Crystallography & Symmetry
patterns derived from their groups
Concept

Rigid unit mode — where it appears

A motion of a framework in which each rigid unit rotates and only the joints between them deform. Such modes carry a wavevector, so a mode found in one cell is one point of a family spread across reciprocal space.

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

the kagome net: one of 1 mechanism. An infinitesimal mechanism of the kagome net, drawn as a velocity at every joint. The vector is an exact solution of the rigidity matrix — a set of joint velocities and a rate of change of the cell's metric under which no bar's length changes to first order — with the two rigid translations projected out so that what is left is a motion rather than a shift. Whether it continues into a finite motion is a separate question that a first-order calculation cannot answer, and this collection answers it for one framework by constructing the motion explicitly.

A fold that keeps its symmetry

The kagome framework has exactly one mechanism, and it does not stop at first order. Every triangle turns, alternate ones the other way, the cell shrinks to half its size, and not one bar changes length — and the count that found the mechanism cannot see how many there really are.

applied · Rigidity the kagome net: 34 of 144 wavevectors carry a mechanism. The zone of the kagome net, with a mark at every wavevector whose rigidity matrix drops rank — which is to say at every wavevector that carries a motion of the bars. There are few of them and they are isolated, so enlarging the cell adds mechanisms slowly. The ranks at the half-integer wavevectors are exact; the others are computed with a stated tolerance, because the matrix there has genuinely complex entries.

A mechanism that is a wave

The framework essays found the kagome net's mechanism count growing with the cell it was looked for in, and recorded it as a finding without an explanation. Here is the explanation: the motions lie along lines in reciprocal space, and a larger cell samples a line at more places.

applied · Rigidity

Named alongside it

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

MechanismSupercellDisplacive transitionFinite flexKagomeKagome netRigidity matrixSelf stressSoft modeWavevector

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