Anions are larger than cations. Large sized anions occupy lattice sites, while small sized cations are found in voids. The ratio of radius of cation to anion is called radius ratio. In

condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...

and

inorganic chemistry Inorganic chemistry deals with synthesis and behavior of inorganic and organometallic compounds. This field covers chemical compounds that are not carbon-based, which are the subjects of organic chemistry. The distinction between the two disci ...

the cation-anion radius ratio (also: radius ratio rule) is the ratio of the

ionic radius Ionic radius, ''r''ion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the catio ...

of the cation to the

of the anion in a cation-anion

compound Compound may refer to: Architecture and built environments * Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall ** Compound (fortification), a version of the above fortified with defensive struct ...

. This is simply given by

r_ / r_

. According to

Pauling's rules Pauling's rules are five rules published by Linus Pauling in 1929 for predicting and rationalizing the crystal structures of ionic compounds. First rule: the radius ratio rule For typical ionic solids, the cations are smaller than the anions ...

for

crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...

s, the allowed size of the cation for a given structure is determined by the critical radius ratio. If the cation is too small, then it will attract the anions into each other and they will collide hence the compound will be unstable due to anion-anion repulsion; this occurs when the radius ratio drops below 0.155. At the stability limit the cation is touching all the anions and the anions are just touching at their edges (radius ratio = 0.155). For radius ratios greater than 0.155, the compound may be

stable A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...

. The table below gives the relation between radius ratio and

coordination number In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central i ...

, which may be obtained from a simple geometrical proof. The radius ratio rule was first proposed by Gustav F. Hüttig in 1920.''The Origin of the Ionic-Radius Ratio Rules'' William B. Jensen, Journal of Chemical Education 2010 87 (6), 587-588 In 1926,

Victor Goldschmidt Victor Moritz Goldschmidt (27 January 1888 in Zürich – 20 March 1947 in Oslo) was a Norwegian mineralogist considered (together with Vladimir Vernadsky) to be the founder of modern geochemistry and crystal chemistry, developer of the Golds ...

extended the use to ionic lattices.V. Goldschmidt, “Crystal Structure and Chemical Constitution,” Trans. Faraday Soc. 1929, 25, 253-283.

References

{{reflist Crystallography Inorganic chemistry Ratios Atomic radius

See also

References