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.