in order, and the whole of it can be derived from the group's matrices — with one genuine convention and one exception, and the exception is orthorhombic.

point-groups · Crystal classes {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.

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.

applied · Forms The crystal classes 3m, 3̅m, 6̅2m. 3m, 3̅m, 6̅2m: the orbit of a general direction under each group, giving 6, 12, 12 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it.

3m1 and 31m are one class

This site has an essay arguing that p3m1 and p31m are genuinely different groups. As point groups the same two objects are one class — and the two subgroups are each normal in the hexagonal holohedry, so nothing in the lattice relates them. What does is a rotation of thirty degrees.

point-groups · Crystal classes The thirty-two, by crystal system. 32 classes in 7 crystal systems. Each column is one crystal system and each cell one class, ordered by the number of operations it holds. Nothing here is tabulated: the classes come from the enumeration and the marking from a character sum over each group.

The holohedry is the ceiling

A crystal never has more point symmetry than its lattice. That single containment decides which system a class belongs to, why there are seven systems and not thirty-two, and why a lattice can be more symmetric than the crystal sitting on it — which is the usual case rather than the exception.

point-groups · Crystal classes 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.

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.

applied · Twinning The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.

The eleven a diffraction pattern reports

A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

point-groups · Diffraction 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, dielectric tensor from 6 down to 1. 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.

Twenty-one, thirteen, nine, three

The number of independent elastic constants runs 21, 13, 9, 7, 6, 5, 3 down the crystal systems. Two of those systems carry two numbers rather than one, and which classes take which is not predicted by counting operations — a class with six of them can have more constants than a class with six of them.

point-groups · Neumann's principle Every property count, in every class. For each of these 32 classes, how many independent components a property may have: piezoelectric moduli from 18 down to 0, 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.

Permitted is not present

A symmetry argument says which components of a property may be non-zero. It is routinely read as saying they are — and the gap between the two is where every practical use of this table either works or quietly fails.

point-groups · Neumann's principle What the inversion does to three sums in 4/mmm. The 16 operations of 4/mmm come in pairs — every operation together with its own negative, because the class contains the inversion — and each row pairs the two terms they contribute. For an even-rank polar property the pair is two equal bars: the inversion changes an even number of indices and the character cannot see it, so the average is whatever it was before the inversion was added. For an odd-rank polar property and for an axial one the pair is a bar and its reflection, and the sum is exactly zero — which is why piezoelectric moduli and gyration tensor are forbidden here outright rather than merely small. Neither statement is about this class: the cancellation is checked over all 11 centrosymmetric classes and all four odd or axial properties every time this figure is drawn.

Twenty of the twenty-one

Twenty-one crystal classes have no centre of symmetry, and twenty of them permit piezoelectricity. The exception is 432, which has twenty-four operations, no inversion, and a character sum that cancels to nothing — and the reason it fails is not that it has too much symmetry in any ordinary sense.

point-groups · Properties 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.

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.

point-groups · Properties 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.

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.

point-groups · Properties 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.

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.

point-groups · Crystal classes 75 rod groups over 27 axial classes. Each axial crystal class, with the number of rod groups it carries: every consistent choice of translation along the axis, in every way the class can sit on the rod, with two groups counted as one when a shift of the origin along the rod or a turn about it carries one onto the other. The total is 75, and every row as well as the total agrees with the International Tables, which are compared with this enumeration rather than used to produce it.

Seventy-five ways to be a thread

Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.

classification · Subperiodic Thirty-two classes, eighteen groups. Every abstract group the thirty-two crystal classes realise, with the classes that realise it. 8 of the eighteen carry more than one class, and the largest collision is the four hexagonal classes that are all the dihedral group of order twelve. Nothing here is looked up: two classes are put in the same row when a search over images of a generating set finds a bijection preserving multiplication, and the search is finite because a generating set is small and the elements it may map to are the ones of the same order.

Thirty-two classes, eighteen groups

An inversion centre, a mirror and a two-fold rotation are three of the most different things a crystal can have, and they are the same group of order two. Forget the matrices and keep the multiplication table, and the thirty-two classes collapse to eighteen.

point-groups · Crystal classes p4m: 4 and 4. The standard motif — three points in no particular arrangement — repeated by p4m, with each copy coloured by the sign of the area of the triangle it makes. 4 copies have one sign and 4 the other, because the group contains an operation that reverses orientation. A structure built from one enantiomer cannot sit here: the group would put its mirror image in the same crystal. The colours were computed from the coordinates rather than assigned.

