# Do we say that phonon has effective mass through its dispersion relation?

The effective mass is proportional to the second derivative of the dispersion relation d2k/dE2. Do we say that phonon have effective mass through it ? Spin wave have.

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Phonon dispersions are generally indicated by a linear spectrum, so the second derivative is 0. Thus phonons are effectively massless, and have a velocity given by the first derivative.

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What about the optical modes they seem quadratic ? en.wikipedia.org/wiki/File:Diatomic_phonons.png –  user12445 Sep 25 '12 at 18:14
@user12445: Optical modes are massive. –  Ron Maimon Sep 26 '12 at 7:24

This figure shows the phonon dispersions of ZnO. It is clear that while some phonons have very roughly linear dispersions many do not (especially close to zone center). The second derivative would be non zero in these regions. I hope this has added to the conversation.

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The effective mass of modes that are linear near the centre of the Brillouin zone will be zero since $\frac{d^{2}k}{d\omega^{2}}$ vanishes, but near extremes of the BZ this often isn't true and they will of course gain an effective mass.