In
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 wo ...
,
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 s ...
, 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 1921,
Lars Vegard discovered that the
lattice parameter 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:
:
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. Wilkins ...
of
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 exhib ...
as used in the fabrication of
MOX nuclear fuel:
:
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 crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns ...
. 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 ...
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 conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
compound . A relation exists between the constituent elements and their associated lattice parameters, , such that:
:
When variations in lattice parameter are very small across the entire composition range, Vegard's law becomes equivalent to
Amagat's law
Amagat's law or the Law of Partial Volumes describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) gases. It is of use in chemistry and thermodynamics. It is named after Emile Amagat.
Overview
Amagat's la ...
.
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 (i ...
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,
, can be written as:
:
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, :
:
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 pr ...
and also makes the link with the
Gladstone–Dale equation:
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
Amagat's law or the Law of Partial Volumes describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) gases. It is of use in chemistry and thermodynamics. It is named after Emile Amagat.
Overview
Amagat's la ...
*
Gladstone–Dale equation
*
Kopp's law
Kopp's law can refer to either of two relationships discovered by the German chemist Hermann Franz Moritz Kopp (1817–1892).
#Kopp found "that the molecular heat capacity of a solid compound is the sum of the atomic heat capacities of the elemen ...
*
Kopp–Neumann law
*
Rule of mixtures
References
Crystallography
Materials science
Metallurgy
Mineralogy
Eponyms
{{crystallography-stub