Crystallography & Symmetry
patterns derived from their groups
Concept

Irreducible representation — where it appears

A representation with no proper subspace the group carries onto itself. Every representation is a sum of them in exactly one way, and their number equals the number of conjugacy classes.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

6mm: 6 irreducible characters on 6 classes. The character table of the plane point group 6mm, constructed rather than quoted. The columns are its 6 conjugacy classes, with the number of operations in each; the rows are its 6 irreducible representations, of dimensions 1, 1, 1, 1, 2, 2. The dimensions satisfy 1² + 1² + 1² + 1² + 2² + 2² = 12, which is the order of the group, and that identity together with the count of classes is what proves the list complete. Every entry is an exact element of the twelfth cyclotomic ring; the decimals shown are the numerical value of an integer vector, not a computed approximation.

What a group does to a function

A symmetry operation moves points about, and two hundred essays here have watched it do so. It also acts on everything defined over those points — a density, a displacement, a wave — and that action is linear, which turns a group of motions into a set of matrices and every question about it into arithmetic.

point-groups · Representations The ten plane classes, and the dimensions they permit. Every crystallographic point group of the plane, with one block per irreducible representation and each block as wide as its dimension. Nine of the ten have only one-dimensional representations; 4mm, 3m and 6mm carry a two-dimensional one, drawn in the measured colour. Nothing is wider than two, and the sum of the squares of the widths in each row is the order of that group — the identity that says the row is complete.

How large a degeneracy may be

Symmetry can force two things to have the same value, and in a crystal it can force three. It can never force five, and the reason is a sum of squares — the same kind of arithmetic that forbids a five-fold axis, arriving at a question about levels rather than about rotations.

point-groups · Representations p3m1 at (0, 0): the levels the little group requires, and the ones measured. The levels of the p3m1 model at (0, 0), with degenerate ones drawn thick. The little group there has order 6, and its characters predict levels of dimensions 1, 1, 2, 2. The measurement is 1, 2, 2, 1, and the account is "unitary". The values are numerical and the multiplicities are read at a stated gap; the prediction they are compared against is exact.

Where two levels must meet

At most wavevectors nothing of a crystal's symmetry survives, and its levels are as unconstrained as any operator's. At a handful of them a whole point group survives, and where that group has a two-dimensional representation, two levels are obliged to coincide — before anything about the material is known.

space-groups · K symmetry 4mm: a degeneracy tuned into existence, and gone at the next weight. Four invariant operators on one orbit of 8 points under 4mm, differing only in the weight given to a single class of pairs. The first column is not a choice: it is the value that weight has to take for two levels of different symmetry to arrive at the same number, found by sweeping the weight and closing on the crossing, and the two levels there agree to 1.0e-9. The character table predicts levels of sizes 1, 1, 1, 1, 2, 2; the tuned column shows 1, 1, 1, 2, 3 and every other column shows the predicted pattern again. That is the whole of what an accidental degeneracy is — a property of one choice of weights, not of the group — and it is why the weights have to be moved before a degeneracy is called forced. A degeneracy the group requires would be in all four columns, because nothing respecting the symmetry can lift it.

A coincidence the group did not ask for

Two levels sitting at the same value look identical whether symmetry required it or not. The difference is testable: move the numbers the symmetry does not decide and watch what survives, because a degeneracy the group forces cannot be shifted by anything the group leaves alone.

point-groups · Representations An orbit of 12 points under 6mm, split into 6 kinds. The functions defined on one orbit of 12 points under 6mm form a space of that dimension, and the group acts on it by permuting the points. The character of that action is the cheapest one in the subject — at each operation it is the number of points left where they are — and decomposing it against the table gives the multiplicities on the right. They are whole numbers, they weight the dimensions to 12 again, and the multiplicity of the trivial representation is the number of orbits, which is Burnside's lemma arriving as a special case.

The shell that splits into kinds

The neighbours of an atom carry a space of functions as large as the shell, and the group does not treat that space as one thing. It splits into pieces of a few kinds, in whole numbers, and the count is the cheapest character in the subject: how many neighbours each operation leaves where they are.

point-groups · Representations The honeycomb's two levels meet at K, exactly. The two levels of the honeycomb net along a line from the centre of the zone to its corner. The off-diagonal entry of its two-by-two matrix is the sum of the phases of three bonds, and at the corner those phases are the three cube roots of unity, whose sum is zero — exactly, as an identity in the ring the phases live in rather than as a number that came out small. So the matrix there is the zero matrix and both levels are zero. It is the shortest exact statement of a crossing in this collection.

