The outer layer, and the one behind it

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.

One essay draws 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.

A surface is not a count of broken bonds

The binding a site receives in each of the first eight layers of an open block, against the value in the bulk. The outer layer is well below it. The layer immediately behind it is slightly above it. Everything deeper is the bulk value to four figures.

Three numbers for one quantity. A surface atom keeps five of its six bonds, which is 83.33 per cent. The second-moment rule says it should keep the square root of that, which is 91.29 per cent. What it actually keeps is 91.24 per cent.

The same profile on a larger block. The outer layer is the same fraction of the way to the bulk value and the layer behind it is nearer, so the reach of the boundary is a length in layers rather than a fraction of the block — which is what makes the surface energy per unit area a well-defined quantity at all.