The groups a single hand may sit in

A protein is built from one enantiomer of every amino acid, and a crystal of it contains nothing else. That single fact deletes most of the classification at a stroke: any operation reversing orientation would put the other hand in the same crystal. The criterion is one line of arithmetic, and in the plane the enumeration is complete — five of the seventeen.

space-groups · Chirality The seven groups a uniform field can have. Five of them have an axis and two do not. A cone has every rotation about its axis and mirrors containing it; a cylinder adds the mirror across the axis and the two-folds that go with it; turning either one destroys the mirrors that would reverse the turn. The sphere and the sphere made of something with a handedness are the two with no axis to speak of. Each drawing is the definition: the group is the set of motions leaving the picture unchanged.

The seven groups a field can have

Every group in this collection so far has been finite, because a lattice forbids the alternatives. A uniform field has no lattice: rotate it about its own axis through any angle at all and nothing has changed. There are exactly seven such groups, and they come out of the same closure argument that turns sixteen frieze candidates into seven.

point-groups · Curie What a crystal keeps of itself in a field. Each class, with what is left of it when a field is applied along the axis of its own setting. The residual is the intersection of the class with the field's own group, computed on matrices and matched against the thirty-two rather than named by hand. Where the residual is the class itself, the field takes nothing away — and for an electric field those are exactly the polar classes.

What a crystal keeps in a field

Curie's principle says the symmetry of an effect contains the intersection of the symmetries of its causes. Applied to a crystal in a field that is an intersection of two groups, one of them infinite — and it comes out exactly, class by class, as a subgroup that decides which effects are permitted next.

point-groups · Curie 12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted.

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

point-groups · Invariants Thirty-two classes, from fourteen Gram matrices. The five hundred and ten subgroups sorted by how many operations of each kind they contain — a determinant and a trace decide which of the ten kinds a matrix is. Thirty-two answers come out, and they are the thirty-two crystal classes: matched against the construction elsewhere in this collection by signature rather than by name, since nothing here names a point group.

Thirty-two from fourteen matrices

Write the fourteen Bravais lattices as Gram matrices, ask each one which integer matrices preserve it, and take every subgroup of every answer: five hundred and ten of them. Sort those by how many operations of each kind they contain — which a determinant and a trace decide — and thirty-two answers come out. They are the crystal classes, from a construction in which no point group is ever named.

point-groups · The fourteen Bravais lattices Seventy-three arithmetic classes, from fourteen groups. Every subgroup of every lattice's own group, split by whether the subgroup's own Bravais group is that lattice's. The ones that are not belong to a lower lattice and are counted there, which is what stops the same class being counted twice. The running total ends at seventy-three, and no conjugacy in GL(3, ℤ) was ever decided.

Seventy-three, without a search

The unit a space group is built from is a point group together with the lattice it acts on, and there are seventy-three of them. Getting there looks like it needs conjugacy in GL(3,ℤ), which is a search this collection tried and abandoned. It does not: every finite group of integer matrices carries a canonical larger group that says which lattice it belongs to, and once that is computed the search has nothing left to do.

point-groups · The fourteen Bravais lattices Cubic means the Sylow 3-subgroup is not normal. The thirty-two sorted two ways at once: by crystal system and by whether the Sylow 3-subgroup is normal. Two of the four boxes are empty, so the two properties coincide exactly. That gives 'cubic' a definition with no geometry in it — a class is cubic when its threefold subgroups are conjugate to each other rather than unique — and it explains why a cubic class has no principal axis: a group cannot single out one member of a conjugate family.

How many axes there are is a Sylow count

Sylow's theorems say that the subgroups of prime-power order in a finite group are all conjugate and that how many there are is congruent to one modulo the prime. Applied to the thirty-two crystal classes that arithmetic counts axes: the number of Sylow 3-subgroups is the number of equivalent threefold directions, it is four in exactly five classes, and those five are the cubic ones. So cubic has a definition with no geometry in it.

point-groups · Crystal classes Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups

Named alongside it

The objects these essays reach for when they reach for this one.

Point groupSubgroupHolohedryNeumann principleCharacterTensorAnisotropyConjugacy classFriedel lawEnumerationLaue classAxial vector

All concepts