Application to Crystals with NaCl Structure

Confining ourselves to nearest-neighbour interactions for the overlap forces in the NaCl structure (Fig.4.4), we find from (4.22,23) 

It is instructive to consider the case q 0. From (4.48), it is found for crystals such as the alkali-halides in which each ion is at a site of at least tetrahedral symmetry, that 

This difference is characteristic for Coulomb interactions. It should be noted that from (4.57) 

We have derived this result in a purely mathematical way and it is desirable to gain some insight into its physical origin. This is possible from a consideration of the effective fields acting on the ions. First we 

The essential point is that the macroscopic field is different for TO and LO-modes. In fact from (4.52), we find that for a TO-mode (q perpendicular to ), the macroscopic field vanishes while for an LO-mode (q parallel to ), the macroscopic field is = - 4ttP. Thus, the effective fields are, according to (4.49), given by 

---