# Vegard's law

In materials science and metallurgy, Vegard's law is the empirical heuristic that the lattice parameter of a solid solution of two constituents is approximately equal to a rule of mixtures of the two constituents' lattice parameters at the same temperature:[1][2]

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

Vegard's law assumes that both components A and B in their pure form (i.e. before mixing) have the same crystal structure. Here, aAxB(1-x) is the lattice parameter of the solution, aA and aB are the lattice parameters of the pure constituents, and x is the molar fraction of B in the solution. Vegard's law is seldom perfectly obeyed; often deviations from the linear behavior are observed. A detailed study of such deviations was conducted by King.[3] However, it is often used in practice to obtain rough estimates when experimental data are not available for the lattice parameter for the system of interest. For systems known to approximately obey Vegard's law, the approximation may also be used to estimate the composition of a solution from knowledge of its lattice parameters, which are easily obtained from diffraction data.[4] For example, consider the semiconductor compound InPxAs(1-x). A relation exists between the constituent elements and their associated lattice parameters, a, such that:

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

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

## Relationship to band gaps in semiconductors

In many binary semiconducting systems, the band gap in semiconductors is approximately a linear function of the lattice parameter. Therefore, if the lattice parameter of a semiconducting system follows Vegard's law, one can also write a linear relationship between the band gap and composition. Using InPxAs(1-x) as before, the band gap energy, ${\displaystyle {\mathit {E_{g}}}}$, can be written as:

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

Sometimes, the linear interpolation between the band gap energies is not accurate enough, and a second term to account for the curvature of the band gap energies as a function of composition is added. This curvature correction is characterized by the bowing parameter, b:

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

## 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.
3. ^ King, H. W. (February 1966). "Quantitative size-factors for metallic solid solutions". Journal of Materials Science (Springer) 1 (1): 79–90. Bibcode:1966JMatS...1...79K. doi:10.1007/BF00549722. ISSN 0022-2461.
4. ^ Cordero, Zachary C.; Schuh, Christopher A. (January 2015). "Phase strength effects on chemical mixing in extensively deformed alloys". Acta Materialia (Elsevier) 82 (1): 123–136. doi:10.1016/j.actamat.2014.09.009.