# Limiting Radius Ratio

The radius ratio rule was first proposed by Gustav F. HĆ¼ttig in 1920. Victor Goldschmidt extended its use to ionic lattices in 1926, and Pauling incorporated it as the first of his rules for crystal structures in 1929.

The ratio of the cationic radius to anionic radius of an ionic solid is called limiting radius ratio. If r^{+} and r^{−} be the cationic and anionic radius respectively, then the radius ratio is r^{+}/r^{−}.

Let us consider a cation (Blue Color) surrounded by three identical anions (Red Color) which are touching the cation and not touching one another. In the limiting case on decreasing the cationic size, a time comes when three anions also touch one another. In this case, the radius ratio r^{+}/r^{−} = 0.155 is called the limiting radius ratio. Hence, the limiting radius ratio is always lower than the corresponding radius ratio.

Thus, knowing the radius ratio, we can predict the coordination number and geometrical shape of crystal. If the cationic size decreases beyond this limiting value, the radius ratio falls below this limiting value and the structure is unstable. The limiting radius ratio corresponding to common coordination number are given below in the table.

Thus, limiting radius ratio is a useful guide but not a reliable method for predicting the actual structure as ions do not addopt the highest coordination number, they are not hard inelastic spheres and their radii are not exactly known as it increases by 3 per cent when the coordinadtion number increases from six to eight but decreases from six to four. Hece, ionic radii can not be measured absolutely but are only estimated. Therefore, for particular radius ratio, the coordination number is always lower but never higher.

For SrO, the radius ratio 0.81 suggest that its coordination number should be eight but in fact, it is six.