# 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] Vergard's law assumes that both components A and B in their pure form (i.e. before mixing) have the same crystal structure and that they form a solid solution upon mixing. Then Vegard's law states the average interatomic distances (or lattice parameters) in the solid solution are given by a linear interpolation at the fraction $\mathit{x}$ of the A component:

$\mathit{a}_\mathrm{AB} = \mathit{x}\mathit{a}_\mathrm{A} + (1-\mathit{x})\mathit{a}_\mathrm{B}$

Here $\mathit{a}_\mathrm{AB}$ is the interatomic distance in the mixture, $\mathit{a}_\mathrm{A}$ and $\mathit{a}_\mathrm{B}$ are the interatomic distances in the pure components. However, Vegard's rule is only seldomly obeyed and often deviations from the linear behavior are observed. Still it may be used to obtain rough estimates. 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.