Theme

The theme: Symmetry is decidable — page 9

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.

$A wavevector of thirds, and the boxes that cannot see it. Which sizes of box can carry the wavevector at the corner of a hexagonal zone. The characters of the box's translation group are its wavevectors, and there are exactly N² of them — the fractions with denominator dividing N. A wavevector of thirds is therefore present in a box of three, six, nine or twelve cells and absent from one of two, four or five: not approximated badly, not resolved coarsely, absent. A mechanism or a level living there is invisible to such a calculation, and that is the practical content of a mechanism count depending on the cell it was looked for in.$ The classification

Crystallography in a box

A calculation over a crystal is not performed on a crystal. It is performed on a finite block with its edges glued, and the block has a symmetry group of its own — finite, complete in one direction and missing something decisive in the other.

square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here.

Symmetry at work

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

Order without repetition

The average is the same wherever it is taken

A measurement is made on a window somewhere, and the question is whether the answer belongs to the chain or to the window. For the Fibonacci chain the error falls as one over the window's length; for a shuffle of the same letters it falls as one over the square root, and the two exponents are fitted rather than asserted.

$the most consistent answer has 100 per cent of the signs. 60 runs of the sign procedure from 60 different random starts, each plotted at its self-consistency — a figure computed without any knowledge of the answer — against the fraction of its signs that are in fact right. Throwing out the 1 run that reached the uniform solution — every sign the same, perfectly consistent and physically a single peak — the highest consistency belongs to a run with 100 per cent of the signs right. The ranking works here, and the reason it works is that the cell is small. Nothing in the plot's horizontal axis knows the answer, which is the only reason a procedure of this kind is a procedure at all.$ How it is known

The formula that has the answer already

The tangent formula rebuilds each phase from all the others, and the true phase set is very nearly a fixed point of it — hand it the answer and it hands the answer back. Start it anywhere else and it does not arrive. Having a fixed point and finding it are different problems, and the second is where the subject spent twenty years.

Order without repetition

One tile, and no period

Every aperiodic pattern in this collection so far needs two shapes. A search over the eight-hundred-and-seventy-three ways of gluing eight kites together, filtered by nothing but whether a shape tiles and whether it repeats, returns exactly one — and it is the shape announced in 2023.

B = 3.45 against 3.4, K = 0.37 against 0.37. The mean intensity of each resolution shell of a cell of 1194 reflections, divided by Σf² computed from the cell's content alone, and logged. The points fall on a line whose slope gives B = 3.45 against the 3.4 put in, and whose intercept gives a scale of 0.37 against 0.37 — both recovered before a single atom has been placed. The atoms here are independent, so the line is straight at every resolution; the shells below the cutoff are marked in the second colour.

How it is known

The average that knows the atoms and not where they are

Square a structure factor and average it over a shell of reflections at one resolution. The cross terms — every one of which carries a fact about the arrangement — cancel, and what is left is a sum over the *content* of the cell with no position in it anywhere. A scale and a temperature factor come out of that before a single atom has been placed.

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.

Into space

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.

monoclinic: 9 twin laws, 1 of them exact. The twin laws of a monoclinic lattice: a two-fold about the row [uvw] paired with the plane (hkl) it is meant to be a mirror in, kept when the twin index is at most six and the obliquity at most six degrees, which are Friedel's own limits and are a convention rather than a theorem. 9 of the 9 are listed. The index n is how many lattice nodes there are per node the operation restores, computed from the integers and checked against the sublattice built from the plane and the row. The obliquity is the angle between the row and the plane's normal: 1 of these laws have none, and for those the operation restores a sublattice exactly.

Symmetry at work

The index and the angle a twin misses by

Whether a crystal will twin on a given operation is decided by its lattice, not by its structure. Two numbers decide it: how many lattice nodes there are per node the operation restores, and how far the operation is from being a symmetry at all. Both are computed from integers, and one of them is a fiction that has to be labelled as one.

4mm: how many independent invariants there are at each degree. The number of independent polynomial invariants of the plane point group 4mm, one bar per degree from 0 to 8. The heights are the coefficients of the Molien series, computed from an integer recursion on the trace and determinant of each operation. A dot above a bar marks a degree where the same number has been computed a second way, by averaging every monomial of that degree over the group and taking the rank of the result — a computation sharing no code with the first. The two agree at every degree checked. Degrees where the bar is absent hold no invariant at all: for 4mm that is every odd degree below the first invariant, and it is a statement about which functions the group refuses to leave alone.

What symmetry decides

How many invariants of each degree

A group moves the plane about, and some polynomials do not notice. How many independent ones there are at each degree is a sequence of integers, computed here by a recursion on traces and again by averaging every monomial — two routes that share no code and agree everywhere.

p6m: freezing Γ2 leaves p31m. The same crystal three times. On the left, a pattern with the full symmetry of p6m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is p31m, of index 2 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical.

