Solid Solutions: The Hume-Rothery Rules

H. K. D. H. Bhadeshia

Hume-Rothery (1899-1968) was a metallurgist who studied the alloying of metals. His research was conducted at Oxford University where in 1958, he was appointed to the first chair in metallurgy.

His research led to some simple and useful rules on the extent to which an element might dissolve in a metal [1-4]. The rules that he derived are paraphrased here. The rules are still used widely. For example, the miscibility gap in Au-Ni is correlated with the fact that the lattice parameter of Au is 1.15 times that of Ni, thus acting maximally according to Hume-Rothery [5].

• If a solute differs in its atomic size by more than about 15% from the host, then it is likely to have a low solubility in that metal. The size factor is said to be unfavourable.
• If a solute has a large difference in electronegativity (or electropositivity) when compared with the host, then it is more likely to form a compound. Its solibility in the host would therefore be limited.
• A metal with a lower valency is more likely to dissolve in one which has a higher valency, than vice versa.

A quasicrystal has long-range order but lacks translational periodicity in three dimensions. Such crystals are in fact periodic in higher dimensions. When these higher dimensions are projected in three, it becomes possible to observe symmetries in diffraction patterns which are not possible in structures with 3-dimensional translational symmetry. Examples include 5-fold and 10-fold rotation axes. A quasicrystal can in three dimensions be imagined as the tesslation of two different unit cells.

Quasicrystals are frequently found in aluminium alloys, and in particular those with electron/atom ratios around 1.75. Hume-Rothery deduced (for example in Cu-Zn alloys) that compound formation occurs at specific electron/atom ratios and this principle has been used to predict quasicrystal formation in aluminium based systems [6].

References

1. W. Hume-Rothery and H. M. Powell, Z. Krist.,, 91 (1935) 23.
2. W. Hume-Rothery, Atomic Theory for Students of Metallurgy, The Institute of Metals, London, 1969 (fifth reprint).
3. W. Hume-Rothery, R. W. Smallman and C. W. Haworth, The Structure of Metals and Alloys, , The Institute of Metals, London, 1969.
4. A. H. Cottrell, An Introduction to Metallurgy, Edward Arnold (publishers), London, 1967, p.192.
5. K. Janghorban, J. S. Kirkaldy and G. C. Weatherly, J. Phys. (Condensed Matter), 13 (2001) 8661.
6. A. P. Tsai, MRS bulletin, November 1997.

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