A one-dimensional chain of identical atoms

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Optics & Photonics · Wave propagation and light in media

Optics and Photonics, Ch Harmonic waves, phasors and group velocity

Problem

A one-dimensional chain of identical atoms, each linked to its neighbours by identical springs and spaced a=2.00×10−10 ma=2.00\times10^{-10}\ \mathrm{m} apart, supports vibrational waves with the dispersion relation ω(k)=ω0sin⁡(ka2),0≤k≤πa\omega(k) = \omega_0\sin\left(\frac{ka}{2}\right), \qquad 0\le k \le \frac{\pi}{a} where ω0=2.00×1013 rad s−1\omega_0 = 2.00\times10^{13}\ \mathrm{rad\,s^{-1}} is the maximum (cut-off) frequency of the chain. Consider a wave packet built from wavenumbers close to those of a wave of wavelength λ=6.00 a\lambda = 6.00\,a. ((a)) Before calculating anything, state whether you expect the packet's envelope to move faster or slower than its individual crests here, arguing from whether the curve ω(k)\omega(k) bends above or below the straight line joining the origin to the point (k,ω(k))(k,\omega(k)). ((b)) Find kk from the given wavelength, then calculate the phase velocity v=ω/kv=\omega/k and the group velocity vg=dω/dkv_g=d\omega/dk, and check that they agree with your prediction.

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