the Hume-Rothery rules

Suppose you want to mix two metals and have them dissolve fully into one another as a substitutional solid solution — like copper and nickel do. When will that actually happen, and when will the two refuse to mix and separate instead? In the 1930s the metallurgist William Hume-Rothery worked out a short checklist of conditions that predict it remarkably well. Think of them as compatibility rules for atomic roommates: the closer two atoms match, the more happily they share a crystal.

There are four rules for extensive substitutional solubility. First, the size rule: the atomic radii should differ by less than about 15 percent — a guest too big or too small strains the lattice too much and pops out. Second, the crystal-structure rule: the two pure metals should have the same crystal structure (both FCC, say), so the mixed lattice has no reason to change form. Third, the electronegativity rule: their chemical pull on electrons should be similar — if one is far more electronegative, the two would rather react and form a compound than dissolve. Fourth, the valence rule: a metal of lower valence tends to dissolve one of higher valence more readily than the reverse.

Copper and nickel pass every rule (radii 0.128 vs 0.125 nm, a 2 percent difference; both FCC; nearly equal electronegativity; valences +1/+2), which is exactly why they show complete solid solubility. Treat these as a guide with exceptions, not a law: satisfying them makes wide solubility likely, but violating even one — most often the size rule — usually caps solubility to a few percent. They are the first thing a metallurgist checks when reading a phase diagram or designing a new alloy.