What holds matter together, per pair

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.

Eight 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.

What holds a solid together

Four kinds of interaction between two units of matter, each computed from the model named beside it, on a logarithmic energy axis. The ordering is computed rather than assumed, and it is not the one usually taught.

The same four interactions with the denominator changed. Per pair of atoms at contact they span a factor of five hundred; per electron pair involved they span far less, because the interactions that look weakest are the ones spread over fewest electrons. Which of the two numbers is the interesting one depends on the question, and the ordering is not the same in both.

The same four on one scale, with the ratio from top to bottom printed. Between an ion pair at contact and two argon atoms at theirs is a factor of 517, and the materials at each end of that range are different enough that they are studied by different people.

The lattice sum that depends on the order of adding

The four kinds of binding on one scale. Three of them are essentially complete as pair quantities. The ionic entry is one term of a series that runs over the whole crystal, and this essay is about why that series is difficult.

The same four interactions with the label taken off, so the magnitudes can be compared directly. Three of the four are bounded sums over neighbours and one is a series that does not converge absolutely — and it is the only one of the four a neighbour sum cannot compute, which is what the essay is about.

What a bounded interaction buys, drawn against the contrast it is computed at. Every one of the three computable columns is a sum with a range, so its value is a number rather than a limit of partial sums — and that is the property the ionic column lacks and the reason its arithmetic is a different subject.

Two structures with the same neighbours

Measured cohesive energies against what a count of bonds would predict. The scatter around the square-root line is the size of everything the second moment does not fix, and this essay’s twenty-two per cent is the same quantity computed rather than fitted.

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 the arithmetic above; the curve is a hundred diagonalisations. Every mixture is more bound than the line, and by 0.42987 per site at the middle of the range.

The departure against the contrast between the two ends, which is the shape of the whole effect. It is never a fixed share: a small contrast gives a departure quadratic in it and a large one gives a departure that saturates, so no single number describes how far a mixture is from the average of its ends.

One composition, three arrangements, three cohesions — ordered by how many bonds join unlike sites, which is a count of edges and not a spectral quantity. Two hundred unlike bonds are worth 0.42987 per site, ninety-four are worth 0.26968 and twenty are worth 0.23690.

Twelve basins where there were two

How many distinct arrangements a steepest-ascent search settles at, for two nets and two contrasts, with the fraction of starts that reach the best of them. The fourth bar is a different kind of landscape from the first three.

The number of arrangements at a fixed composition against the size of the net, on a logarithmic axis, with the number of diagonalisations one ascent actually used marked where it is known. The two lines cross before thirty-six sites.

All 8,008 arrangements of six raised sites on a wrapped square net of sixteen at a contrast of four, as a histogram of binding per site, with the two energies a local search stops at marked. The right-hand one is the best of all 8,008.

The composition that is hard is not the full one

The share of random starts reaching the best arrangement found, against the fraction of sites raised, for two nets at two contrasts. Open rings are half filling.

The number of distinct local optima the ascents find, against the fraction of sites raised.

The number of arrangements at each composition, which peaks at half filling, beside the share of starts that reach the best one.

A net with no two-colouring

Sixteen sites wrapped into a square net and into a triangular one, with the wrapping bonds left undrawn.

The ceiling on unlike bonds in a triangular net, from counting triangles alone.

The half-filled composition on each net at each contrast, with the unlike-bond count and the ceiling.

The gap follows the winner late

Every basin found at half filling on the triangular net, at eleven contrasts, placed by its gap; winners filled, squares with thirty-two unlike bonds and circles with thirty.

For both nets at two contrasts and five compositions: basins found, the winner’s gap and the widest gap among the basins.

The binding per site of the thirty-two-bond contender minus the thirty-bond one, across the contrast.