The radius ratio $r^{+}$/$r^{-}$ determines the coordination number and crystal structure:

Derivation of Limiting Ratios

For octahedral (CN 6) — critical ratio = 0.414: In the equatorial plane, 4 anions form a square. Each anion-anion distance = 2$r^{-}$. Square diagonal = 2$r^{-}$√2 = 2($r^{+}$ + $r^{-}$).

$r^- \sqrt{2} = r^+ + r^- \implies \frac{r^+}{r^-} = \sqrt{2} - 1 \approx 0.414$

For tetrahedral (CN 4) — critical ratio = 0.225:

$\frac{r^+}{r^-} = \sqrt{\frac{3}{2}} - 1 \approx 0.225$

Quick Memory Table

• CsCl (0.93) > NaCl (0.56) > ZnS (0.40) by radius ratio
• Higher ratio → higher CN → more anions can surround the cation