The crossing at the corner

The honeycomb's two levels meet at the corner of its zone, and the meeting is not approximate. Three phases sum to zero there — an identity between cube roots of unity — so the matrix is the zero matrix, and making the two sites differ opens a gap of exactly that difference.

space-groups · K symmetry Compatibility at (0, 0) in p4m. Every representation of the little group at (0, 0) in p4m, and what it becomes along two lines out of that point. A one-dimensional representation stays one level and acquires a label; a two-dimensional one splits into two levels of opposite label. The rows where the two columns differ are the point: the same level is even under the mirror that survives along one line and odd under the mirror that survives along the other, so which bands may cross and which must repel is different in the two directions out of one point. Labels are the characters on the classes, computed rather than named.

Which levels join which, on the way out of a point

A degeneracy at a symmetry point is forced by the little group there. Move off the point and the little group shrinks, the degeneracy is free to split, and which pieces it splits into is decided by restricting a character. That restriction is what joins a table of isolated points into a band structure.

space-groups · K symmetry 4mm: 17 of 25 transitions forbidden. Every pair of irreducible representations of 4mm, with the number of times the identity occurs in the product of the two with the vector operator. A zero is a prohibition: the integral that would give the transition rate vanishes for every choice of functions carrying those representations, whatever the material is made of. A positive number is a permission and nothing more. Rows are final states and columns initial ones; the labels are the dimensions of the representations, so the twos are the degenerate levels. Every one of the 17 prohibitions here was checked again against explicit polynomials.

What a group forbids to happen

Two levels and a thing that might carry a crystal from one to the other. Whether it can is one sum over the group — and a zero there is a prohibition that no material, no temperature and no intensity of light gets round.

point-groups · Representations One framework has a count of zero, one mechanism and one self-stress. Every net this collection has a placement for, as a periodic bar-and-joint framework in a fixed cell: its point group, the joints and bars of one cell, the scalar Maxwell count 2n − e − 2, and the mechanisms and self-stresses found exactly from the rank of the rigidity matrix. The scalar count is always the difference of the last two, which is Maxwell's identity — and the bathroom net is the row that shows what the identity costs: nought equals one minus one, and a framework that reads isostatic moves.

The mechanisms a count cannot see

Maxwell's count subtracts constraints from freedoms, and a mechanism and a state of self-stress cancel in the subtraction — so a framework with one of each reports the same number as a rigid one. The bathroom net reports nought and moves. Doing the same subtraction with representations instead of numbers separates them, because a mechanism and a self-stress cancel only when they belong to the same representation.

applied · Rigidity The window in which A₄ and its dual are both minima. The exact second derivative of the Gaussian sum at A₄ and at A₄ along their five-dimensional family of changes of shape, for widths within six hundred-thousandths of one. A₄'s turns positive at t = 0.99998110 and A₄'s turns negative at 1.00001890, the reciprocal. Between them both are positive, so both lattices are local minima; at t = 1 itself both equal 8.59e-6.

The self-dual width is not flat

A search over four-dimensional lattices found that A₄ and its dual stop being traps for the Gaussian sum almost exactly at the self-dual width, and read that as a flat of the sum sitting on the width itself. Computed exactly rather than by finite differences, the curvature there is not zero. It is six parts in a million of the other curvature, positive, and the same for both lattices, because duality makes their Hessians one matrix at that width. Both are minima, in a window four parts in a hundred thousand wide, with basins a few millionths across.

lattices · Lengths Where a gradient term is allowed at a special point. For each of the seventeen plane groups and each special point of its zone — the centre, the edge midpoints, the corner of a rectangular or square zone and the corner K of a hexagonal one — every physically irreducible order parameter, by its real dimension, marked where it admits a Lifshitz invariant. They occur at 10 points in 7 groups: the edges of the glide groups pg, pmg, pgg and p4g, and the corner K of p31m, p6 and p6m. Nowhere else.

