Atoms in a crystal oscillate about equilibrium positions. Because neighboring atoms are coupled, the natural motions are collective normal modes rather than independent atomic vibrations.

For a one-dimensional chain of identical masses $m$ separated by lattice spacing $a$ and connected by springs of constant $K$, try a wave-like displacement $$u_n(t)=Ue^{i(kna-\omega t)}.$$ The equations of motion give the dispersion relation $$\omega(k)=2\sqrt{\frac Km}\left|\sin\frac{ka}{2}\right|.$$ At long wavelengths, $ka\ll1$, this becomes approximately linear, $$\omega\approx a\sqrt{\frac Km}|k|,$$ so disturbances propagate like sound.

Quantum mechanically, each normal mode is a harmonic oscillator with energy $$E_n=\hbar\omega\left(n+\frac12\right).$$ A quantum of lattice vibrational excitation is a phonon. Phonons are quasiparticles: they are convenient quantized excitations of the collective lattice, not additional fundamental particles inside the material.

Phonons carry energy and crystal momentum and therefore play central roles in heat capacity, thermal conductivity, sound propagation and electron scattering. Acoustic branches correspond to neighboring cells moving nearly together at long wavelength; crystals with multiple atoms per primitive cell can also support optical branches with relative motion inside each cell.

The phonon description is the bridge between ordinary mechanical normal modes and quantum statistical mechanics of solids.