Theory of Harmonic Vibrations

and 33 being the vectors defining the primitive cell. The primitive cell may contain several atoms, S3y N, whose positions with respect to the origin of the cell can be labeled as R (A ), k=. N, Any particle in the crystal can thus be located by giving the particle index k and the cell index L. The vibrational problem is of large dimension (3NM), M being the number of primitive cells in the crystal, and can be reduced to dimension 3N by use of the translational symmetry of the primitive cell. 

The forces between particles in crystals arise from coulombic, van der Waals, hydrogen, or metallic bonds. The instantaneous position of the atoms can be written in cartesian coordinates as [100,101]  

The potential energy is a function of the displacement of the atoms and can be written as an expansion about the atomic equilibrium positions  

The form that (p p, the harmonic force constant, takes depends on the nature of the interaction between atoms k and k in the crystal. Although the interactions are, in fact, quite complex, the assumption of effective two-body interactions such as a Born-Mayer potential 

The 3NM equations of motion are solved preserving the periodicity of the crystal and under the assumption that the vibrations are plane waves yielding 

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We shall apply here the theory of harmonic vibrations, given in TGN and also in Part Two of this review, for calculation of dielectric response. For that we should replace the simplified spectral functions (19) by more rigorous ones, namely by... 

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