Crystallography & Symmetry
patterns derived from their groups
Concept

Kagome net — where it appears

The net with three vertices and six edges, obtained by putting a vertex at each midpoint of the triangular net's edges. It carries a level that does not move with the wavevector and a line of mechanisms in reciprocal space, both from the same hexagon.

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

The kagome net's level that does not move. Three levels of the kagome net across the zone, one of them flat. The reason is drawn beside it: a state that alternates in sign round one hexagon and vanishes everywhere else is an exact eigenvector of the adjacency operator at −2, because every site outside the hexagon that touches it touches exactly two of its vertices and those two carry opposite signs. The check is integer arithmetic in a supercell of 27 sites, with a residual of exactly zero. A state confined to one hexagon has no wavevector, and a level made of such states cannot depend on one — which is what a flat line across a zone means.

The level that does not move

Three levels cross the kagome net's zone and one of them is a horizontal line. The reason is a state that alternates in sign round a single hexagon and is exactly zero everywhere else — a solution with no wavevector in it at all, which is why no wavevector can move it.

applied · Nets 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.

WavevectorBloch stateCrystal netCycle spaceFlat bandLocalised stateMechanismRigid unit modeRigidity matrixSelf stressSupercell

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