Into space

An order parameter is a representation

The quantity that measures a lost symmetry is not a number the physics chooses freely: the parent group mixes its components, so it carries a representation, and the symmetry that survives is what leaves its value alone. Every prediction here is checked by moving the atoms and asking the detector.

656 sets, every one decided. Every set of one, two, three and four tiles over two colours — sixteen tiles exist in all, so these are complete lists rather than samples — reduced by relabelling the two colour alphabets, and each set decided by the two half-searches. The last column is the one that matters: it is empty. At these sizes there is no room for a set that tiles the plane and admits no periodic tiling, which is the residue undecidability lives in. The smallest aperiodic set is known to have eleven tiles and four colours.

The classification

How much room a hard question needs

No algorithm decides whether a set of tiles covers the plane. Every set of four or fewer tiles over two colours is nevertheless decided here, exhaustively, in under a second — because the sets that defeat the two half-searches have nowhere small to live.

[001]: 8 of 24 faces. The 24 faces of the form {210} of a cubic crystal, stereographically projected, with one zone marked. The great circle is the set of directions perpendicular to the axis [001]; the 8 faces on it are the zone, and each is on it because the integer sum hu + kv + lw is exactly zero. No length and no angle enters that test. Faces of the lower hemisphere are drawn faintly.

Symmetry at work

A zone is a vanishing dot product

Look at a crystal and the obvious thing about it is that the faces run in bands — sets whose edges are all parallel. A face belongs to such a band exactly when three integers multiplied by three others sum to zero. No length enters, no angle enters, and that is why every index on a nineteenth-century mineral specimen is still the index used now.

Six of the ten plane classes have a free invariant ring, and four do not. Every plane point group, with the degrees of the generators of its invariant ring, whether the ring is free, and the relation where it is not. The six generated by their own reflections — 1, m, 2mm, 4mm, 3m and 6mm — have two generators whose degrees multiply to the order of the group, which is Chevalley's theorem checked rather than quoted. The four without reflections — 2, 4, 3 and 6 — need three generators in two variables, so one polynomial relation ties them together, and the degree that relation appears at is printed at the right of its row. Nothing here is a lookup: the generators are found degree by degree as the invariants the earlier ones do not reach, and the relation is the kernel of the map back to polynomials.

What symmetry decides

The groups whose invariants are free

Six of the ten plane classes have an invariant ring generated by two polynomials with no relation between them, and the six are exactly those generated by their own reflections. The degrees of those generators multiply to the order of the group, and their excess counts the reflections.

4mm: which phase depends on where the order parameter points. The plane of a two-dimensional order parameter, with each sampled direction marked by the symmetry that survives when the parameter points that way. The directions along which some operation is preserved are drawn large with a spoke to the centre; the general directions, where nothing survives, are the small faint marks between them. One representation, several phases — and the symmetry does not say which of them a crystal takes. That is decided by terms in an energy, which no symmetry argument supplies: what symmetry supplies is the list a material must choose from.

Into space

Which way the order parameter points

A two-component order parameter has a direction as well as a size, and the symmetry that survives depends on where it points. One representation therefore offers several low-symmetry phases — and symmetry, having produced the list, has nothing to say about which one a crystal takes.

342 unlabelled spots, cell volume 52. A bag of 342 reflection positions with no indices on them, collected out to a bound of 3 on each index. Their pairwise differences generate the reciprocal lattice; a basis of that is taken by integer elimination and then reduced, and the reduced basis is printed. Its determinant is 52, which is the volume of the cell the reflections were computed from — so the cell has been recovered from positions alone, with no intensity used anywhere.

Symmetry at work

A cell from a bag of spots

A single-crystal experiment returns a list of directions with no labels on them. Recovering the cell is recovering the lattice those directions generate, and the whole of it is take differences, reduce, read the answer. What no quantity of data settles is whether the lattice found is the true one or a sublattice of it.

4: three generators in two variables, and the one relation between them. The invariant ring of 4 needs 3 generators, of degrees 2, 4, 4, and three functions of two variables cannot be algebraically independent. The relation between them is found rather than quoted: every monomial in the generators of the degree at which they can first be dependent is written out, the map back to polynomials in x and y is formed, and its kernel is the relation. It is then evaluated at points of the lattice, where all three generators take integer values and the combination comes to exactly zero. A group with a reflection has no such relation, which is the same statement as its ring being free.

What symmetry decides

Three invariants and one relation

Four of the ten plane classes need three invariants where two variables can only support two, so exactly one polynomial identity ties them together. The identity is not recognised or recalled: it is the kernel of a linear map, computed and then checked at lattice points where every term is an integer.

