Structure & Transformations · Hume-Rothery rules

Hume-Rothery rules: when does one metal dissolve in another?

Pick two metals and see, atom by atom, which of the four rules they pass and how much one really dissolves in the other.

The checkerDarken-Gurry mapElectron countTakeawaysQuestionsReferences

Which metals can replace each other in a crystal?

A solute atom (coral) sits on a site of the host metal (the solvent); a big size gap squeezes its neighbours.

Atom sizes to scale (12-neighbour radii, Teatum et al. 1968). All neighbour shifts: schematic.
Solvent (host)
Solute (added metal)
  • 1. Size gap within 15%
  • 2. Same crystal structure
  • 3. Electronegativity gap within 0.4
  • 4. Solute valence higher

Try it: Cu with Ni passes rules 1 to 3 and mixes at every ratio above about 335 °C. Cu with Ag passes the size rule and sits inside the ellipse, yet dissolves only 4.5 at.%.

What is a Darken-Gurry map?

Each metal is a dot placed by its size and its electronegativity (how strongly it pulls electrons); dots inside the ellipse are expected to dissolve.

Click a dot to load it in the checker. Filled dots: a measured solubility is shown above.
Solvent (centre of the map)
Inside the outer ellipse
Picked solute

Try it: switch the solvent to Ag: Zn, Al and Cd move close to the star, while Pb stays outside.

Why do brasses change structure as zinc is added?

Count the outer (valence) electrons per atom, e/a: each new crystal structure appears at its own count.

Schematic atoms and cells (cells show sites only). The cell shown is the phase whose e/a is nearest (the page's choice).
Alloy
pure Cumore solute
Electrons per atom, e/a
Nearest textbook phase
Example of that phase

Try it: press Cu-Al: the bcc β phase now sits at 25% Al (Cu3Al), the same e/a of 1.50 as CuZn.

What to take away

Size first

More than 15% size gap: little solubility. This test is right most of the time.

Passing is not enough

Under 15% does not promise wide solubility. Cu-Ag passes, yet mixes only a few percent.

Valence matters

Copper, silver and gold usually dissolve higher-valence metals more than the reverse.

Electrons pick the phase

CuZn, Cu3Al and Cu5Sn are all bcc at 3/2 electrons per atom.

More detail: the rules as written, the data and how well the rules work

The four rules

1. Size. "If the atoms differed in size by more than 15% extensive solid solutions would not form" (Joseph 1965, quoting Hume-Rothery and Raynor). The reason given: a large size difference makes the lattice strain energy high, and the solid solution may split into two phases. The page measures the gap against the solvent radius, Δr/r = (rsolute − rsolvent)/rsolvent, as Darken and Gurry did. Hume-Rothery, Mabbott and Channel-Evans (1934) measured the solid solubility limits of silver and copper alloys, the kind of data behind the rule.

2. Crystal structure. The two metals need the same crystal structure for considerable solubility (Lohwongwatana 2007). Zn is hcp, yet fcc Cu holds 38.3 to 39.4 at.% Zn near 450 °C (Tang et al. 2022), so this rule is not a hard limit.

3. Electronegativity. A gap close to zero gives the most solubility (Lohwongwatana 2007). The page draws its pass line at 0.4, the height of the Darken-Gurry ellipse.

4. Valence. "Other things being equal, a metal of lower valency should be regarded as more likely to dissolve one of higher valency than vice versa." Hume-Rothery later limited this "relative valency effect" to copper, silver and gold (Yao 1961).

The data

Radius, electronegativity and valence all come from one table, Teatum, Gschneidner and Waber (1968), Table I. The radii are for 12 neighbours, calculated from lattice constants. The electronegativities are averages from several sources, so they differ a little from the Pauling scale. For transition metals (Ni, Pd, Fe) valence has no settled value: Teatum lists 10 and 8 (electrons outside filled shells), Pauling used 6 for Ni and Raynor −0.61, and Mizutani and Sato (2017) call it "unresolved". Rule 4 was stated for Cu, Ag and Au with higher-valence metals (Yao 1961), so the page does not apply it to transition metals. Structures are from the same report: Al, Ni, Cu, Pd, Ag, Au, Pb fcc; Be, Mg, Zn, Cd hcp; α-Fe bcc.

