Borns model and later developments

The hard-sphere model, presented above in its most simplified form, was later refined in order to account for the cohesion energy and the elastic properties of ionic crystals. 

Short-range repulsion A first improvement is related to the repulsive forces which become effective at short inter-ionic distances. In the original model, steric - or hard-core - repulsive forces prevent two ions i and J from coming closer than the sum of their ionic radii r,- and rj. The short-range repulsion energy is infinite if Rij r, + ry, and zero otherwise, which may be written in the form  

Each of these contains two parameters which are functions of the interacting ions i and j A and n in (1.1.6), with 5 n 12 B and p in (1.1.7), with p of the order of 0.2 to 0.3 A. In both cases, the repulsion energy decreases strongly when the inter-atomic distance gets larger, so that only first neighbour interactions are relevant. The Lennard-Jones form is often 

The crystal structure determines the Madelung constant a and the number Z of anion-cation bonds per formula unit. In binary compounds, Z is equal to the coordination number of the ion which has the larger number of first neighbours (e.g. Z = 6 for Ti02). The minimization of E with respect to R yields the equilibrium distance Ro between first neighbours in a given structure. Rq is the solution of the implicit equation  

The cohesion energy, necessary to separate the system into independent ions, is equal to  

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