Definition of the Model and Dynamical Matrix

2 The Rigid-Ion Model for Ionic Crystals 4.2.1 Definition of the Model and Dynamical Matrix 

In the rigid-ion model, an ion of type k carries a point charge z e, where e is the magnitude of the electronic charge thus, the ions are not polarizable. The interaction energy of two ions i = ( ) and k = (, ) separated by a distance R. is 

Instead of (4.19), the repulsive energy could also be approximated by an expression of the form B/R as has been used for the solid inert gases (Sect.4.1). In any case, is a short-range energy which in a first approximation may be extended to nearest-neighbour ions only. On the other hand, cp represents the long-range Coulomb forces and it is this term which will complicate the lattice dynamical problem considerably. 

Since we are dealing with central forces, the force constants can be obtained directly from Appendix J. According to (4.18) and (J.4 and 7), the total force constants are 

Similarly, we obtain for the force constants of the Coulomb forces for i + k 

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