Ionic crystals: polyhedra and radius ratios

Sodium chloride and caesium chloride have almost the same formula — a singly charged cation with chloride. Yet in the crystal each $\ce{Na+}$ is surrounded by 6 $\ce{Cl-}$ and each $\ce{Cs+}$ by 8. What decides is not charge but size: compared at the same six-coordination, $\ce{Cs+}$ (167 pm) is much larger than $\ce{Na+}$ (102 pm), so more anions fit around it. This lesson reads ionic crystals through their coordination polyhedra and tests a classic rule of thumb — where it holds and where it breaks.

Join an ion to its nearest shell of oppositely charged ions and you get its coordination polyhedron: a tetrahedron for coordination number 4, an octahedron for 6, a cube for 8, a cuboctahedron for 12. An ionic crystal can be read as these polyhedra sharing corners, edges or faces:

• rock salt (NaCl): $\ce{NaCl6}$ octahedra sharing edges, 6 : 6 coordination;
• caesium chloride: $\ce{CsCl8}$ cubes sharing faces, 8 : 8;
• zinc blende (ZnS): $\ce{ZnS4}$ tetrahedra sharing corners, 4 : 4;
• fluorite (CaF₂): $\ce{Ca^2+}$ inside a cube of 8 $\ce{F-}$, $\ce{F-}$ inside a tetrahedron of 4 $\ce{Ca^2+}$, 8 : 4;
• rutile (TiO₂): $\ce{TiO6}$ octahedra sharing edges in chains along c, the chains joined by corners, 6 : 3;
• perovskite (SrTiO₃): $\ce{TiO6}$ octahedra sharing all their corners, with $\ce{Sr^2+}$ in the large cavity between 12 oxide ions.

The ratio of coordination numbers is the inverse of the ratio of ions in the formula: $\ce{CaF2}$ has twice as many $\ce{F-}$ as $\ce{Ca^2+}$, so $\ce{Ca^2+}$ has twice the coordination number of $\ce{F-}$.

Pauling’s first rule says that anions form a coordination polyhedron around each cation, with the cation–anion distance set by the sum of their radii. Picture the anions as large spheres and the cation as a small one: the cation must touch all its anions, or it “rattles” in its cage and the structure is unstable. The limit comes when the anions just touch each other. Working out that limit geometrically gives the smallest radius ratio $r_+/r_-$ each polyhedron allows.

Take the octahedron: a slice through its centre shows four anions in a square with the cation in the middle. When the anions touch along the edges, the edge is $2r_-$ and the diagonal $2\sqrt2\,r_-$; the diagonal holds two anion radii and one cation diameter, so $2r_- + 2r_+ = 2\sqrt2\,r_-$, giving $r_+/r_- = \sqrt2 - 1 = 0.414$ — exactly the size of the octahedral hole in the previous lesson.

Test it with Shannon ionic radii, all taken at six-coordination: NaCl (0.564, predicts 6, actually 6), CsCl (0.923, 8, 8), ZnS (0.402, 4, 4), $\ce{CaF2}$ (0.752, 8, 8) and $\ce{TiO2}$ (0.432, 6, 6) all agree. But there are clear counter-examples:

• KCl: ratio 0.762 predicts the eight-coordinate CsCl type, yet KCl has the six-coordinate rock-salt structure; so does RbCl;
• ZnO: ratio 0.529 predicts six-coordination, yet ZnO is four-coordinate wurtzite — the Zn–O bond is substantially covalent and prefers tetrahedral directions;
• perovskite’s $\ce{Sr^2+}$: predicted 8, actually 12. Its site is dictated by the framework of $\ce{TiO6}$ octahedra and is better described by the tolerance factor $t \approx 1$.

A systematic test on about 5000 oxides found that the radius-ratio rule gets only about two-thirds of coordination environments right. There are three reasons: ions are not hard spheres and their radii change with coordination; bonds have covalent character; and the energies of competing structures often differ very little. The rule’s value lies in pointing intuition the right way, not in precise prediction.