One-Phonon Effect

Like molecules, crystals can also vibrate as a whole. Their vibrations can be excited thermally, and they can display a residual vibrational motion at zero 

Kelvin (the zero point motion). This latter effect is explained by quantum mechanics, and it can in turn explain absorption features of impurities in crystalline matrices. The presentation of the fundamental vibrational modes of crystals is based on the harmonic approximation, where one only considers the interactions between an atom or an ion and its nearest neighbours. Within this approximation, an harmonic crystal made of N ions can be considered as a set of 3N independent oscillators, and their contribution to the total energy of a particular normal mode with pulsation ivs (q) is  

As already mentioned, because of the lattice periodicity of the crystals, the dispersion curves are studied for propagation vectors lying only in the first 

In elemental (homonuclear) crystals with cubic symmetry, the LO and TO branches are degenerate at q = 0 (the T point of the BZ), and the phonons at that point are denoted as O(T). The situation is different in compound crystals, where the energy of the LO branch is larger than that of the TO branch. This difference in compound crystals is attributed to the contribution of an electric field effect to the restoring forces, and it can be shown that at q = 0  

This expression is known as the Lyddane-Sachs-Teller relation [61]. 

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