Phonons in a 3D crystal

Alternatively, one can use the formalism discussed in chapters 4 and 5 (DFT/LDA and its refinements) for calculating total energies or forces, as follows. The atomic displacements that correspond to various phonon modes can be established 

In phonon nomenclature L stands for longitudinal, T for transverse, O for optical and A for acoustic. 

Bottom atomic displacements associated with the phonon modes in Si at the T and X points in the BZ inequivalent atoms are labeled 1.  

Frequencies of four high-symmetry modes are given (in THz) at the center (F) and the boundary (X) of the Brillouin Zone, as obtained by theoretical calculations and by experimental measurements. DFT results are from Refs. [65, 67]. The force-constant model is based on nearest neighbor interactions only, with values of Kr and ko chosen to reproduce exactly the experimental frequencies marked by asterisks. The atomic displacements corresponding to these modes are shown in Fig. 6.4. 

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