Crystallography & Symmetry
patterns derived from their groups
Concept

Cleavage — where it appears

The splitting of a crystal along particular planes, whose faces carry the symmetry its space group leaves on those planes. Which symmetry that is depends on where the cut was made, and two conditions decide it.

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

The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it.

Why a crystal face carries small whole numbers

A crystal face is flat because it lies on a plane of lattice points, and its orientation is therefore named by three integers rather than by two angles. That the integers exist is a theorem about lattices; that they are usually smaller than four is a separate claim about growth, and the two are routinely run together.

applied · Forms The symmetry of a cut through Pnma. Every height in one cell, and the number of operations of Pnma that map the plane at that height to itself. The answer is 2 almost everywhere and rises to 4 at the special heights, where the plane group named above the spike is what a reader looking down at that surface would see. The rule is two conditions and no more: the operation must not tilt the plane, and the plane must come back to its own height — so a twofold axis lying in the plane survives at two heights per cell and a screw axis along the normal survives nowhere.

What a cleave leaves

A surface is a crystal that has been cut, and the symmetry it presents is what the space group leaves of itself on that plane. Two conditions decide it — the plane must not tilt, and it must come back to its own height — and the answer changes with where the cut was made.

classification · Subperiodic

Named alongside it

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

CentringCrystal habitFace poleLattice planeLayer groupMiller indicesRational indicesReciprocal latticeScrew axisSectional layer groupSite symmetrySubperiodic group

All concepts