The collective mode: an electron ring sliding over an ion ring
n protons on an inner ring and
n electrons on an outer ring, neutral overall. The
electron ring is rotated relative to the ions and the energy recorded.
E(φ) is the pinning
energy, and its amplitude is the resistance.
Why this and not a single carrier. Conduction in a metal is not one electron moving past
stationary others — it is the whole distribution shifting while the ions stay. Every previous
attempt here imposed a single localised carrier and measured what it costs to move; the ring
experiment showed the framework has no such object, so those costs were the price of an artificial
constraint. In the collective mode each electron moves only a fraction of a spacing and nothing
squeezes past anything.
What it is, in standard terms. This is the pinning of a charge-density wave to a lattice, and
the Frenkel–Kontorova picture of friction. Flat → the sea slides freely, a conductor.
Corrugated → pinned, with the amplitude the barrier to sliding.
Built-in check. By symmetry E(φ) must repeat when the electron ring is rotated by one
site spacing, so phi=0 and phi=1 must agree. If they do not, the
calculation is wrong.
URL: ?n=6 ions and electrons, &a=2.2 ion arc spacing,
&dr=1.2 electron-ring radius minus ion-ring radius,
&phi=0 relative rotation in site spacings (0 = electrons over ions,
0.5 = electrons between ions), &bs=0.25, &reset clears.
relaxing…