Where a gradient term survives at the edge of the zone

A transition that freezes a mode at the edge of the zone doubles the cell, but only if the energy has no term linear in the mode's gradient. That condition was stated here for the zone boundary and never computed. Computed for every order parameter at every special point of the seventeen zones, it fails in exactly ten places: the edge points of the four glide groups where one glide acts and no second glide crosses it, and the zone corner of three hexagonal groups that have no glide at all.

space-groups · Landau What Kleinman's rule removes from the second harmonic, class by class. The 20 crystal classes that permit second-harmonic generation, each with its independent susceptibility components drawn as squares: those that survive when the tensor is made totally symmetric, and those Kleinman's rule removes, split into the traceless part of the gyration tensor and a polar vector. In all, 83 components become 48. Two classes, 422 and 622, lose everything; four, 6̅, 6̅2m, 23, 4̅3m, lose nothing.

Two classes go dark far from absorption

The second-harmonic tensor of a crystal has as many components as its piezoelectric tensor, and far from any absorption it gains a symmetry of its own that removes eight of the eighteen. What it removes is not arbitrary: it transforms as a polar vector plus the traceless part of the tensor that carries optical activity. So the classes that lose the most are the polar ones and the handed ones, and two of them, 422 and 622, lose everything.

point-groups · Properties How many resonances each crystal class needs for a second-order response. The 32 crystal classes sorted by the number of distinct oscillators a model needs before the class can have any second-order optical response, with the dimensions of the class's invariants among the cubic couplings: fully symmetric (one oscillator), mixed (two), alternating (three). The eleven centrosymmetric classes have none. Eighteen need one; four of those (23, 4̅3m, 6̅, 6̅2m) have no mixed invariant and obey Kleinman's rule at every frequency. 422 and 622 need two. 432 needs three.

A crystal in 422 needs two resonances

Kleinman's rule silences classes 422 and 622 far from absorption, and the obvious guess is that dispersion brings them back near it. In the simplest model of dispersion, one charged oscillator with an anharmonic potential, it does not: they stay dark at every frequency. Two oscillators with different resonances light them, with a response growing as the square of the frequency. 432 needs three, and then only mixes two different frequencies, growing as the sixth power. The number of resonances a class needs is a count of invariants, and it sets how fast the forbidden part can grow.

point-groups · Properties The first order of stress that can magnetise each magnetic class. The ninety magnetic point groups that are not grey, in four columns by the lowest power of a stress that can give them a magnetisation: 31 have one at rest, 35 acquire one in proportion to the stress, 3 only in proportion to its square, and 21 under no power of it at all. The three that need the square are the ordinary cubic classes 432, 4̅3m and m3̅m, and the twenty-one that no stress magnetises are exactly the classes holding the inversion combined with time reversal.

A squeeze magnetises a cubic crystal across itself

Three ordinary cubic classes forbid a magnetisation proportional to a stress. At the square of the stress they each admit exactly one constant, and nothing else joins, at that order or any higher: sixty-nine classes is the final list. The one constant has a shape no table of coefficients shows. A squeeze along any direction magnetises the crystal at right angles to the squeeze, and along each of the cube's twenty-six rotation axes it magnetises nothing, which is the least a field on a sphere is allowed to vanish.

point-groups · Magnetic Eight of the thirty-two classes allow a gradient term at the zone centre. The thirty-two crystal classes, marked by whether any order parameter at the centre of the zone has a term linear in its gradient. In the plane none does. In space eight do: 3, 4, 6, 32, 422, 622, 23 and 432. All eight are among the eleven classes with no operation that reverses hand, and the other three, 1, 2 and 222, have only one-dimensional order parameters.

A screw keeps a line and a glide keeps a plane

The census of the seventeen zones left two predictions for space. A transition at the zone boundary should meet a gradient term where one screw or glide is active, along that operation's direction, and a second should remove it. Across all 219 types, the first half holds without exception in the monoclinic groups and the direction is wrong: a glide's term fills its plane. The second half fails from the orthorhombic groups on. And the zone centre, empty in the plane, is not empty in space.

space-groups · Landau

Named alongside it

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

CharacterDegeneracyCharacter tableWavevectorLittle groupRepresentationBloch stateConjugacy classCrystal netGlide reflectionGyration tensorOptical activity

All concepts