Ionic Radius Ratio Calculator

The radius ratio rule treats ions as hard spheres and asks how many anions of radius r-minus can pack around a cation of radius r-plus while all of them stay in contact with the cation and out of contact with each other. The boundaries are pure geometry rather than empirical fits. Three anions in a plane require a ratio of 2 divided by the square root of 3, minus 1, which is 0.155; a tetrahedron requires the square root of 6 over 2, minus 1, or 0.225; an octahedron requires the square root of 2 minus 1, or 0.414; and a cube requires the square root of 3 minus 1, or 0.732. Below the limiting value the anions would touch and repel, so the structure drops to a lower coordination number. Sodium chloride, with Shannon radii of 102 and 181 picometres, gives 0.564 and correctly predicts the six-coordinate rock salt structure; zinc sulfide at 0.402 predicts the four-coordinate zinc blende arrangement; caesium chloride at 0.939 predicts eight-coordinate cubic packing. The interionic distance is simply the sum of the two radii, 283 picometres for sodium chloride. The rule is a first approximation and fails often enough that it should never be treated as decisive, since it assumes ions are spherical, incompressible and bonded purely by electrostatics. Silver iodide adopts the zinc blende structure despite a ratio predicting octahedral coordination, because directional covalent bonding and the polarisability of iodide override the packing argument, the situation Fajans's rules describe. Layered compounds such as cadmium iodide have bonding intermediate between ionic and covalent, and transition metal compounds are influenced by crystal field preferences. Surveys of simple binary halides and oxides have found the rule assigns the observed coordination number in only around half to two thirds of cases. The radii themselves are a choice rather than a fact: they depend on how the interionic distance is partitioned.

Assigning a size to an individual ion required splitting a measured interionic distance between two partners, a problem J. A. Wasastjerna addressed in 1923 by apportioning the distance according to molar refraction. Victor Moritz Goldschmidt, working in Oslo, used those values as a starting point and built an empirical table of ionic radii from crystallographic distances, publishing them within his Geochemische Verteilungsgesetze der Elemente series from 1926. Goldschmidt correlated radius ratios with observed coordination polyhedra as part of his effort to explain how elements distribute themselves among minerals in the Earth. Linus Pauling derived a competing set of radii in 1927 from theoretical screening constants, and in 1929 published his rules for the structures of complex ionic crystals in the Journal of the American Chemical Society. The first of those rules states that a coordination polyhedron of anions forms around each cation with the cation-anion distance set by the sum of the radii and the coordination number set by the radius ratio, which is the principle this calculation applies. The modern reference table came from Robert Shannon and Charles Prewitt, who analysed roughly a thousand oxide and fluoride structures and published effective ionic radii in Acta Crystallographica in 1969, revised by Shannon in 1976; unlike earlier sets these vary explicitly with coordination number and spin state, listing sodium at 99 picometres in four-coordination and 102 in six. Critical examinations published in the Journal of Chemical Education during the 1980s documented the rule's substantial failure rate, and it is now taught as a useful geometric heuristic rather than a predictive law.