radiusratio

Radius ratio is a simple, heuristic quantity used in solid-state chemistry and crystallography to predict the coordination environment of ions in an ionic solid. It is defined as the ratio of the radius of a central cation to the radius of the surrounding anions, r+/r-, where the radii are effective ionic radii that depend on factors such as oxidation state and coordination number.

In practice, the radius ratio is computed using standard sets of ionic radii, such as the Shannon radii, and serves as a rough guide to the likely coordination number (CN) of the cation in the lattice. The idea is that the cation must fit into the voids formed by the anions without overlapping them, so larger cations favor higher coordination numbers.

Typical ranges, though approximate, map to common coordinations: about 0.225 to 0.414 for tetrahedral coordination (CN = 4); about 0.414 to 0.732 for octahedral coordination (CN = 6); and values above roughly 0.732 favor cubic or higher coordination (CN ≈ 8). Values outside these ranges can occur depending on lattice geometry, covalent character, and the specifics of the ionic radii used. Examples include NaCl, which has CN 6 and a radius ratio around 0.56; ZnS in tetrahedral coordination with a smaller ratio; and CsCl with CN 8 and a ratio around 0.9.

Limitations include the fact that ionic radii are not fixed and vary with environment; the radius ratio is a simplification that often does not capture covalency, distortions, or complex structures. Nevertheless, it remains a useful first-order rule of thumb for predicting feasible coordination and crystal structures.