How well do the rules predict?

Waber, Gschneidner, Larson and Prince (1963) tested 1455 binary pairs, calling more than 5 at.% "extensive" solubility. The size rule alone was right 90.3% of the time when it predicted limited solubility, but only 50.1% when it predicted extensive solubility, "only slightly better than a purely random choice". Their modified Darken-Gurry map scored 84.8% and 61.7%. So the rules are good at saying when solubility is restricted; meeting them does not guarantee wide solubility.

The Darken-Gurry ellipse

Darken and Gurry (1953) plotted electronegativity against size for solvents silver, magnesium and aluminium and drew ellipses ±15% of the solvent radius wide and ±0.4 units tall. Solutes inside are expected to dissolve more than 5%, solutes outside less than 5 at.%. Waber et al. added the inner ellipse, ±8% and ±0.2. The page counts a dot inside when (Δr/r ÷ 0.15)2 + (Δχ ÷ 0.4)2 ≤ 1.

Electron concentration

e/a = (1 − x)·1 + x·v, with x the solute atom fraction and v its valence (Cu 1, Zn 2, Al 3, Sn 4). In 1926 Hume-Rothery noted that CuZn, Cu3Al and Cu5Sn are all bcc at e/a = 3/2. In 1928 Westgren and Phragmén found Cu5Zn8 and Al4Cu9 both have 52 atoms per cubic cell at 21/13 (Mizutani and Sato 2017). The fcc α phase ends at about e/a 1.4 (Mizutani and Sato); Degtyareva and Afonikova (2017) give the sequence fcc, bcc, γ, hcp at 1.35, 1.5, 1.62, 1.75 and link the end of fcc near 1.36 to the Fermi sphere touching the Brillouin zone. ε is hcp near 7/4 = 1.75 (Degtyareva and Afonikova); real ε brass holds about 78 to 87 at.% Zn, e/a 1.78 to 1.87 (Tang et al. 2022, Table 7). The β and γ phases exist only in narrow ranges around their counts (Berger et al. 2010), and real phase limits shift with temperature: β in Cu-Zn orders near 450 °C (Tang et al.). The picture only shows the nearest textbook value.

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Questions people ask

What are the Hume-Rothery rules?

They are four empirical guides to whether one metal can replace another on its crystal sites (a substitutional solid solution): a size gap under about 15%, the same crystal structure, a small electronegativity gap, and the relative valence effect. Hume-Rothery built them from studies of copper and silver alloys in the 1920s and 1930s.

Why is the size limit 15%?

It is an empirical limit from measured alloys, not a derived number. A large size difference strains the lattice, and the solid solution may then split into two phases (Joseph 1965).

If two metals pass all the rules, will they mix completely?

Not necessarily. In a test on 1455 pairs, a prediction of extensive solubility from the size rule was right only 50.1% of the time; a prediction of limited solubility was right 90.3% of the time (Waber et al. 1963). Cu-Ni passes and is fully soluble above about 335 °C; Cu-Ag passes the size rule but dissolves only 4.5 at.% Ag.

What is the relative valence effect?

Other things being equal, a metal of lower valence dissolves one of higher valence more than the reverse. Hume-Rothery later limited this to copper, silver and gold with higher-valence metals (Yao 1961). It has no clear meaning for transition metals such as Ni, whose valence is unsettled (Mizutani and Sato 2017).

What are electron compounds (Hume-Rothery phases)?

They are phases whose structure is set by the number of valence electrons per atom, not by a fixed formula. β (bcc) appears at 3/2, as in CuZn, and γ (52 atoms per cubic cell) at 21/13, as in Cu5Zn8 (Mizutani and Sato 2017). ε (hcp) comes near 7/4 (Degtyareva and Afonikova 2017); in Cu-Zn it holds about 78 to 87 at.% Zn (Tang et al. 2022).

