Ionic Radius - The Soft-sphere Model

The Soft-sphere Model

For many compounds, the model of ions as hard spheres does not reproduce the distance between ions, to the accuracy with which it can be measured in crystals. One approach to improving the calculated accuracy is to model ions as "soft spheres" that overlap in the crystal. Because the ions overlap, their separation in the crystal will be less than the sum of their soft-sphere radii.

The relation between soft-sphere ionic radii, and, and, is given by

,

where is an exponent that varies with the type of crystal structure. In the hard-sphere model, would be 1, giving . In the soft-sphere model, has a value between 1 and 2. For example, for crystals of group 1 halides with the sodium chloride structure, a value of 1.6667 gives good agreement with experiment. Some soft-sphere ionic radii are in the table. These radii are larger than the crystal radii given above (Li+, 90 pm; Cl-, 167 pm).

Inter-ionic separations calculated with these radii give remarkably good agreement with experimental values. Some data are given in the table. Curiously, no theoretical justification for the equation containing has been given.