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I have a question about determining the phase and group velocity for a monoatomic lattice.

I know from various reference texts that

$$v_p = \frac {\omega}{q}$$ $$v_g = \frac {\partial \omega}{\partial q}$$

Where q is the wave vector and $\omega$ is the angular frequency.

My question is, why? This seems to be derived out of thin air, and I'm not sure what the physical/mathematical explanation for this is.

Thank you,

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So the derivation of the first is just that you look at a plane wave $\exp(i (qx-\omega(q) t))$ and the speed that it is going is just $\omega(q)/q.$

The derivation of the second is more subtle, you consider a Gaussian wave packet in the $q$-space $$f(x,t)=\int\mathrm dq~e^{-aq^2}e^{i (qx-\omega(q) t)}$$ where this is a valid solution of the wave equation due to it being a linear combination of valid solutions to the wave equation. Now Taylor expand $\omega(q)$ out to second order to find a Gaussian which is moving at speed $\omega'(q)$ and maybe also diffusing with a rate that has something to do with $\omega''(q).$ Inside it still has the same phase factor $\omega/q$ but the Gaussian envelope itself is moving at a different speed than the wave it contains inside of it.

Another derivation, as Pieter mentions in comments below, is to just look at a sum $$e^{i(qx-\omega(q) t)} + e^{i((q+\delta q)x-\omega(q+\delta q) t)} =e^{i(qx-\omega(q) t)} (1 + e^{i \delta q(x - \omega'(q) t)})$$ This thing on the right can be interpreted as a sort of envelope in which the wave on the left lives, and it has this pattern of $f(x-vt)$, a function traveling forward with speed $v$, for $v=\omega'(q).$

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  • $\begingroup$ Derivation of group velocity is easiest by considering the propagation of the "beats" between two sines with a small difference in frequency. $\endgroup$
    – user137289
    Oct 6, 2020 at 19:10
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    $\begingroup$ @Pieter Thanks! Added! $\endgroup$
    – CR Drost
    Oct 6, 2020 at 19:19
  • $\begingroup$ Thank you! I understand this answer a lot better now. $\endgroup$
    – o's1234
    Oct 7, 2020 at 13:39

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