Crystallography & Symmetry
patterns derived from their groups
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The theme: The group acts on functions — page 3

A rotation carries an atom onto an atom, and it carries a density, a displacement or a wave onto another one. The second action is linear, so the group becomes a set of matrices, and questions that look analytic — which levels must coincide, how many independent components a property has, how a shell of neighbours splits — become counts of whole numbers.
Every fibre texture is a point in one plane. The two numbers that decide every coefficient of a polar fibre, the mean first and third Legendre polynomials of the grains' tilt, as a plane. A cone of grains at a single tilt is a point on the curve; any other texture is an average of such points and lies in the curve's convex hull. Spreads about the axis trace the inner path. For a grain with a large shear coefficient the largest d33 of any texture is at a point of the curve, the cone at 51°, 223 pC/N; the best spread about the axis reaches 168. What symmetry decides

The best texture is a cone

Every fibre texture of a polar grain, however its orientations are distributed about the axis, reduces for piezoelectric purposes to two numbers, the averages of the first and third Legendre polynomials of the tilt. The textures fill a region of a plane bounded by one curve, and a coefficient is a straight line across that plane, so its best texture sits on the curve: a cone of grains at one angle. For a grain with a large shear coefficient the best cone lies at 51° and gives 223 pC/N, where the best spread about the axis gives 168.

 Every crystal class, as a reflection class and its relative invariants. The thirty-two crystal classes, each with the smallest reflection class containing it after a relabelling of the axes, the index, and the degrees of the secondary invariants that make its invariant ring a free module over the reflection class's polynomial ring. Twelve classes are reflection classes themselves; seventeen have index two and one secondary invariant besides 1; three — 1̄, 4̄ and 3̄ — have index four and three. What symmetry decides

What the twenty lack is a product of mirrors

Twelve crystal classes have invariant rings generated by three independent polynomials, and they are the twelve generated by their own mirrors. The other twenty were left as the classes whose counts need a correction. The correction has a simple shape. Each of the twenty sits inside one of the twelve, at index two or four, and its invariants are the larger class's invariants times one or three extra polynomials. Each extra polynomial is a product of the mirror planes the smaller class lacks. For the rotations of the cube it is a single polynomial of degree nine, the product of all nine mirrors of the full cubic class.

 Almost no homometric pair on the torus is a reversed factor. For ten plane groups at several numbers of atoms on a 6 × 6 torus, the structures, the pairs of structures sharing one Patterson, how many of those pairs reversing one factor of a sumset explains, and how many are a patchwork of Galois classes. Factor reversal explains 2 of 3522; the patchwork explains all 3522. How it is known

A homometric twin is the structure moved class by class

On a ring, every homometric pair a small search found came from one construction: factor the structure as a sumset and reverse one factor. Imposing a plane group on a torus made homometry more common, and the obvious guess was that symmetry makes structures easier to factor. Tested pair by pair, factor reversal explains two of 3,522 homometric pairs on a 6 × 6 torus, and none at all at six atoms. What explains all 3,522 is a different mechanism: the partner is the structure's own Fourier transform moved independently on each Galois class of characters, by a shift, a sign or an inversion.

 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. Into space

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.

 The fifty-eight black-and-white classes, sorted by reflection class. Every black-and-white magnetic point group, grouped by the smallest reflection class containing its underlying crystal class, with the degrees of the mirror products that generate its relative invariants for a polar and for an axial time-odd quantity. Every class's two series are exactly those sums of mirror products, and the number of generators is the class's index in its reflection class: one, two or four. What symmetry decides

Every magnetic class counts its properties by mirrors

The invariants of every crystal class turned out to be products of mirrors of a reflection class, times that reflection class's own free invariants. A magnetic class asks for polynomials that some of its operations negate, which is harder in principle. It is not harder in fact: all fifty-eight black-and-white classes, for a time-odd quantity that is polar and for one that is axial, are counted by mirror products — provided the reflection class contains the whole class and not only its unprimed part, which for thirty of them it does not.

 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. What symmetry decides

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.

 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. What symmetry decides

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.

