Lattice vibrations in a one-dimensional crystal are often introduced with a linear chain of masses connected by nearest-neighbor springs. In the harmonic approximation, assuming displacements u_n from equilibrium, the equations of motion yield a dispersion relation ω(k) relating angular frequency to crystal wavevector k in the first Brillouin zone. For a monoatomic chain of mass m and spring constant K, with lattice constant a (distance between equivalent sites), ω(k) = 2√(K/m)|sin(ka/2)| — a single acoustic branch that is linear near k = 0 with sound speed v_s = a√(K/m) and flattens to zero group velocity at the zone boundary k = ±π/a. A diatomic chain with alternating masses m and M (μ = M/m) produces two branches: a low-frequency acoustic branch (atoms move in phase at long wavelength) and a high-frequency optical branch (out-of-phase motion on the two sublattices). With a as the repeat distance between equal masses, the analytic result is ω²±(k) = (K/m)((1+μ)/μ)[1 ± √(1 − 4μ sin²(ka/2)/(1+μ)²)]. When μ ≠ 1, a frequency gap opens at k = ±π/a between the acoustic and optical branches; when μ = 1 the folded diatomic spectrum has no gap. Group velocity v_g = dω/dk describes the propagation speed of a wave packet and is shown alongside ω(k). This simulator plots both branches, marks a probe wavevector, animates a qualitative snapshot of the lattice motion, and reports ω and v_g at the probe — ideal preparation for Debye theory, neutron scattering, and the link between mechanical springs and phonons in solids.
Who it's for: Undergraduate solid-state physics, materials science, or physical chemistry students studying lattice dynamics after crystal structures and before Debye heat capacity.
Key terms
Phonon dispersion
Acoustic branch
Optical branch
Brillouin zone
Group velocity
Diatomic chain
Mass–spring model
Lattice vibrations
How it works
Phonon dispersion in a 1-D mass–spring chain: monoatomic acoustic branch or diatomic acoustic and optical branches, Brillouin zone edges, and group velocity v_g = dω/dk — the standard lattice-vibration introduction beside band-structure models.
Frequently asked questions
Why does the acoustic branch go to ω = 0 at k = 0?
At k = 0 all masses oscillate in phase — a uniform translation of the whole chain costs no spring energy in the harmonic model, so the mode is a zero-frequency Goldstone mode (acoustic phonon).
What is the difference between acoustic and optical branches?
Acoustic modes have neighboring atoms moving in the same direction at long wavelength (sound waves). Optical modes have the two sublattices moving opposite to each other; at k = 0 the center of mass is fixed and ω is finite — typical of infrared-active vibrations in ionic crystals.
Why is there a gap at k = π/a for m ≠ M?
At the zone boundary a standing wave can localize on light or heavy atoms. Different masses split the two eigenfrequencies; equal masses restore the degeneracy and the diatomic formula collapses to the monoatomic |sin| band.
Is this the same as the Kronig–Penney electron bands?
Same mathematical structure (periodicity → Bloch theorem → dispersion curves in k), but phonons are collective atomic displacements obeying Newton’s law with spring forces, whereas Kronig–Penney describes electron waves in a periodic potential.