Crystallography & Symmetry
patterns derived from their groups
Concept

Elastic constants — where it appears

The coefficients relating a crystal's stress to its strain, and the second derivatives of its energy with respect to strain. Which of them are independent is fixed by the crystal class, and whether the atoms inside the cell are allowed to relax changes their values without changing which ones there are.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

How many components a property may have, at each rank and in each class. One row per plane point group, one column per rank of a fully symmetric property tensor, with the number of independent components in each cell — counted by averaging the tensor over the group index by index, and equal at every entry to the Molien coefficient of that degree. A symmetric property of rank r is a form of degree r, so Neumann's principle and the invariant ring are the same arithmetic in two notations. The last column is the elastic tensor, which is not fully symmetric — symmetric within each pair of indices and under exchanging the pairs — and its counts are not in the table to its left. A property with its own symmetries needs its own average, and that is why the elastic constants are not read off a degree.

The parts a property splits into

A symmetric property of rank r is a polynomial of degree r wearing indices, so the number of components a class permits it is a coefficient of an invariant ring's series. The elastic tensor is not a polynomial in disguise, and its counts are not in that table — which is the most useful thing about it.

point-groups · Neumann's principle Five strains, and which of them move the atoms. A honeycomb of harmonic bonds, strained five ways, with the internal coordinate minimised at fixed cell each time. The shuffle is how far the second atom moves away from where the strain alone would have put it. Which strains produce one is decided before any energy is computed: a strain that leaves the site's three-fold axis intact forces the shuffle to vanish, because the only vector a three-fold rotation of the plane fixes is the zero vector. The prediction and the measurement are in adjacent columns.

The strain the atoms do not follow

The rule that makes an elastic constant computable — deform the cell and move every atom by the same map — is exact for a lattice with one atom in it and wrong for every other, and the reason is a site symmetry rather than a mechanical one. Which strains move the atoms inside the cell is decided by what survives of the site's own group, and a wrong answer here is a constant that is too stiff by a third.

point-groups · Properties Diffuse scattering gathered under every Bragg peak. The one-phonon thermal diffuse scattering of a square lattice with springs to nearest and next-nearest neighbours, over the reciprocal plane from minus 2.5 to 2.5 in each index, with the Bragg reflections marked. The intensity rises as one over the squared distance to every reflection, because the acoustic phonons' frequencies vanish there, and it grows with the square of the scattering vector, so the lobes are strongest far from the origin. The lobes are not round: a small step from (1,0) along k gives 2.7 times the intensity of the same step along h, because it probes transverse phonons, which are softer, and that ratio is what an experiment reads the elastic constants from.

A warm plane crystal has no Bragg peaks

Thermal motion is disorder that moves, and in space it costs a crystal a little of each Bragg peak and gathers it into diffuse scattering underneath. In a plane it costs more than a little. Computed exactly from the phonons of a harmonic square lattice, an atom's mean-square wandering grows by the same amount every time the crystal doubles in size, so no Debye–Waller factor survives, and every Bragg peak fades as a power of the crystal's size — a power proportional to the temperature times the square of the reflection's distance from the origin.

diffraction · Disorder 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.

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.

point-groups · Neumann's principle 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.

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.

point-groups · Neumann's principle

Named alongside it

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

Independent componentsNeumanns principleProperty tensorLaue classPhotoelastic tensorAverage structureAveraging projectorAxial vectorBragg peakChange of basisCharacterDiffuse scattering

All concepts