the critical radius ratio

The critical radius ratios are the exact numbers where one coordination gives way to the next, the boundary lines on the radius-ratio map. They are not measured; they are pure geometry, worked out by asking a simple question: for a given cage of touching anions, what is the smallest cation that can hold those anions apart without them overlapping? At that smallest size the cation just touches all its anion neighbours while the anions just touch each other, and that borderline ratio is the critical value.

The classic thresholds are 0.155 (below which coordination 2), 0.155 to 0.225 (triangular, CN 3), 0.225 to 0.414 (tetrahedral, CN 4), 0.414 to 0.732 (octahedral, CN 6), and 0.732 to 1.0 (cubic, CN 8), with 1.0 and above allowing the highest close-packed coordination 12. Each comes from a little trigonometry: for the octahedron you place six anions on the corners of a cage, demand cation-touches-anion and anion-touches-anion together, and out drops r_cation / r_anion = sqrt(2) - 1 = 0.414. The tetrahedral limit is sqrt(3/2) - 1 = 0.225; the cubic limit is sqrt(3) - 1 = 0.732.

These critical values are why the ionic model can predict structure at all: they turn a size ratio into a definite coordination number and hence a candidate crystal structure. But right at a boundary the prediction is genuinely on a knife-edge, and because the whole derivation assumes rigid, purely ionic spheres, real compounds routinely straddle or defy the lines when bonding is covalent or ions are soft. Use the criticals as sharp geometric guideposts, not as hard walls nature is obliged to respect.