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The radius ratio rules are a first approximation which have some success in predicting coordination numbers, but many exceptions do exist. [3] In a set of over 5000 oxides, only 66% of coordination environments agree with Pauling's first rule. Oxides formed with alkali or alkali-earth metal cations that contain multiple cation coordinations are ...
The radius ratio rule was first proposed by Gustav F. Hüttig in 1920. [ 5 ] [ 6 ] In 1926, Victor Goldschmidt [ 5 ] extended the use to ionic lattices. [ 7 ] [ 8 ] [ 9 ] In 1929, the rule was incorporated as the first of Pauling's rules for crystal structures .
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 cation and anion gives the distance between the ions in a crystal lattice .
For interstitial solid solutions, the Hume-Rothery Rules are: Solute atoms should have a smaller radius than 59% of the radius of solvent atoms. [5] [6] The solute and solvent should have similar electronegativity. [7] Valency factor: two elements should have the same valence.
The crystal structures of simple ionic solids have long been rationalised in terms of Pauling's rules, first set out in 1929 by Linus Pauling. [3] For metals and semiconductors one has different rules involving valence electron concentration. However, prediction and rationalization are rather different things.
[54] [36] These rules concern (1) the ratio of cation radius to anion radius, (2) the electrostatic bond strength, (3) the sharing of polyhedron corners, edges and faces, (4) crystals containing different cations, and (5) the rule of parsimony.
The bond valence method is a development of Pauling's rules. In 1930, Lawrence Bragg [11] showed that Pauling's electrostatic valence rule could be represented by electrostatic lines of force emanating from cations in proportion to the cation charge and ending on anions. The lines of force are divided equally between the bonds to the corners of ...
[34] [35] Depending on the stoichiometry of the salt, and the coordination (principally determined by the radius ratio) of cations and anions, a variety of structures are commonly observed, [36] and theoretically rationalized by Pauling's rules.