crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

, materials science and

metallurgy Metallurgy is a domain of materials science and engineering that studies the physical and chemical behavior of metallic elements, their inter-metallic compounds, and their mixtures, which are known as alloys. Metallurgy encompasses both the sc ...

, Vegard's law is an empirical finding (

heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...

approach) resembling the

rule of mixtures In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material . It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, mass density, ultimate te ...

. In 1921,

Lars Vegard Lars Vegard (3 February 1880 – 21 December 1963) was a Norwegian physicist who is a pioneer in crystallography. He also made contributions to materials science and the science of aurora borealis. The Vegard's law in solid state chemistry and m ...

discovered that the

lattice parameter A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has o ...

of a

solid solution A solid solution, a term popularly used for metals, is a homogenous mixture of two different kinds of atoms in solid state and have a single crystal structure. Many examples can be found in metallurgy, geology, and solid-state chemistry. The word ...

of two constituents is approximately a

weighted mean The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...

of the two constituents' lattice parameters at the same temperature: :

a_ = (1-x)\ a_\mathrm + x\ a_\mathrm

e.g., in the case of a

mixed oxide In chemistry, a mixed oxide is a somewhat informal name for an oxide that contains cations of more than one chemical element or cations of a single element in several states of oxidation.Advanced Inorganic Chemistry, F. A. Cotton, G. Wilkinson ...

uranium Uranium is a chemical element with the symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. Uranium is weak ...

and

plutonium Plutonium is a radioactive chemical element with the symbol Pu and atomic number 94. It is an actinide metal of silvery-gray appearance that tarnishes when exposed to air, and forms a dull coating when oxidized. The element normally exhibi ...

as used in the fabrication of MOX nuclear fuel: :

a_\mathrm = 0.93\ a_\mathrm + 0.07\ a_\mathrm

Vegard's law assumes that both components A and B in their pure form (i.e. before mixing) have the same

crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...

. Here, is the lattice parameter of the solid solution, and are the lattice parameters of the pure constituents, and is the

molar fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...

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

semiconductor A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...

compound . A relation exists between the constituent elements and their associated lattice parameters, , such that: :

a_ = x\ a_\mathrm + (1-x)\ a_\mathrm

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 solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in ...

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 as before, the band gap energy,

E_g

, can be written as: :

E_ = x\ E_+(1-x)\ E_

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, : :

E_ = x\ E_+(1-x)\ E_-bx\ (1-x)

Mineralogy

The following excerpt from Takashi Fujii (1960)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 summarises well the limits of the Vegard’s law in the context of

mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proces ...

and also makes the link with the Gladstone–Dale equation:

References

Crystallography Materials science Metallurgy Mineralogy Eponyms {{crystallography-stub

Relationship to band gaps in semiconductors

Mineralogy

See also

References