# Vegard's law

In metallurgy, Vegard's law is an approximate empirical rule which holds that a linear relation exists, at constant temperature, between the crystal lattice parameter of an alloy and the concentrations of the constituent elements.[1][2]

For example, consider the semiconductor compound InPxAs1-x. A relation exists between the constituent elements and their associated lattice parameters, $\mathit{a}$, such that:

$\mathit{a}_\mathrm{InPAs} = \mathit{x}\mathit{a}_\mathrm{InP} + (1-\mathit{x})\mathit{a}_\mathrm{InAs}$

One can also extend this relation to determine semiconductor band gap energies. Using InPxAs1-x as before one can find an expression that relates the band gap energies, $\mathit{E_g}$, to the ratio of the constituents and a bowing parameter $\mathit{b}$:

$\mathit{E_{g,\mathrm{InPAs}}} = \mathit{x}\mathit{E_{g,\mathrm{InP}}}+(1-\mathit{x})\mathit{E_{g,\mathrm{InAs}}}-\mathit{bx}(1-\mathit{x})$

When variations in lattice parameter are very small across the entire composition range, Vegard's law becomes equivalent to Amagat's law.

## References

1. ^ Vegard, L.  (1921). "Die Konstitution der Mischkristalle und die Raumfüllung der Atome". Zeitschrift für Physik 5 (1): 17–26. Bibcode:1921ZPhy....5...17V. doi:10.1007/BF01349680.
2. ^ Denton, A. R.; Ashcroft, N. W. (1991). "Vegard’s law". Phys. Rev. A 43 (6): 3161–3164. Bibcode:1991PhRvA..43.3161D. doi:10.1103/PhysRevA.43.3161.