Lattice Vibrations and Phonons: The Quantised Sound of a Crystal

A crystal is a giant network of coupled springs

Atoms in a solid are not frozen in place — they sit in a potential energy well created by their neighbours' bonds and constantly vibrate around their equilibrium positions. To first approximation, each bond behaves like a small spring obeying Hooke's law, so a crystal is mechanically a vast three-dimensional network of masses connected by springs. Displace one atom and its motion couples to its neighbours through those springs, its neighbours' motion couples to the next shell of atoms, and the disturbance propagates through the lattice as a wave.

The monatomic chain and its dispersion relation

The simplest model is a one-dimensional chain of identical masses m connected by identical springs of stiffness K, spaced a apart. Writing Newton's second law for one atom in terms of its neighbours' displacements and looking for travelling-wave solutions gives a direct relationship between a mode's wavevector k and its angular frequency omega — the dispersion relation:

omega(k) = 2 * sqrt(K/m) * |sin(k*a/2)|      (monatomic chain, one branch)

small k (long wavelength):  omega =~ a*sqrt(K/m) * k     -- linear, like sound
k = pi/a (zone boundary):   omega reaches its maximum, group velocity = 0

At long wavelengths this reduces to a linear relationship between frequency and wavevector, exactly like an ordinary sound wave, with the constant of proportionality equal to the material's speed of sound. Near the edge of the first Brillouin zone the curve bends over and flattens — neighbouring atoms end up moving in exactly opposite directions, group velocity drops to zero, and the wave stops propagating energy through the lattice.

Quantising the vibration: enter the phonon

Each vibrational mode of the lattice, treated quantum mechanically, behaves like a quantum harmonic oscillator, and its energy comes in discrete steps of h-bar times omega rather than varying continuously. A phonon is simply the name given to one quantum of vibrational energy in a particular mode — not a physical particle that could exist outside the crystal, but a bookkeeping unit (a quasiparticle) that behaves like one in every calculation that matters: phonons carry momentum, scatter off each other, off electrons, and off crystal defects, and obey Bose-Einstein statistics just as photons do. Nearly all thermal transport, thermal expansion and much of electrical resistance in ordinary metals ultimately comes down to counting and tracking phonon populations and their collisions.

Two atoms per cell: acoustic and optical branches

Real crystals like sodium chloride or gallium arsenide have two different atom types in each repeating unit cell, and the one-branch dispersion picture splits into two. In the acoustic branch, the two atoms in a cell move essentially together, in phase, the same collective sound-wave-like motion as the monatomic chain. In the optical branch, the two atoms move against each other, out of phase, oscillating around their shared centre of mass even at long wavelength. If the two atom types carry opposite charge — as in an ionic crystal — that out-of-phase motion creates an oscillating electric dipole that can couple directly to an electromagnetic wave of matching frequency, which is exactly why the branch is called optical: it is the mode responsible for a crystal's strong infrared absorption and reflection bands.

diatomic chain, masses m1 and m2, spring constant K:

acoustic branch:   starts at omega = 0 for k = 0, atoms move in phase
optical branch:    starts at a nonzero omega for k = 0, atoms move
                    out of phase against each other

a frequency GAP separates the top of the acoustic branch from the
bottom of the optical branch -- no lattice wave can exist in that
gap, regardless of k

Between the top of the acoustic branch and the bottom of the optical branch lies a forbidden frequency gap where no propagating lattice wave exists at any wavevector — a direct analogue of the electronic band gap that separates a semiconductor's valence and conduction bands, arising from the same underlying periodicity of the crystal.

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