Ionic crystals geometric requirements

A factor of great importance in ionic crystals is the difference in size of the positive and negative ions. Pauling has shown how the geometric requirements for close packing of spheres of different sizes can be simply expressed in terms of the radius ratio p = rjvi, defined as the ratio of the radius of the smaller ion, r, to that of the larger ion,... 

The relationship between the ionic radii and the ionic volumes vi leads directly to Vi = (4 r/3)(ri) and a set of such volumes, denoted as vm is reported in [17]. The ionic volumes obtained from the formula unit volumes in crystals [28], denoted as vj, is also shown in [ 17], with vj = (1.258 0.01 6)vm having been established with a correlation coefficient of 0.995 for 55 cations. The larger size of vj takes into account the void spaces between the (more or less spherical) ions. Another way to express this, based on the suggestion of Mukerjee [29] regarding such ions in solution, is to use the factor =1.213 multiplying the radius of univalent monatomic ions vi = (4jr/3)( ri). This factor is near the value, 1.159, that is geometrically required for close-packed spheres of arbitrary but comparable sizes. 

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