Molecular Geometry & Bonding
what the shape decides
Figure

A half-filled ring's cheapest excitation goes to zero

The energy of the smallest available excitation of a half-filled ring, against the number of atoms, on log axes. Every doubling at least halves it, so in the limit there is no smallest excitation — which is what a metal is, before any band picture is drawn.
A half-filled ring's cheapest excitation goes to zero. The energy of the smallest available excitation of a half-filled ring, against the number of atoms, on log axes. Every doubling at least halves it, so in the limit there is no smallest excitation — which is what a metal is, before any band picture is drawn.

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.

Four 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

What a metal actually is

The smallest excitation available to a half-filled ring, at five sizes. Each doubling of the ring at least halves it: one β at twelve atoms, 0.065β at a hundred and ninety-two. There is no lower bound, which is the definition being met rather than illustrated.

The metal a thermometer cannot find

The sequence the words were defined by, taken out to three hundred and eighty-four sites. A half-filled ring’s cheapest excitation falls with no lower bound, and every number in this essay is a Fermi–Dirac occupation of a finite list of levels from a finite matrix — with no periodicity assumed anywhere.

The ratio of the carriers in a uniform ring to those in an alternating one of the same size, at four sizes and six temperatures. The top-left corner and the bottom row are both places where the ratio is small, for opposite reasons, and the word only means anything in between.

Carriers per site against temperature for the two rings of three hundred and twenty-two, on a logarithmic scale. Twelve orders of magnitude apart at the cold end; a factor of four at the hot end. The temperature at which the gapped ring carries half what the gapless one does is a quarter of its gap, and nothing to do with its size.

The carriers a distortion was hiding

The rigid-ring measurement: carriers per site against temperature for a ring with a gap and one without. Everything in it is right about the structures it describes, and this essay is about whether a ring of 4m is one of them.

The exponent was the window's

What the distortion was doing to the carriers, which is what the original measurement was about. The exponent is a detail of how the alternation goes; the carriers going with it is the finding that measurement made.

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