crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...

materials science Materials science is an interdisciplinary field of researching and discovering materials. Materials engineering is an engineering field of finding uses for materials in other fields and industries. The intellectual origins of materials sci ...

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 ...

, Vegard's law is an empirical finding (

heuristic A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...

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, ultimate tensile strengt ...

. In 1921, Lars Vegard discovered that the lattice parameter of a

solid solution A solid solution, a term popularly used for metals, is a homogeneous mixture of two compounds in solid state and having a single crystal structure. Many examples can be found in metallurgy, geology, and solid-state chemistry. The word "solutio ...

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 of

uranium Uranium is a chemical element; it has chemical symbol, 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. Ura ...

and

plutonium Plutonium is a chemical element; it has symbol Pu and atomic number 94. It is a silvery-gray actinide metal that tarnishes when exposed to air, and forms a dull coating when oxidized. The element normally exhibits six allotropes and four ...

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 ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...

. 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 (unit), mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity (science), quantity defined as the ratio between the amount of substance, amount of a constituent substance, ''ni'' (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 with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...

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 parameters 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 and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to t ...

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 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 mineralogy, optical) properties of minerals and mineralized artifact (archaeology), artifacts. Specific s ...

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

References

{{Reflist Crystallography Materials science Metallurgy Mineralogy Eponymous chemical rules

Relationship to band gaps in semiconductors

Mineralogy

See also

References