Molecular Geometry & Bonding
what the shape decides
Concept

Dimensionality — where it appears

The number of independent directions a structure extends in: one for a chain, two for a plane, three for a solid. It sets how many sites lie within a given distance, growing as a length, an area or a volume, while the coupling law, the density of levels and their spacing stay the same — so a resonance a chain reaches only at absurd lengths saturates a small patch of a plane.

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

On a plane the law is an exponential of a square. The share of runs of six with a twin inside a given distance, measured in generated square nets and drawn against the plane's own closed form. Along a chain there are s sites within a separation s, so the chance of no twin is (1 − ρ)ˢ; on a plane the count within a distance grows as the area, so it is exp(−ρπs²). The curve is that form with nothing fitted, and the marks are three nets of a quarter of a million sites each.

A plane spends the same reach on a disc

Everything this argument has computed about defect bands is one-dimensional, and the one-dimensional part is not the coupling law or the level spacing — it is the count of how many candidate twins lie within a given distance. On a plane that count grows as the area, so the same reach that finds one twin in a chain finds a disc full of them, and half the runs of six are in a resonant pair at 10³·⁹ sites rather than 10⁴⁰·⁸. That is thirty-seven orders of magnitude, it is a patch ninety-two sites square, and it means no two- or three-dimensional material has a partly resonant defect band at any size.

solids · Defect Where a twin couples strongly enough, drawn rather than assumed. Every displacement of a second run of six from a first, in a square net with the chain's site energies, as a dot whose size grows with the logarithm of the splitting their two levels make. Dark dots couple above the level spacing at the size where half the runs have such a twin; there are 174 of them, the measured area. The dashed circle is the disc the plane's half-way length was computed with, radius 7.52 sites. The measured region is a wide band along the row and a ragged one across it, with holes where an offset's parity cancels the coupling.

The disc was a band with holes in it

The plane's defect band was built from a chain's coupling law spent on a disc of twins. Computed instead from the plane's own Green's function, two runs of six couple with a direction and a parity: a twin in the same row couples thirty times as strongly as one stacked above at the same distance between centres, and offsets across rows alternate between strong and weak. The region of twins strong enough to matter is a ragged band eleven sites long and six high, and half the runs are resonant at 10³·⁰⁹ sites rather than 10³·⁹³. The thirty-seven decades between a chain and a plane survive and grow.

solids · Defect

Named alongside it

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

Closed formDefect stateDisorderModel limitTight-binding modelsThermodynamic limit

All concepts