From Voids to Formulas: The Logical Chain

Understanding ionic crystal structures reduces to a single logical chain: count anion FCC atoms → count available voids → determine how many are filled → derive the formula.

For NaCl: 4 $Cl^{-}$ in FCC → 4 octahedral voids → all 4 filled by $Na^{+}$ → 4 NaCl per cell → formula NaCl. Coordination 6:6.

For ZnS: 4 $S^{2-}$ in FCC → 8 tetrahedral voids → only 4 filled by $Zn^{2+}$ (alternate voids) → 4 ZnS per cell → formula ZnS. Coordination 4:4.

For $CaF_{2}$: 4 $Ca^{2+}$ in FCC → 8 tetrahedral voids → all 8 filled by $F^{-}$ → $Ca_{4}F_{8}$ → $CaF_{2}$. Coordination 8:4.

For $Na_{2}O$ (antifluorite): 4 $O^{2-}$ in FCC → 8 tetrahedral voids → all 8 filled by $Na^{+}$ → $O_{4}Na_{8}$ → $Na_{2}O$. Coordination 4:8.

The radius ratio determines which void is geometrically accessible to the cation:

• $r^{+}$/$r^{-}$ < 0.225: too small for any void (CN 2)
• 0.225–0.414: tetrahedral void (CN 4) → ZnS structure
• 0.414–0.732: octahedral void (CN 6) → NaCl structure

• 0.732: cubic position (CN 8) → CsCl structure

A critical trap: in $CaF_{2}$, the cation ($Ca^{2+}$) forms the FCC sublattice and the anion ($F^{-}$) fills tetrahedral voids — this is the opposite of NaCl and ZnS where the anion forms FCC and the cation fills voids. Always identify which ion is in FCC first.