Vegard's law

In crystallography, materials science and metallurgy, Vegard's law is an empirical finding (heuristic approach) resembling the rule of mixtures. In 1921, Vegard discovered that the lattice parameter of a solid solution of two constituents is approximately a weighted mean of the two constituents' lattice parameters at the same temperature:[1][2]

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

e.g., in the case of a mixed oxide of uranium and plutonium as used in the fabrication of MOX nuclear fuel:

${\displaystyle a_{\mathrm {U_{0.93}Pu_{0.07}O_{2}} }=0.93\ a_{\mathrm {UO_{2}} }+0.07\ a_{\mathrm {PuO_{2}} }}$

Vegard's law assumes that both components A and B in their pure form (i.e. before mixing) have the same crystal structure. Here, a_A(1-x)Bx is the lattice parameter of the solid solution, a_A and a_B are the lattice parameters of the pure constituents, and x is the molar fraction of B in the solid 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 InP_xAs_(1-x). A relation exists between the constituent elements and their associated lattice parameters, a, such that:

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

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 InP_xAs_(1-x) as before, the band gap energy, ${\displaystyle E_{g}}$, can be written as:

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

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:

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

Mineralogy

The following excerpt from Takashi Fujii (1960)[5] summarises well the limits of the Vegard’s law in the context of mineralogy and also makes the link with the Gladstone–Dale equation:

In mineralogy, the tacit assumption for the linear correlation of the density and the chemical composition of a solid solution is twofold: one is an ideal solid solution and the other identical or nearly identical molar volumes of the components. …
Coefficients of thermal expansion and compressibilities of the ideal solid solution can be discussed in the same manner. But when the solid solution is ideal, the linear correlation of molar heat capacities and chemical composition is possible. The linear correlation of refractive index and chemical composition of an isotropic solid solution can be derived from the Gladstone–Dale equation, but it is required that the system must be ideal and the molar volumes of the components are equal or nearly equal. If the concept of the volume fraction is introduced, density, coefficient of thermal expansion, compressibility and refractive index can be correlated linearly with the volume fraction in an ideal system.“[6]

See also

When considering the empirical correlation of some physical properties and the chemical composition of solid compounds, other relationships, rules, or laws, also closely resembles the Vegard's law, and in fact the more general rule of mixtures:

• Amagat's law
• Gladstone–Dale equation
• Kopp's law
• Kopp–Neumann law
• Rule of mixtures

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. S2CID 120699637.
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. PMID 9905387.
3. King, H.W. (1966). "Quantitative size-factors for metallic solid solutions". Journal of Materials Science. 1 (1): 79–90. Bibcode:1966JMatS...1...79K. doi:10.1007/BF00549722. ISSN 0022-2461. S2CID 97859635.
4. Cordero, Zachary C.; Schuh, Christopher A. (2015). "Phase strength effects on chemical mixing in extensively deformed alloys". Acta Materialia. 82 (1): 123–136. doi:10.1016/j.actamat.2014.09.009.
5. Fujii, Takashi (1960). Correlation of some physical properties and chemical composition of solid solution. The American Mineralogist, 45 (3-4), 370-382. http://www.minsocam.org/ammin/AM45/AM45_370.pdf
6. Zen, E.-AN (1956). Validity of Vegard’s law. American Mineralogist (1956) 41 (5-6), 523-524. https://pubs.geoscienceworld.org/msa/ammin/article-abstract/41/5-6/523/539644