How much zinc dissolves in copper?

About 38 to 39 at.% Zn in the fcc α phase near 450 °C (Tang et al. 2022, with earlier assessments). That is an e/a of about 1.39, close to the 1.4 where the α phase ends.

References

Show the 14 references
  1. W. Hume-Rothery, G. W. Mabbott and K. M. Channel-Evans, The freezing points, melting points, and solid solubility limits of the alloys of silver and copper with the elements of the B sub-groups, Philosophical Transactions of the Royal Society A 233, 1 (1934). doi:10.1098/rsta.1934.0014
  2. W. Hume-Rothery and G. V. Raynor, The Structure of Metals and Alloys, 4th ed., Institute of Metals, London (1962), p. 100, as quoted by Joseph (ref. 3).
  3. R. R. Joseph, Solid solubility of magnesium in the close-packed modifications of some rare earth metals, thesis, Iowa State University (1965).
  4. L. S. Darken and R. W. Gurry, Physical Chemistry of Metals, McGraw-Hill, New York (1953).
  5. J. T. Waber, K. Gschneidner, A. C. Larson and M. Y. Prince, Prediction of solid solubility in metallic alloys, AIME Institute of Metals Division (1963).
  6. Y. L. Yao, A mutual solid solubility scale for metals, AIME Institute of Metals Division (1961).
  7. B. Lohwongwatana, Development, characterization, and applications of gold and platinum bulk metallic glasses, PhD thesis, California Institute of Technology (2007), ch. 2. doi:10.7907/F215-BD47
  8. E. T. Teatum, K. A. Gschneidner Jr. and J. T. Waber, Compilation of calculated data useful in predicting metallurgical behavior of the elements in binary alloy systems, Los Alamos report LA-4003 (1968), Table I.
  9. U. Mizutani and H. Sato, The physics of the Hume-Rothery electron concentration rule, Crystals 7(1), 9 (2017). doi:10.3390/cryst7010009
  10. V. F. Degtyareva and N. S. Afonikova, Simple metal and binary alloy phases based on the fcc structure: electronic origin of distortions, superlattices and vacancies, Crystals 7(2), 34 (2017).
  11. R. F. Berger, P. L. Walters, S. Lee and R. Hoffmann, Connecting the chemical and physical viewpoints of what determines structure: from 1-D chains to γ-brasses, Chemical Reviews (2010).
  12. Y. Tang, J. Ma, D. Han, J. Wang, H. Qi and L. Jin, Critical evaluation and thermodynamic optimization of the Cu-Zn, Cu-Se and Zn-Se binary systems, Metals 12(9), 1401 (2022), Table 7.
  13. NIST Metallurgy Division, Phase diagrams and computational thermodynamics, solder systems: Ag-Cu (after F. H. Hayes, H. L. Lukas, G. Effenberg and G. Petzow, Z. Metallkde. 77, 1986), Cu-Pb (after A. Bolcavage et al., TMS 1995) and Ag-Pb (after B.-Z. Lee, C.-S. Oh and D. N. Lee, J. Alloys Compd. 215, 293 to 301, 1994). Mass % converted to at.% with IUPAC (CIAAW) standard atomic weights.
  14. W. D. Callister and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 8th ed., Wiley, ch. 9 (Cu-Ni diagram after P. Nash, Phase Diagrams of Binary Nickel Alloys, ASM); and R. Duran, P. Stender, S. M. Eich and G. Schmitz, Atom probe study of the miscibility gap in CuNi thin films and microstructure development, Microscopy and Microanalysis 28, 1359 to 1369 (2022).
Cite this page: Tripathy, Manisha. “Hume-Rothery Rules Lab.” untethered atom, 2026, https://untetheredatom.com/phase-transformations/hume-rothery-rules.
BibTeX 
@misc{tripathy2026humerothery,
  author = {Tripathy, Manisha},
  title  = {Hume-Rothery Rules Lab},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/phase-transformations/hume-rothery-rules}},
  note   = {Interactive web tool}
}
Last updated 1 October 2026.