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The radius ratio rule defines a critical radius ratio for different crystal structures, based on their coordination geometry. [1] 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 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.
In inorganic chemistry, Fajans' rules, formulated by Kazimierz Fajans in 1923, [1] [2] [3] are used to predict whether a chemical bond will be covalent or ionic, and depend on the charge on the cation and the relative sizes of the cation and anion. They can be summarized in the following table:
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.
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 .
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r B is the radius of the B cation. r O is the radius of the anion (usually oxygen). In an ideal cubic perovskite structure, the lattice parameter (i.e., length) of the unit cell (a) can be calculated using the following equation: [ 1 ]
A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio, [ 59 ] or their approximations generated from Fibonacci numbers, [ 60 ] often depicted inscribed within a spiraling pattern of squares ...