phonon

A phonon is a quantized normal mode of vibration in a crystalline lattice, describing the collective excitations of the atoms in the solid. Phonons are quasiparticles that obey Bose-Einstein statistics and carry energy and momentum.

In a crystal, vibrational modes form branches labeled by wavevector k within the Brillouin zone. Dispersion relations ωs(k) describe the frequency of each branch s. There are acoustic branches, in which ω → 0 as k → 0, and optical branches, with finite ω at k = 0; modes can be longitudinal or transverse.

The conventional theory uses the harmonic approximation, treating each mode as a quantum harmonic oscillator with energy E = ℏω(n + 1/2). Phonons are created and annihilated by operators that raise or lower the number of quanta in a given mode. In the Debye model, a single acoustic-like spectrum is assumed up to a Debye frequency ωD, with the Debye temperature ΘD = ℏωD/kB.

Phonons govern thermal properties and lattice dynamics. They contribute to the heat capacity and to thermal conductivity through phonon transport; anharmonicity leads to phonon-phonon scattering, including Umklapp processes that limit heat flow. Phonons also mediate interactions between electrons, playing a key role in conventional superconductivity via electron-phonon coupling and in determining electrical resistivity.

Phonons are probed experimentally by neutron scattering, Raman and infrared spectroscopy, and inelastic X-ray scattering, which reveal dispersion relations, lifetimes, and interaction strengths. The concept extends to nanostructures and low-dimensional materials, where altered phonon spectra influence thermoelectric performance and thermal management.