# Hume-Rothery rules

The Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid solutions, and the other refers to interstitial solid solutions.

## Substitutional solid solution rules

For substitutional solid solutions, the Hume-Rothery rules are follows:

1. The atomic radius of the solute and solvent atoms must differ by no more than 15%:
${\displaystyle \%{\mbox{ difference}}=\left({\frac {r_{solute}-r_{solvent}}{r_{solvent}}}\right)\times 100\%\leq 15\%.}$
2. The crystal structures of solute and solvent must be similar.
3. Complete solubility occurs when the solvent and solute have the same valency.[1] A metal dissolves a metal of higher valency to a greater extent than one of lower valency.
4. The solute and solvent should have similar electronegativity. If the electronegativity difference is too great, the metals tend to form intermetallic compounds instead of solid solutions.

## Interstitial solid solution rules

For interstitial solid solutions, the Hume-Rothery rules are:

1. Solute atoms must be smaller than the interstitial sites in the solvent lattice.
2. The solute and solvent should have similar electronegativity.[2]
3. They should show a wide range of composition.
4. valency factor: two elements should have same valence .Greater the difference in valence between solute and solvent atoms, the lower the solubility.