$The twin fraction, recovered from a moment and nothing else (12 atoms). A structure of 12 atoms twinned at each of 6 fractions, with the second moment of its intensity distribution measured and the fraction solved back out of it. The recovery is within a few hundredths as far as thirty per cent — 4 rows here — and 4 of the 6 fractions get a number at all. Beyond thirty per cent the relation flattens: the derivative of 2α(1−α) vanishes at a half, the two roots meet, and a small error in the moment becomes a large one in the fraction. Where the sampled moment falls below 1.5 the quadratic has no real root and the estimate refuses rather than clamping, which is why a nearly perfect twin is the hard case in practice rather than the easy one.$ How it is known

A twin hides in the statistics

A twinned crystal scatters as two orientations at once and the detector cannot separate them. What arrives is a sum of two intensities — and adding two independent quantities narrows a distribution, which is a signature no model of the structure is needed to read.

p4m: freezing Γ3 leaves pmg. The same crystal three times. On the left, a pattern with the full symmetry of p4m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is pmg, of index 4 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical.

Into space

The cell a zone-boundary mode doubles

An order parameter that alternates from cell to cell keeps only half the translations, so the frozen structure has a cell twice as large and reflections that were never there before. The phases at such a wavevector are ±1, so the whole computation stays in exact integers.

The diffuse intensity of an alloy with α₁ = -0.46. The diffuse part of the scattering across the wavevectors an 12 × 12 block can be asked about, one square per wavevector with darkness the intensity. The marked squares are where the average structure scatters — the sharp part, which is what a Bragg reflection is. The diffuse maximum here is at (0.50, 0.50), which is the zone boundary: the alloy is trying to alternate, and a crystal that succeeded would put a sharp reflection exactly there. Summed over every wavevector, the intensity is exactly one per site whatever the correlations are — order moves scattering about, it does not create it.

How it is known

The average scatters sharply and the rest does not

A crystal whose lattice is perfect and whose occupation is not scatters in two parts: the average structure gives Bragg reflections, and the variance is spread over everything between them. The split is exact, the total is one unit per site whatever the disorder does, and an ensemble of n arrangements mislays exactly a fraction 1/n of it.

the kagome net: 4, 8, 14, 18 at the first four shells. The vertices of the kagome net at graph distance one, two, three and four from a chosen vertex, each marked with its distance. Distance here is a number of edges and nothing else — no length enters, and the shells are drawn on the barycentric placement only so that they can be seen. The counts are 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, which is the net's coordination sequence.

Symmetry at work

Counting outwards

How many vertices lie one step from a vertex, two steps, three? The counts settle into a straight line — but for some nets only along the even distances, with a different line along the odd ones, alternating for ever. The period is measured, and it is not always one.

Into space

How chiral, as a number

A group answers one bit: a set either has an improper symmetry or it does not. Two shapes can both be chiral and one of them be a mirror-symmetric thing with a substituent out of place while the other is a helix, and nothing in the classification says which is which. A distance does.

Which order parameters carry a cubic invariant, and therefore cannot grow from zero. Every order parameter of every plane class, with the number of independent cubic invariants it admits. The count is the degree-three coefficient of the Molien series of the representation's image — the same computation the invariant-ring figures make for a different reason — and Landau's condition is that it be zero. Where it is not, a free energy in that order parameter has a term of odd degree, which puts its minimum away from zero the moment the quadratic coefficient does anything at all, so the parameter jumps rather than growing. In the plane exactly two order parameters carry one, and both are the two-dimensional representation of a class with a threefold axis and no sixfold.

What symmetry decides

The cubic term that forbids a continuous change

A crystal may lose a symmetry gradually only if the quantity measuring the loss admits no cubic invariant. Whether it does is the third coefficient of a Molien series — so a question about how a material changes is answered by counting polynomials.

How much one site knows about another, by separation. The Warren–Cowley parameters: the average of the product of the occupations of two sites a given vector apart, over every pair in every configuration. The value at the origin is exactly one — a site always agrees with itself — and it falls away with distance, alternating in sign where the alloy prefers unlike neighbours. These numbers are the whole of what the diffuse scattering measures: its intensity at a wavevector is their Fourier transform, computed here separately and agreeing to the last bits of the arithmetic. Nothing about them requires the crystal to be ordered, and their falling away is what short-range order means.

How it is known

The order a diffuse pattern measures

Where a diffuse maximum sits says what the crystal is trying to become, and its shape is the Fourier transform of how much each site knows about its neighbours. The correlations are a small array of numbers, the intensity is their transform, and neither route to the other loses anything.

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.

Into space

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.

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