Crystallography & Symmetry
patterns derived from their groups
Concept

Spontaneous strain — where it appears

The distortion a crystal acquires at a transition without being pushed, being a strain component the child permits and the parent did not. Its size is a fact about the material and its existence is a difference of two symmetry counts.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Which descents change the shape of the cell, and into how many shapes. Each descent the modes produced, with the number of independent strain components the parent class permits and the number the child permits. A transition is ferroelastic exactly when the second is larger — the child leaves alone a distortion the parent moves — and the difference is a spontaneous strain the crystal acquires without being pushed. The count of distinct shapes is the orbit of that strain under the parent, which can be smaller than the number of domains: two domains may differ in something a change of shape cannot show. Every count here is a rank of an averaged set of quadratic forms, computed twice — once by averaging, once from a character.

The strain that arrives with the transition

A crystal that loses symmetry usually changes shape, and whether it does is a subtraction: how many strain components the child permits, minus how many the parent did. The difference is a distortion nobody applied, and it is what makes a domain visible in a microscope.

applied · Domains 3m → 1: the two directions a wall between domains may take. The difference between the strains of two domains, sampled around a circle of directions: the first colour where that direction is stretched, the second where it is compressed. The two solid lines are the directions where it is neither, and those are the only orientations a straight wall between the two domains can take without straining itself — Sapriel's condition, one dimension down from the planes it is usually written for. There are exactly two, and that is not luck: the two domains are images of one another under the parent group, so their strains have the same area change and their difference changes no area at all. A form that changes no area takes both signs, and its zero set is a pair of directions.

The walls a strain permits

Two domains of different shape can only meet along a line neither of them stretches. That condition is a quadratic in a direction, so a pair of domains has exactly two permissible walls — and the reason there are always two rather than sometimes none is that their strains differ by no area at all.

applied · Domains Every wall is a frieze. Each ferroelastic descent, with the frieze group of each of the two walls its domains permit. A wall is periodic along its length and bounded across it, so its symmetry group is one of the seven — the classification this collection derived early as the same argument on a strip, arriving here as a fact about interfaces. The last column counts the operations in the wall's group that exchange the two domains rather than fixing them: a wall is unchanged by having its sides swapped, so those belong to it, and they are why a wall is often more symmetric than either domain.

The wall has a group of its own

A boundary between two domains is periodic along its length and bounded across it, so its symmetry is a frieze. The seven, derived here early on as an exercise on a strip, turn out to be the classification of interfaces.

applied · Domains

Named alongside it

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

Domain stateDomain wallFerroelasticityStrain tensorTwin lawAizu criterionCompatibilityCosetCouplingFerroelasticFrieze groupIndependent components

All concepts