a ferroic domain

When a crystal cools through a symmetry-lowering transition, the low-symmetry structure can orient itself in several equally good ways, and different parts of the crystal make different choices. Each uniform region — everywhere pointing the same way — is a domain, and the thin interfaces between them are domain walls. In a ferroic crystal these domains differ in some switchable property (spontaneous polarisation, strain, or magnetisation), and applying the matching field (electric, stress, or magnetic) can flip one domain state into another. Those are ferroic domains.

They form because the high-symmetry parent had operations — say the equivalence of three cube axes — that the low-symmetry daughter loses. Each lost operation maps one domain variant onto another, so the number of distinct domain states equals the index of the group-subgroup relation. In tetragonal barium titanate cooling from cubic, the polarisation can point along +x, minus x, +y, minus y, +z, or minus z: six ferroelectric domain variants, separated by 90-degree and 180-degree walls. Left to itself the crystal splits into many domains because that lowers the electrostatic and elastic energy (a single giant domain would carry huge stray fields or strain).

Ferroic domains are the working parts of a huge class of devices. Poling a ferroelectric (aligning its domains with a field) is what makes a piezoelectric actuator or a ferroelectric memory work; magnetic domains and their walls are the physics of magnetic recording; ferroelastic domain switching underlies shape memory. Understanding a material often means understanding its domains: how many variants symmetry allows, how the walls move, and how a field re-writes the pattern. A single honest reminder: domains are a consequence of broken symmetry, not defects to be eliminated — the parent's lost symmetry guarantees they exist.