Normal Modes of a Mass-Spring Chain
A fixed-end chain of $N$ masses on springs oscillating in a chosen normal mode, with the dispersion relation alongside. The monatomic chain has exactly $N$ modes at $\omega_n = 2\sqrt{K/m} \sin\left(\frac{n\pi}{2(N+1)}\right)$, each a standing wave with $n-1$ internal nodes. Switching to a diatomic chain with two alternating spring constants splits the spectrum into an acoustic and an optical branch separated by a zone-boundary band gap that closes exactly when the springs are equal. The scene animates the chain in the selected mode while the side panel draws $\omega(k)$ with the gap shaded; clicking the curve picks a mode. This is the one-dimensional origin of phonon bands and the acoustic/optical split in real crystals.
WHAT TO TRY
- Click the dispersion curve to pick a mode: long-wavelength points on the acoustic branch rock the whole chain together, while the optical branch has neighbouring atoms beating out of phase.
- Mind the band gap: between the top of the acoustic branch and the bottom of the optical branch lies a forbidden frequency range, the phononic analogue of an electronic band gap.
- At the zone boundary the two atom species move oppositely and the group velocity drops to zero, a standing wave that transports no energy.