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 State of matter, phases, that arise from electromagnetic forces between atoms and elec ...

and

inorganic chemistry Inorganic chemistry deals with chemical synthesis, synthesis and behavior of inorganic compound, inorganic and organometallic chemistry, organometallic compounds. This field covers chemical compounds that are not carbon-based, which are the subj ...

, the cation-anion radius ratio can be used to predict the

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

of an

ionic compound In chemistry, a salt or ionic compound is a chemical compound consisting of an assembly of positively charged ions (Cation, cations) and negatively charged ions (Anion, anions), which results in a compound with no net electric charge (electrica ...

based on the relative size of its atoms. It is defined as 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 cati ...

of the positively charged cation to the

of the negatively charged anion in a cation-anion compound. Anions are larger than cations. Large sized anions occupy lattice sites, while small sized cations are found in voids. In a given structure, the ratio of cation radius to anion radius is called the radius ratio. This is simply given by

r_ / r_

Ratio rule and stability

The radius ratio rule defines a critical radius ratio for different crystal structures, based on their

coordination geometry The coordination geometry of an atom is the geometrical pattern defined by the atoms around the central atom. The term is commonly applied in the field of inorganic chemistry, where diverse structures are observed. The coordination geometry depen ...

. The idea is that the anions and cations can be treated as incompressible spheres, meaning the crystal structure can be seen as a kind of unequal sphere packing. 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 the critical radius ratio for that particular structure. At the stability limit the cation is touching all the anions and the anions are just touching at their edges. For radius ratios greater than the critical ratius ratio, the structure is expected to be

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

. The rule is not obeyed for all compounds. By one estimate, the crystal structure can only be guessed about 2/3 of the time. Errors in prediction are partly due to the fact that real chemical compounds are not purely ionic, they display some covalent character. The table below gives the relation between critical radius ratio,

R

, 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 ion ...

CN

, which may be obtained from a simple geometrical proof. :

R = \sqrt

History

The radius ratio rule was first proposed by Gustav F. Hüttig in 1920. In 1926,

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

extended the use to ionic lattices. In 1929, the rule was incorporated as the first of

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 ani ...

for

References

{{reflist Crystallography Inorganic chemistry Ratios Atomic radius

Ratio rule and stability

History

See also

References