phonons

Phonons are quantized collective excitations of the lattice vibrations in crystalline solids. In quantum mechanics, vibrational modes of a periodic lattice are described as harmonic oscillators; the quanta of these normal modes are phonons. They behave as quasi-particles carrying energy ħω(q) and crystal momentum ħq, defined within the first Brillouin zone. Phonons are organized into branches: acoustic phonons, for which ω→0 as q→0 and atoms move in phase, and optical phonons, with finite frequencies at q=0 and neighboring atoms moving out of phase. Each branch can be longitudinal or transverse.

The spectrum ω(q) is obtained from the dynamical matrix derived from interatomic forces. Dispersion relations determine how frequency depends on wavevector. In many models the long-wavelength limit gives linear acoustic modes, while optical modes lie at finite frequencies independent of q near zero. Debye model provides a simplified, linear spectrum up to a cutoff to estimate thermodynamic quantities.

Phonons play a central role in determining lattice heat capacity and thermal conductivity. At low temperatures, nonmetallic crystals exhibit a Debye T^3 law for heat capacity; in metals, electron-phonon interactions contribute to specific heat and resistivity. Phonons scatter from defects, boundaries, and interact via Umklapp processes that impede heat flow. Electron-phonon coupling can lead to phenomena such as conventional superconductivity and charge-density waves in some materials.

Experimentally, phonons are studied with inelastic neutron scattering, Raman spectroscopy, infrared spectroscopy, and inelastic X-ray scattering, which reveal phonon dispersion relations and lifetimes.

Applications of phonon physics include thermoelectrics, thermal management, and phononics, where engineered nanostructures and materials manipulate phonon transport to control heat flow.