🔗 Lattice Vibrations & Phonons
A real 3D spring-mass chain, driven by an exact analytic normal mode at your chosen wavevector k. Switch between a monatomic chain (single acoustic branch) and a diatomic chain (acoustic + optical branches with a real frequency gap), and watch the live omega(k) dispersion plot track the current mode.
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Monatomic Chain — Acoustic Branch Only
A 1D chain of identical masses M connected by identical springs K has one vibrational branch: omega(k) = 2·sqrt(K/M)·|sin(ka/2)|. As k → 0 this becomes a sound wave (omega proportional to k); at the Brillouin-zone boundary k = π/a neighbouring atoms move exactly out of phase and the group velocity drops to zero.
Reading the dispersion plot
Acoustic branch — starts at omega = 0 when k = 0
(uniform translation, no restoring force) and rises like a sound wave for small k. In a
diatomic chain, at k = π/a it caps out at omega = sqrt(2K / M_heavy) because only the
heavy sublattice is moving.
Optical branch — exists only for the diatomic
chain. At k = 0 the two sublattices vibrate against each other (their centre of mass is
fixed) at omega = sqrt(2K·(1/M1 + 1/M2)). At k = π/a only the light sublattice moves, at
omega = sqrt(2K / M_light).
The gap — because sqrt(2K/M_heavy) <
sqrt(2K/M_light), there is a forbidden frequency band between the top of the acoustic
branch and the bottom of the optical branch at the zone boundary: no propagating wave of
that frequency can exist in the lattice.