Molecular Geometry & Bonding
what the shape decides
Concept

Doping — where it appears

Changing the number of electrons in a structure without changing the structure. It moves the filling, which decides whether a chain is a metal and which period of distortion it is unstable to.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The state on a defect at site 31 of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 31 has its energy raised by 1.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

A defect is a level in the gap

Change one site's energy in a chain of a hundred and sixty and a level leaves the band, carrying a state that lives on a handful of atoms. In one dimension it happens for any change however small — the threshold measured on chains of forty, eighty and a hundred and sixty halves with every doubling, so there is no threshold at all.

solids · Defect Two impurities at 2 to 14 sites apart. The splitting between the two levels a pair of impurities of strength -2β pulls out of a chain of 61, against how far apart they are, on a logarithmic scale. It falls by a constant factor per site of separation, and that factor is the decay of the isolated bound state computed from its energy alone. The two levels close on the single impurity's level as the pair separates.

Two defects, and the level between them

One impurity pulls a level out of a band at exactly −√(h²+4). Two of them pull out a pair, split by an amount that falls by a factor of 0.41421 for every site of separation — which is √2 − 1, predicted from the isolated level's energy alone and measured to six figures.

solids · Defect One impurity is a level; many are a band. The impurity levels of a ring of 160 with sites of depth -3, drawn as a bar from the lowest to the highest, against the fraction of sites that are impurities. At the lowest concentration every level is at the same energy and the bar has no height at all. By 30 per cent the levels span 2.44 and have closed to within 0.25 of the host band, which is shaded.

One defect is a level, many are a band

A single deepened site in a chain pulls one state out of the band to −√(h² + 4), exactly, and holds it on 1.42 sites. Put in more and the levels spread: at one site in ten they span 1.45 in the same units and have closed to within 0.69 of the host band, and above one site in eight the count of levels stops matching the count of defects, because two defects on neighbouring sites push one of their pair back into the band.

solids · Defect A mixture is not the average of its ends. The binding per site of a square net whose sites are of two kinds, against how many of each. The straight line is arithmetic rather than a fit: a structure of one kind only has every level shifted by ±δ, so the two ends and the line between them are known before anything is diagonalised. Every mixture lies above it — more bound — by as much as 0.43 per site at the middle, and that departure is the whole of what makes an ordered compound worth forming.

A mixture is not the average of its ends

Half of a structure's sites raised and half lowered, and the line between the two pure ends is arithmetic — no diagonalisation needed. Every mixture lies below it, by 0.42987 per site for the ordered arrangement and 0.23690 for the segregated one at the same composition, so composition fixes neither the binding nor the gap. And the departure is not quadratic in the contrast: it grows as its 1.79 power in a chain and its 1.28 power on a square net.

solids · Cohesion How many places a local search can stop. The number of distinct arrangements a steepest-ascent search settles at, for each net and contrast, with the fraction of starts reaching the best of them written beside it. 16 sites, contrast 1: 2 from 200 starts, best reached 67 per cent of the time; 16 sites, contrast 4: 2 from 200 starts, best reached 59 per cent of the time; 36 sites, contrast 1: 4 from 20 starts, best reached 80 per cent of the time; 36 sites, contrast 4: 12 from 20 starts, best reached 10 per cent of the time. The larger net at the larger contrast is a different kind of landscape.

Twelve basins where there were two

Sixteen sites can be searched exhaustively and thirty-six cannot, so the only instrument available past about twenty sites is a local search — and how much weaker it is has never been measured against the answer on a net where both can be run. On sixteen it is barely weaker at all. On thirty-six it fails nine starts in ten, and the arrangement it is beaten by has fewer unlike bonds than the one it starts from.

solids · Cohesion The coupling at the closest pair, and at the typical one. For runs of four to eight low sites at half concentration: the splitting between two runs one site apart, which is the largest coupling any chain can produce, and the splitting at the separation two runs typically have. Both on a logarithmic axis spanning 173 decades. A band width taken from the second is not a small number; it is not a number.

A band that is a hundred and seventy decades of nothing

The chain length at which a defect's levels become a band can be located by asking when the coupling between the closest pair exceeds the level spacing. A band also has a width, and a width is set by the typical coupling rather than the closest one. For runs of eight at half concentration those two numbers differ by a hundred and seventy-three orders of magnitude.

solids · Defect

Named alongside it

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

Bands in a solidTight-binding modelsLocalisationBand edgeDefectLevel spacingParticipation ratioThresholdBand widthClosed formCohesionDefect state

All concepts