Molecular Geometry & Bonding
what the shape decides
Figure

Where the two halves of an alloy come apart

The gap at the centre of a binary alloy's band against the contrast between its two components, for a chain of 2048 sites arranged three ways. The ordered arrangement's gap is exactly twice the contrast and opens at once; the segregated one's is exactly the contrast less the band width and opens at 2; and the random one, which has no closed form, opens a little before the segregated one and stays a little wider. Below the openings the curves sit at one level spacing rather than at zero, which is what a finite chain has instead of a gap.
Where the two halves of an alloy come apart. The gap at the centre of a binary alloy's band against the contrast between its two components, for a chain of 2048 sites arranged three ways. The ordered arrangement's gap is exactly twice the contrast and opens at once; the segregated one's is exactly the contrast less the band width and opens at 2; and the random one, which has no closed form, opens a little before the segregated one and stays a little wider. Below the openings the curves sit at one level spacing rather than at zero, which is what a finite chain has instead of a gap.

One of the figures on extended structures: Chains and rings taken far enough to behave like solids — bands, gaps, fillings, defects and ends — every one of them a finite matrix diagonalised, with no lattice anywhere in the argument.

Three essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

In the essays

Two bands, if the chain is short enough

The gap at the centre of a binary alloy’s band against the contrast between its components, on a chain of 2,048 sites arranged three ways. Two of the three curves are straight lines with known slopes; the third is not, and it is the one this essay is about.

The same three arrangements, with the variance they share against the fourth moment they do not. A measurement that pins the second moment has said something about the composition and nothing about the structure; the shape is where the difference is, and it is a factor of nearly two across the three.

The three densities of states at a contrast of 1.6, drawn on one energy axis. Same centre, same width in the sense of a second moment, three different distributions of the states within it — and only the ordered one has opened a gap at this contrast.

The arrangement a count cannot pick

The same point in the other direction: three arrangements of one composition with the same variance and three different shapes. A summary statistic that cannot tell arrangements apart and a count that ranks them wrongly are two halves of one complaint.

A particle in a box the alloy made

The forty levels nearest the gap centre of five random alloys, against the length of the run of like sites each sits on. The line is a closed form with nothing fitted in it. Two of the forty are somewhere else, and they are the interesting ones.

Two of them drawn on the sites they live on, with the raised sites shaded. The first is a half-sine on a run of thirteen and stops where the run stops. The second is on a run of six and is nothing like a half-sine.

The weight a gap state keeps inside its own run, against the contrast between the components. A gap eleven times deeper, and a flat line.

Every figure · Every orbital, by what it encloses · All essays