 Photoelastic constants of the eleven Laue classes, split into four parts. The number of independent photoelastic constants each Laue class permits, from 36 for 1̅ to 3 for m3̅m, drawn as squares and split into the elastic tensor's part, which is symmetric under exchanging the two index pairs, and the extra fifteen-dimensional part, which divides into an axial vector, a second quadrupole and an octupole. The axial vector survives only in 1̅, 2/m, 4/m, 3̅ and 6/m; the octupole is what separates m3̅ from m3̅m; and in m3̅m nothing extra survives, so the two counts agree at three. What symmetry decides

Fifteen constants an energy would have forbidden

A strain changes a crystal's refractive index through a fourth-rank tensor with the elastic tensor's index pairs and without its one remaining symmetry, because no energy is differentiated twice to make it. The missing symmetry is worth fifteen components. They split into an axial vector, a quadrupole and an octupole, and each does something the elastic count cannot: the octupole gives 23 and m3̅ a fourth constant, and the axial vector is a part of the tensor no turn of the axes removes.

 The ninety magnetic classes that are not grey, sorted by their centre. The ninety magnetic point groups that are not grey, in three columns by whether they contain the inversion, the inversion combined with time reversal, or no centre at all, each tile shaded by what a stress can do: magnetise a crystal that is already magnetised (31 classes), magnetise one that has no moment at rest (35), or nothing. A dot marks the classes that also permit the magnetoelectric effect. No class with the primed inversion is piezomagnetic and no class with the plain one is magnetoelectric. What symmetry decides

Thirty-five classes a squeeze can magnetise

A magnetisation proportional to a stress is permitted in sixty-six of the hundred and twenty-two magnetic classes. Thirty-one are the classes that may be magnetised already; the other thirty-five may have no moment at rest and acquire one when squeezed, and in them a stress tells apart two domains that differ only by the direction of time. Among the ninety classes that are not grey, it is forbidden only by a primed centre and in the three ordinary cubic classes with four-fold operations.

 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. What symmetry decides

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.

 The constants a turn of the axes removes, class by class and property by property. The thirteen crystal classes whose operations all commute with some turn of the axes, against six properties: pyroelectric, dielectric, optical activity, piezoelectric, elastic and photoelastic, and all six together. Each cell gives the printed number of constants, and where a turn removes some, the number left. Triclinic classes have three turns and lose up to three; the others have one and lose at most one. Classes 6 and 6/m lose nothing in any property, and the last column loses the turns once, however many properties could spend them. What symmetry decides

A six-fold crystal hides its frame until rank six

Thirteen crystal classes can be turned without moving any of their operations, and the tables print each property's constants in one frame among the many such a turn allows. Spending the turn removes a constant only where a property has a part that turns with the frame, and an n-fold axis keeps such a part only at rank n or above. So a monoclinic crystal's free angle shows in its dielectric tensor, a tetragonal one's first in its elastic constants, and a hexagonal one's in nothing anybody measures. And a crystal has one frame, not one per property: the angle is spent once.

 Where the two-fold axis sits decides what the coloured optic axes keep. Interference-figure positions of the two optic axes from 400 to 700 nm for four model crystals seen down the acute bisectrix. Orthorhombic: both mirrors survive and only the angle between the axes changes. Two-fold axis along the optic normal: the axes stay on one line and slide unequally (inclined dispersion), one mirror along that line. Along the obtuse bisectrix: the axial plane stays parallel and shifts (horizontal), one mirror across it. Along the acute bisectrix: the axial plane turns about the centre (crossed), a half-turn and no mirror. What symmetry decides

Three places for the two-fold axis

A monoclinic crystal fixes one direction of its indicatrix, the two-fold axis, and an ellipsoid has three principal directions for it to be. Which one it is decides what an interference figure keeps in white light: a single mirror along the plane of the optic axes, a single mirror across it, or a half-turn and no mirror at all. Mineralogy calls the three inclined, horizontal and crossed dispersion. They are one subgroup computed three times, and in light of a single colour all three disappear.

 Down one optic axis, at three colours, for two crystals. Interference figures of two model monoclinic crystals, inclined above and crossed below, each a plate cut across the optic axis at the sodium line, at 450, 589 and 650 nm. In the inclined crystal the melatope slides along the horizontal mirror and the figure stays symmetric across it. In the crossed crystal the melatope leaves the line away from sodium, and the pattern loses the mirror. What symmetry decides

The mirror a plate keeps in one colour

Cut a plate of a crystal across one optic axis, and in the light the plate was cut for its interference figure has a perfect mirror, whatever the crystal's class. Change the colour and the mirror survives only if the crystal really has one. Computed ring by ring, five model crystals give the prediction exactly, including the part the eye would miss: the mirror does not come back when the line is moved to follow the melatope.

 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. Into space

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.

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