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We are discussing about coupling of phonons and dielectric materials and the need to match frequencies with light dispesrion {$\omega=ck$} indicate that only long wavelength IR waves may couple to phonon oscillations. Consider nearly infinite wavelength, {$k\approx0$}, i.e. the nearly dispersionless mode of frequency {$\omega_{LO}(0)=\omega_L$}.
Think of an infinite slab of dielectric material with {$\tau=1,\cdots,r$} inequivalent ions per unit cell. Assume the normal to the slab is the direction {$\hat z$} of the longitudinal optical (LO) phonons modes, i.e. all modes of frequency {$\omega_{LO}(\mathbf k)$} displace the ions of charge {$q_\tau$} along {$\hat z\parallel\mathbf k$}, each by a different amount. The oscillation is given by the eigenfunction of the mode
{$$ u_{n\tau \mathbf k}= \left(\frac{\hbar}{NM_\tau \omega_L}\right)^{\frac 1 2} A_\tau e^{i\mathbf k \cdot \mathbf R_{0n}}$$}
that oscillates at frequency {$\omega_L$} longitudinally in the slab. It produces a non vanishing total dipole moment:
{$$ P_{L\mathbf k} \hat z = \sum_{n,\tau} q_\tau u_{n\tau \mathbf k}\hat z$$}
in the absence of any external charges or fields. Since {$\varepsilon_0\mathbf E$} and {$\mathbf P $} share the same sources (polarization charges on the surfaces of the slab), but with opposite divergence
{$$\boldsymbol\nabla \cdot \mathbf P=-\varepsilon_0\boldsymbol\nabla \cdot \mathbf E$$}
they must be equal and opposite.
{$$\varepsilon_0\mathbf E(\omega_L,\mathbf k) = -\hat z \sum_{n,\tau} q_\tau u_{n \tau\mathbf k} e^{i\omega_L t}$$}
By definition the displacement {$\mathbf D=\varepsilon_0\mathbf E + \mathbf P = \varepsilon\mathbf E$} vanishes, and since {$\mathbf D=\varepsilon\mathbf E$} the dielectric function must also vanish.
This mode has {$\mathbf E_0\cdot\mathbf k=-kP_L/\varepsilon_0$} oscillating along the wavevector direction and it cannot couple to any electromagnetic wave travelling in the vacuum that surrounds the slab, that must be a transverse wave, obeying {$\mathbf E_0\cdot\mathbf k=0$}. The reverse is also true, i.e. the longitudinal mode does not irradiate.
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