Radius Ratio Rule/Limiting Radius Ratio

When two oppositely charged ions in solids are bonded together, the bond-length (d) is the sum of the two radii (r⁺ and r^-) of the two different ions as shown in Figure 3138a. Therefore, we can have d = r⁺ + r^-.

Figure 3138a. Relations between bond-length and ionic radii.

According to Radius Ratio Rule, we have:

i) As the size (ionic radius) of a cation increases, more anions with a particular size can pack around it.
ii) For a specific structure, the limiting radius ratio can be calculated. The limiting radius ratio is the minimum allowable value for the ratio of ionic radii (ρ=r⁺/r^-) for this structure to be stable. Here, r⁺ is the radius of the cation and r^- is the radius of the surrounding anions. Note that the anions are usually larger than cations.

If all of the atoms in a crystal are the same size, the maximum number of atoms that are coordinated around any individual is 12 (also called 12-fold coordination) in which there are two ways that atoms can be packed. For a single layer of atoms with an equal size, there are two kinds of voids between the atoms as shown in Figure 3138b (a):

i) B voids with triangles pointing up.
ii) C voids with the triangles pointing down.

As shown in Figure 3138b (b), if the second layer of atoms is added and its atoms occupy the space above the B voids and the third layer of atoms is added and its atoms occupy the positions over the C voids in the A layer, this will result in cubic closest packing with a stacking sequence of ABC ABC ABC.... etc. As shown in Figure 3138b (c), if the second layer of atoms is added and its atoms occupy the space above the B voids followed by the third layer above the A atoms, this will result in hexagonal closest packing (HCP) with a stacking sequence of AB AB AB ....etc and with the c-axis perpendicular to the AB AB layers.