Crystallography & Symmetry
patterns derived from their groups
Concept

Ferroelasticity — where it appears

The property of a crystal having two or more orientation states differing in spontaneous strain, which an applied stress can switch between. Whether a transition produces it is a difference of two counts of permitted strain components.

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

m3̅m → 4mm: 6 domain states. The transition from class m3̅m to class 4mm loses 40 of the parent's 48 operations, so the child has index 6 and the crystal comes apart into 6 domain states. Each colour is one state — one coset of 4mm in m3̅m — and each holds the same 8 poles. The lost operations are what carries one state onto another, and they survive in the crystal as the relation between its domains rather than as symmetries of any part of it. The descent changes the crystal system, so the states differ in shape as well as in orientation and the transition is ferroelastic.

How many domains a transition makes is an index

Cool a crystal through a symmetry-lowering transition and it has to choose one of several equally good low-symmetry arrangements. Different parts of it choose differently, and the number of available choices is the index of the new group in the old one — a number available before any crystal is grown, and one of the few predictions in this field that is a count rather than a bound.

applied · Domains 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

Named alongside it

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

Domain stateDomain wallSpontaneous strainStrain tensorAizu criterionCompatibilityCosetCouplingFerroelectricityIndependent componentsIndexInvariant metric

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