Crystallography & Symmetry
patterns derived from their groups
Concept

Strain — where it appears

The part of a deformation that changes lengths and angles, as opposed to the rotation that does not. Every invertible deformation splits uniquely into a rotation and a symmetric positive-definite stretch, and only the stretch can obstruct two regions from meeting across a plane.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Three variants, and not one undistorted plane. The three tetragonal variants a cubic parent produces, with the principal stretches of each. Every one of them has the same three numbers in a different order, and the middle one is not one — so none of the three leaves any plane undistorted, and none of them can meet the parent phase across an interface. That is the difficulty the whole of the crystallographic theory of martensite exists to resolve, and it is visible in one column.

The plane a deformation leaves alone

Two differently deformed regions can meet across a plane only if that plane is deformed identically from both sides — which forces the two deformations to differ by a rank-one term. Multiplying each side by its own transpose removes the rotation and leaves a condition on a signature: one positive eigenvalue, one negative, one exactly zero. In that form the classical rule that the middle principal stretch must be one is not quoted but derived, and it says that no single variant of a cubic-to-tetragonal transition can meet its parent at all.

applied · Domains Where the laminate's middle eigenvalue crosses zero. The middle eigenvalue of FᵀF − I for the average deformation of a twinned laminate, against the volume fraction of one variant. At both ends the laminate is a single variant and the value is well away from zero; in between it crosses, twice, and each crossing is a volume fraction at which the laminate can meet the parent phase across a plane. The two roots are complementary, which is the same plate with the two variants exchanged.

The plate that only fits when it is twinned

No single variant of a cubic-to-tetragonal transition can meet its parent across a plane. A fine mixture of two variants can, because its average deformation carries a free parameter — the volume fraction — and that parameter passes through the compatibility condition twice. Sweeping it gives the two fractions, the two habit planes, and one inequality: the plate exists exactly when the two principal stretches satisfy η₁² + η₃² ≤ 2.

applied · Domains 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

Named alongside it

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

CompatibilityDomain wallTwinEigenvalueElastic constantsHabit planeSite symmetryTensor

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