The state on the end of a chain of 60

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.

Seven 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.

A band with no structure in it

The advantage, drawn. One eigenvector of a sixty-site chain whose end site differs, computed directly, localised on the end and outside the band. A calculation organised around a wavevector has to work to describe this state, because it is not a state of definite wavevector at all — it is a superposition of them, and its whole character is the localisation that the label was chosen to make invisible.

A defect is a level in the gap

One eigenvector of a sixty-site chain whose thirtieth site has had its energy raised. The state has left the band, and its amplitude is concentrated on a handful of sites around the one that was changed. The figure prints how many sites it lives on and whether it is inside the band or outside.

The same measurement at the end of a chain rather than in the middle. The decay is visible directly: a large amplitude on the first site, smaller on the second, and essentially nothing past the sixth. The number of sites the state occupies and the distance over which it decays are both printed on the figure.

The same defect at twice the depth. The level has gone further out of the band and the state has drawn in around the site — about four atoms rather than nine — and both changes are continuous in the depth. Nothing here has a value a theorem fixes, which is the property that separates a defect from a vacancy.

The end is the hardest place to bind

The same end site at a well only half as deep, which is the case the whole essay turns on. Nothing has left the band: the lowest state still runs the length of the chain and merely leans towards the end, because a well at an end has to be about one β deep before it binds at all. In the middle of the same chain, a well of any depth whatever binds something.

The state the end site has to compete with: the lowest level of an unmodified chain, with the same sign everywhere and its amplitude tapering to nothing at both ends. That taper is the whole reason the end has a threshold — it is the part of the chain where the most-bonding state is weakest, so it is the part a competing state gains least by sitting on.

Above the threshold: an end site raised by 1.2β, which is enough. The state has left the band and lives on about five and a half sites, decaying by a factor of e every two and two-thirds sites away from the end. At 0.9β nothing at all would be outside the band and this figure would show a state spread over the whole chain.

Two defects, and the level between them

The bound state of a single impurity at the middle of a chain of 61. The amplitude falls geometrically away from the impurity site, and the participation ratio — how many sites the state effectively occupies — is 1.9, so the state lives on about two atoms.

A vacancy is not an impurity

An impurity’s state: one site’s energy changed by 1.2, and a state that lives on a handful of atoms round it. Everything about this picture moves with the strength — how far out of the band the level lies, how tightly the state is bound, how much of the structure it occupies. Nothing in it has a value a theorem fixes.

One defect is a level, many are a band

The state itself, on a chain of sixty with one site deepened by three. The amplitude falls off geometrically from the defect and alternates in sign, which is what the closed form says it must.

A particle in a box the alloy made

The simplest case: one site changed, one level pulled into the gap, and a state on that site and its neighbours. Everything above is that picture with the single site replaced by a run the arrangement made.