# Lyddane–Sachs–Teller relation

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In condensed matter physics, the Lyddane–Sachs–Teller relation (or LST-relation) determines the ratio of the natural frequency of longitudinal optic lattice vibrations (phonons) ($\omega_L$) of an ion crystal to the natural frequency of the transverse optical lattice vibration ($\omega_T$) for k=0.[1] The ratio is equal to the ratio of the static permittivity $\varepsilon_{st}$ (Often also $\varepsilon(0)$) for the permittivity for frequencies in the visible range $\varepsilon_{\infty}$ (often also $\varepsilon(\omega_s)$).[2]
$\frac{\omega_L^2}{\omega_T^2} = \frac{\varepsilon_{st}}{\varepsilon_{\infty}}$