Q.The ionic radii of $A^+$ and $B^-$ are $0.98 \times 10^{-10}$ m and $1.81 \times 10^{-10}$ m. The coordination number of each ion in AB is

The Intuition: Packing Spheres of Different Sizes

Imagine you are trying to fit a small marble (the cation) into a cavity formed by larger marbles (the anions). If the small marble is too small, it will rattle around — the structure is unstable because the cation cannot touch all the surrounding anions. If it is just the right size, it touches every neighbouring anion perfectly. If it is too large, it will push the anions apart, forcing a different arrangement.

That is the entire physical idea behind the Radius Ratio Rule. The ratio $r_{\text{cation}} / r_{\text{anion}}$ determines how many anions can pack around a cation — the coordination number — and therefore the geometry of the crystal.

The Precise Statement

For an ionic crystal, the radius ratio is defined as

The rule states: For a given coordination number and geometry, there is a minimum value of $\rho$ below which the cation is too small to remain in contact with all the surrounding anions. Above that minimum, the geometry is stable.

The key idea is cation–anion contact. In a stable ionic crystal, the cation should touch as many anions as possible. If $\rho$ falls below the critical value, the cation loses contact with some anions, and the structure becomes unstable — it will adopt a lower coordination number instead.

The Critical Limits and Geometries

Here are the most important ranges you need for exams:

Where Do These Numbers Come From? (The Octahedral Case)

Take an octahedral hole. Four anions lie in a square plane, with the cation at the centre. The diagonal of the square is $2r_{\text{anion}} + 2r_{\text{cation}}$. But the diagonal of a square of side $2r_{\text{anion}}$ is $2\sqrt{2}\, r_{\text{anion}}$. Equating:

Divide through by $2r_{\text{anion}}$:

So

$\rho = \sqrt{2} - 1 \approx 0.414$ …