Phonon dispersion relation

A phonon dispersion relation gives its angular frequency $\omega$ as a function of its wavevector $k$. For an isotropic power law, $\omega =C|k{|}^{\alpha }$.

An acoustic phonon branch has $\omega (k)\to 0$ as $k\to 0$. Its long-wavelength group velocity is the speed of sound.

An optical phonon branch has a nonzero frequency as $k\to 0$, corresponding to relative motion of degrees of freedom within one primitive cell.

For equal masses $m$ separated by $a$ and alternating spring constants $\lambda$ and $\alpha \lambda$, the primitive-cell length is $2a$ and the two branches areThe minus and plus signs give the acoustic phonon branch and optical phonon branch, respectively.

For two polarizations in area $A$ and $\omega =C|k{|}^{\alpha }$, counting wavevector states in a disk gives