19.9: A Note on the Physics of These Waves
- Page ID
- 30314
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For wavelength long compared to the interparticle spacing, \(\Omega \cong v k\) these are like sound waves (and indeed they are what are called acoustic phonons in a crystal). As the wavelength shortens, the wave eigenstates are moving more slowly, remember the group velocity of a wavepacket goes as \(d \Omega / d k\). This is because there is some Bragg reflection of the waves by the lattice. At \(k a=\pi\), we have a standing wave. This is the highest energy mode, with even numbered sites all in sync with each other, and the odd numbered sites all half a cycle behind, so the restoring force experienced by an atom as a function of displacement is the maximum possible.
