the phonon

A phonon is a quantum of sound, the same idea as a photon but for the trembling of a crystal lattice instead of the electromagnetic field. When the atoms of a solid vibrate, they do not jiggle independently; they move together in collective waves that ripple through the whole crystal. Nature only allows the energy in each such wave to change in discrete lumps, and one lump is a phonon. Heat, sound, and much of electrical resistance in a solid are, at bottom, stories about phonons.

Formally, one starts with the atoms coupled by springlike interatomic forces and finds the normal modes of the lattice, the independent patterns of collective oscillation, each labelled by a wavevector k and a branch, with a frequency given by the dispersion relation omega(k). Each normal mode is a quantum harmonic oscillator, so its energy is quantized as (n + 1/2) hbar omega(k); adding one quantum of energy hbar omega(k) means creating one phonon. There are acoustic branches, where omega goes to zero as k goes to zero and neighbouring atoms move nearly in phase (ordinary sound), and optical branches, where atoms in the basis move against each other with a finite frequency even at k = 0. Phonons are bosons, so any number can occupy a mode, following the Bose-Einstein distribution.

Phonons matter everywhere heat and vibration meet quantum mechanics: they give the temperature dependence of a solid's heat capacity (the Debye model, with its T^3 law at low temperature), they carry heat and limit thermal conductivity, they scatter electrons to produce electrical resistance, and, remarkably, they provide the attractive glue that binds electrons into Cooper pairs in conventional superconductors. The honest caveat: a phonon is a quasiparticle, a quantized collective excitation of the medium, not a particle that can exist in empty space. No lattice, no phonon.