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In my condensed matter book it says 'For the acoustic mode, all atoms in the unit cell move in-phase with each otehr (at $k=0$) whereas for optical modes they move out of pahse with each other (at $k=0$)'.

I saw that in the example given this is true, but is it always the case? Is it easy to show from the definition of an acoustic mode ($\omega \rightarrow0$ as $ k \rightarrow 0$) that the relative ampltiudes of atoms within the unit cell will always be in phase?

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At $k=0$, acoustic phonons correspond to macroscopic translation of the entire crystal, which naturally are completely in phase and cost zero energy. This behavior is essentially by definition and is always true as acoustic phonons are the Goldstone modes of translation. In other words, the energy of the crystal should be the same if it is located in London or Leiden.

If there was some out of phase component at $k=0$ that cost zero energy, it would mean the crystal would spontaneously deform and reorganize with zero energy cost. Such behavior would imply the material's structure is unstable, which can happen at a structural phase transition. However, structural transitions are limited to very specific temperatures and pressures - under general conditions you do not have any out of phase motion that costs zero energy

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  • $\begingroup$ Thanks, could you elaborate a bit about it being by construction? $\endgroup$
    – Alex Gower
    Jan 8, 2021 at 17:23
  • $\begingroup$ I think "construction" is the wrong word, pardon me. By definition is more appropriate. Acoustic modes are by definition those modes which are connected to a complete translation of the solid in the long wavelength limit $\endgroup$
    – KF Gauss
    Jan 8, 2021 at 19:07
  • $\begingroup$ And (although sort of obvoius) why exactly is that the most natural definition for sound waves in particular? $\endgroup$
    – Alex Gower
    Jan 8, 2021 at 19:12
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    $\begingroup$ Well it's useful to think back to sound in gases or liquids. There, the only important quantity is the local density $n(x)$ which characterizes the sound wave. This is a scalar quantity and accordingly all motion on the atomic scale is "in-phase". So it makes sense to identify the analogous mode in solid state systems as acoustic modes where all motion is "in-phase". For comparison, optical modes don't really exist in gases or liquids because they require a phase relationship between atoms that can only exist in ordered systems. $\endgroup$
    – KF Gauss
    Jan 9, 2021 at 4:23

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