Solid State Physics

Generally higher substitutional solubility when solvent and solute atoms are similar in atomic size (15% according to Hume-Rothery rules) and have same crystal structure in pure form.

Hume-Rothery rules are used to determine the solid solubility:

• The atomic radius of the solute and solvent must be within 15% \[ \frac{r_{\rm solute} - r_{\rm solvent}}{r_{\rm solvent}} \le 15\%\]
• The crystal structures of solute and solvent must be similar
• Complete solubility occurs when the solvent and solute have the same valency. A metal is more likely to dissolve a metal of higher valency, than vice versa.
• 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 solubility is high when the solute atom is small enough (less than 57% in radius).

Hume-Rothery rules

• Solute atoms should have a smaller radius than 59% of the radius of solvent atoms
• The solute and solvent should have similar electronegativity.
• Valency factor: two elements should have the same valence. The greater the difference in valence between solute and solvent atoms, the lower the solubility.

\[ \Delta \tau = Gb\epsilon^{\frac{3}{2}} \sqrt{c} \] where \(\Delta\tau\) is the increase in the , \(c\) is the concentration of the solute atoms, \(G\) is the , \(b\) is the magnitude of the Burger's vector, and \(\epsilon\) is the lattice strain due to the solute.