Clausius–Mossotti relation
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The Clausius–Mossotti relation expresses the dielectric constant (relative
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
, ''ε''r) of a material in terms of the atomic
polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
, α, of the material's constituent atoms and/or molecules, or a homogeneous mixture thereof. It is named after Ottaviano-Fabrizio Mossotti and
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
. It is equivalent to the Lorentz–Lorenz equation. It may be expressed as: \frac = \frac where *\varepsilon_r = \varepsilon/\varepsilon_0 is the dielectric constant of the material, which for non-magnetic materials is equal to n^2 where n is the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
*\varepsilon_0 is the permittivity of free space *N is the number density of the molecules (number per cubic meter), and *\alpha is the molecular polarizability in SI-units (C·m2/V). In the case that the material consists of a mixture of two or more species, the right hand side of the above equation would consist of the sum of the molecular polarizability contribution from each species, indexed by ''i'' in the following form: \frac = \sum_i \frac In the CGS system of units the Clausius–Mossotti relation is typically rewritten to show the molecular polarizability ''volume'' \alpha' = \alpha/(4\pi\varepsilon_0) which has units of volume (m3). Confusion may arise from the practice of using the shorter name "molecular polarizability" for both \alpha and \alpha' within literature intended for the respective unit system. The Clausius-Mossotti relation assumes only an induced dipole relevant to its polarizability and is thus inapplicable for substances with a significant permanent dipole. It is applicable to gases such as N2, CO2, CH4 and H2 at sufficiently low densities and pressures. For example, the Clausius-Mossotti relation is accurate for N2 gas up to 1000 atm between 25°C and 125°C. Moreover, the Clausius-Mossotti relation may be applicable to substances if the applied electric field is at a sufficiently high frequencies such that any permanent dipole modes are inactive.


Lorentz–Lorenz equation

The Lorentz–Lorenz equation is similar to the Clausius–Mossotti relation, except that it relates the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
(rather than the dielectric constant) of a substance to its
polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
. The Lorentz–Lorenz equation is named after the Danish mathematician and scientist
Ludvig Lorenz Ludvig Valentin Lorenz (; 18 January 1829 – 9 June 1891) was a Danish physicist and mathematician. He developed mathematical formulae to describe phenomena such as the relation between the refraction of light and the density of a pure transpar ...
, who published it in 1869, and the Dutch physicist Hendrik Lorentz, who discovered it independently in 1878. The most general form of the Lorentz–Lorenz equation is (in CGS units) : \frac = \frac N \alpha_\mathrm where n is the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
, N is the number of molecules per unit volume, and \alpha_\mathrm is the mean
polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
. This equation is approximately valid for homogeneous solids as well as liquids and gases. When the square of the refractive index is n^2 \approx 1 , as it is for many gases, the equation reduces to: : n^2 - 1 \approx 4 \pi N \alpha_\mathrm or simply : n - 1 \approx 2 \pi N \alpha_\mathrm This applies to gases at ordinary pressures. The refractive index n of the gas can then be expressed in terms of the
molar refractivity Molar refractivity,W. Foerst et.al. ''Chemie für Labor und Betrieb'', 1967, ''3'', 32-34. https://organic-btc-ilmenau.jimdo.com/app/download/9062135220/molrefraktion.pdf?t=1616948905 A, is a measure of the total polarizability of a mole of a subs ...
A as: : n \approx \sqrt where p is the pressure of the gas, R is the
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, and T is the (absolute) temperature, which together determine the number density N.


References


Bibliography

* * * * * Lorenz, Ludvig, "Experimentale og theoretiske Undersogelser over Legemernes Brydningsforhold", Vidensk Slsk. Sckrifter 8,205 (1870) https://www.biodiversitylibrary.org/item/48423#page/5/mode/1up * * * O. F. Mossotti, Discussione analitica sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso, Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena, vol. 24, p. 49-74 (1850). {{DEFAULTSORT:Clausius-Mossotti Relation Electrodynamics Electromagnetism Equations of physics Electric and magnetic fields in matter