Vegard's law

In <a href="/facts/Crystallography/9B5toeqG">crystallography</a>, <a href="/facts/Materials_science/N5HrXRYW">materials science</a> and <a href="/facts/Metallurgy/1SANyxX4">metallurgy</a>, Vegard's law is an empirical finding (<a href="/facts/Heuristic/BcBEkv9c">heuristic</a> approach) resembling the <a href="/facts/Rule_of_mixtures/PoNlqesu">rule of mixtures</a>. In 1921, <a href="/facts/Lars_Vegard/fQcQgb7H">Lars Vegard</a> discovered that the <a href="/facts/Lattice_constant/h3gMKk6V">lattice parameter</a> of a <a href="/facts/Solid_solution/Mla7z8dc">solid solution</a> of two constituents is approximately a <a href="/facts/Weighted_mean/VHyJfPxb">weighted mean</a> of the two constituents' lattice parameters at the same temperature:

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    {\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 <a href="/facts/Mixed_oxide/v8mSHnRJ">mixed oxide</a> of <a href="/facts/Uranium/n2sGWBzU">uranium</a> and <a href="/facts/Plutonium/4bUw9zH3">plutonium</a> as used in the fabrication of <a href="/facts/MOX_fuel/3tKw5I8q">MOX nuclear fuel</a>:

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    {\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 <a href="/facts/Crystal_structure/bJ4bCeHd">crystal structure</a>. Here, aA(1-x)Bx is the lattice parameter of the solid solution, aA and aB are the lattice parameters of the pure constituents, and x is the <a href="/facts/Mole_fraction/nW4EepmJ">molar fraction</a> 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. 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. For example, consider the <a href="/facts/Semiconductor/zFf3mGTE">semiconductor</a> compound InPxAs(1-x). A relation exists between the constituent elements and their associated lattice parameters, a, such that:

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

When variations in lattice parameters are very small across the entire composition range, Vegard's law becomes equivalent to <a href="/facts/Amagat%2527s_law/v9G5S3wN">Amagat's law</a>.

Vegard's law open-in-new

Vegard's law